## 1.Electric Charges

Electric Charges

Take away from previous Knowledge:

You must have read about the Four Fundamental forces in the Universe - Strong force, Weak force, Gravitation force and Electromagnetic force.   Out of these four, like Gravitation force arises due to the mass of the particles, Electromagnetic force is actually due to another intrinsic property of matter that is “ELECTRIC CHARGE”

Some matters have an intrinsic property other than mass which is “Electric charge”. For example electrons, and protons.

INTRODUCTION TO ELECTRIC CHARGES

An electric charge is an intrinsic property of matter which is caused by electromagnetic interaction. Many physical phenomena like lightning, all phenomena related to charges and magnets can be explained by electromagnetic interactions.

Types of charges:  There are two kinds of charges in the universe. Electric charges can either be positive or negative.  Like charges (same charges either both positive or both negative) repel each other and unlike charges (one positive and one negative) attract each other.

S.I units and Dimension of Electric Charges

S.I units of charge is Coulomb and Dimension is  [ M1]

“Electron has the fundamental charge of   - 1.6 * 10-19 C

Conductors and Insulators:

Free charges are responsible for the electric and thermal conductivity of the materials.

Materials with free charges can conduct electricity and hence are called conductors like metals. Example - Copper, silver etc. Materials that don’t have free charges cannot conduct electricity and hence are insulators.  For example glass, Plastic.

Electrostatic Induction:

Electrostatic Induction is the phenomenon of temporary electrification of a conductor in which opposite charges appear at these closer ends and similar charges appear at its farthest end in the presence of a nearby charged body.

In the above situation, we initially have a neutral metal sphere. When we place a negatively charged rod near it, electrons surface near the rod will get repelled from the negative charge from the rod and move to another side of the sphere this the surface near the rod will become positive ( due to deficiency of electron ) and farther part becomes negative( due to excess of electrons). So that metal ball is temporarily charged by induction.

If we ground the sphere such that the excess electrons will move to the earth and thus leaving the metal as a positively charged metal sphere. This is called electrostatic induction.

Basic Properties of electric charges

Additive property of electric charges:

Electric charges can be added with their signs just like we add numbers.

The additive nature of electric charges means that the total charge of a system is the algebraic sum of all the individual charges located at different points inside the system.

Example: If inside a cavity we have some positive charges of  2 coulombs and negative charges of 4 coulombs then we can say that total charge in the cavity is   “-2 C”.

Qtotal = Q1 + Q2 =  +2 C +  (-4 C) = -2 C.

In the same way, we can add many charges together along with their signs to get the total charge.

Let’s try to understand more about it.

Some Important facts: Conceptual questions

A body can be made negatively charged by giving some electrons to it. It can be made positively charged by removing some electrons from it.

Explanation:  Since in an atom we have electrons and protons, electrons are comparatively free than the protons which are strongly bounded inside the nucleus. So to make a body negatively charged we cannot extract positive charge from it. But it can be done by giving excess electrons to it.

And similarly, when we wish to make a body positively charged we cannot give excess protons to it but we can remove some electrons from it so there would be net positive charge.

Remember Atom as a whole is always neutral as it contains an equal number of electrons and protons.

Can charge exist without mass?

One of the basic properties of Electric charge is that a charge is always associated with mass. A charge doesn’t exist without mass.

Earth is a source of an infinite positive and negative charge.

Earth can be considered an infinite source of positive and negative charges. This can be justified by the fact that if we connect any positive or negatively charged body to the ground, all of its charges will go to earth.

If a system has zero overall charge. Is it true there are no charges present in the system?

No. This is not true. There is a possibility that all positive charges cancel out all negative charges and the overall system has zero charges. For example, in any atom, we have an equal number of positive and negative charges so the net charges become zero, but there are still charges inside the atoms.

Conservation of Charge

In physics, charge is a conserved quantity and it cannot be destroyed or produced. Total charge in an isolated system remains conserved and can never change.

In other words, the net quantity of all the charges in the universe (positive-negative) is always conserved.

Quantisation of charge

In the Universe, we have some fundamental particles like electrons.  Fundamental particle means we cannot divide that particle. Like we can have either one electron, 2 electrons or so on. We cannot have anything like 2.5 electrons (any fraction) as we cannot have a fraction of an electron. So we can have only an integer number of electrons.

Now we know that charge is an intrinsic property of the particle along with its mass.  An electron has an elemental charge which is  1.6* 10-19C . So we can have only charge in the integer number of charges of electrons.

Q = n e

Where  Q is charge; n is any integer ±1, ±2,±3,…and so on, e is a charge of 1 electron.

The quantization of any physical quantity means that it cannot vary continuously to have any arbitrary value but it can change discontinuously to take any one of only a discrete set of values.

The energy of the electron in an atom or the electric charge of a system is quantized.

## 1.Electric Charges

### Chapter 1: Electric Charges and Fields

Electric Charges
The name electricity is coined from the Greek word electron meaning amber. Many such pairs of materials were known which on rubbing could attract light objects like straw. “The properties of matter, atoms and molecules are determined by the magnetic and electric forces present in them. There are also only 2 kinds of an entity called the electric charge.”

Conductors and Insulators

Those substances which readily allow the passage of electricity through them are called conductors , just like metals, and the earth and those substances which offer high resistance to the passage of electricity are called insulators When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor. In contrast, if some charge is put on an insulator  it stays at the same place.

Properties of Electric charge?

That there are two types of charges, namely

1.positive and

2.negative

An electric charge has three fundamental properties :-

Conservation-   there is transfer of electrons from one body to the other; no new charges are either created or destroyed , The total charge of an isolated system is always conserved, initial and final charge of the system will be same.

Quantization - it is established that all free charges are integral multiples of a basic unit of charge denoted by e. Thus charge q on a body is always given by

q = ±ne

Additive- This property of electric charges represents the total charge of a body as the algebraic sum all the singular charges acting on the system. If a system contains n charges q1, q2, q3, …, qn, then the total charge of the system is q1 + q2 + q3 + … + qn .

## 1. Electrostatic Potential

Electrostatic Potential

Introduction

The concept of potential can only be understood only when you first understand conservative forces. As we can define potential with only those forces that are conservative in nature. For example, we have the gravitational potential for gravitational forces, and electrostatic potential for electrostatic ( Coulomb’s ) force, as these forces are conservative forces. Have anyone heard of frictional potential or drag potential?. There is no way we can have these potentials as frictional forces and drag forces are not conservative.

Now the next question is what exactly is a conservative force and how do we define it? Let's start it.

Conservative forces

There are many forms of energy: light energy, heat energy, sound energy, potential energy, kinetic energy, chemical energy etc.

But in mechanics, we define mechanical energy. Mechanical energy basically is the sum of potential energy and kinetic energy.

In mechanics when we say that total energy must remain the same; that is we are referring to the mechanical energy of the system

We say that under the effect of conservative forces the mechanical energy of the isolated system remains conserved.

Significance of conservative forces

The work done is independent of the path taken.

Suppose we have two points A and B, there can be many ways to reach B from A. The three possible paths are shown in the figure. Suppose the work done in reaching B from A via path 1 is W1  and similarly, we have W2 and W3 as work done for paths 2 and 3 respectively. So If the force would be conservative then W1=W2=W3.

• Under conservative forces, the mechanical energy remains conserved.

If a body is subjected to conservative force and if its kinetic energy increases by some amount, its potential energy will decrease by the same amount to keep the total mechanical energy constant.

Let us try to understand this with the help of an example

•  Suppose we have a body of mass m, kept at height H.  At this point, the body possesses potential energy “mgH”. Suppose you drop that object and it starts coming down.
• Now which force will be acting on this object falling down? Gravitational force. Agree? From our discussion, we know the gravitational force is a conservative force.
• So due to gravity, there will be a downward force on the object, so there is acceleration in the downward direction too. We call this acceleration “acceleration due to gravity g “.
• So Due to acceleration in the downward direction, the speed of the object increases as it falls down and hence the Kinetic energy of the particle also increases.
• But Since the gravitational force is a conservative force, mechanical energy under this force must be constant so as Kinetic energy increases, its potential energy decreases by the same amount.
• At the lowest position, just before hitting the surface it will have maximum kinetic energy and have velocity say “V” and potential energy will be zero, as all of the Potential energy gets converted into Kinetic energy.

mgH= (½)mV^2

By the above equation, we can find the maximum velocity attained by the object of mass “m” when it is dropped from height “H”.

The above example is to demonstrate how the energy remains conserved in conservative force and what is conservative force?

Now we can move further. Our next discussion is about Potential energy.

Potential energy

When we do some work against the conservative force the work we do gets converted into the potential energy of the body.

Example

•  lifting an object to a height “H” against gravitational force stores potential energy of “mgH”.
• Compressing or expanding a spring by distance “x” introduces potential energy of (½)kx^2 In the spring of spring constant.

The way we have gravitational P.E  we have electric potential energy as well.

Electrostatic potential energy.

Suppose we have a charge Q placed at the origin. This Charge will have an electric field  E around it. Now imagine we have a test charge “q” and we have to bring the test charge from infinity to Point  R which is at distance r from the Q.  Imagine both charges are of the same sign (Say positive).

Now while bringing the test charge “q” at point R, we have to work against the repulsive force due to charge Q as both are positive signs.

Assumptions:

• The test charge q is so small that it does not disturb the original configuration, here charge Q at the origin.
• We assume that we are bringing the test charge without acceleration, with constant speed so the external force just balances the repulsive force due to Q.

Points to be considered here:

• The external force must be in the opposite direction to the electrostatic repulsion force.
• The work done by an external force is negative to the work done by the electric force.
• If an external force is removed, the electric force will take charge and push the test charge q away and thus it will get some kinetic energy.
• The gain in kinetic energy after the removal of external force happens at the cost of loss of potential energy.

So we have discussed potential energy due to two charges  Q and q separated by a distance “r”.

We notice here that electric potential energy and electric potential are proportional to charge “q” and inversely proportional to distance “r”.

I can surely say that at this point you might be wondering what is the difference between electric potential energy and electric potential.

Electric potential energy is the work done in bringing the charge from infinity to a point in the electrostatic field of a charge.

The Unit of Electric potential energy is the same as work done and which is Joules. It is a scalar quantity.

Electric Potential

Electric potential is the work done per unit charge in order to bring that charge from infinity to a point in the electrostatic field.

Electric potential is referred to as Voltage and it is a scalar quantity.

Now let’s try to understand the electric potential difference

Electric Potential difference

Imagine we have a charge Q. In the figure above you can see two points A and B. A is nearer to charge and B is father to charge. As we know Potential is inversely proportional to “r”. So we can conclude that A is at the higher potential here and B is at the lower potential. The difference in the Potential of A and B is called the Potential difference between A and B.

Suppose we have two points A and B at a distance of r1 and r2 from the charge Q. Then the potential difference between A and B will be given as

Va - Vb = K Q( (1/r1)- (1/r2))

S.I. unit of potential difference is the same as potential.

So In the discussion so far, I have tried to give you a detailed explanation of conservative force, the origin of the potential concept. We have discussed Electric potential energy, Electric Potential and its difference in quite a descriptive manner with illustrations of the diagrams.

Electric potential due to multiple charges: Superposition principle.

Like electric force and electric field, Electric potential also follows the principle of superposition. The only difference is that electric potential is scalar, therefore the Electric potential due to multiple charges at a point is simply the algebraic sum of the electric potential due to all individual charges.

Suppose we have three charges  Q1, Q2, Q3  which is at distance r1, r2, r3 respectively from a point P and we wish to find electric potential due to these charges at that point; Electric potential due to three charges will be simply the sum due to individual charges Like shown in the figure.

Electric potential due to Electric dipole.

In the previous chapter, we have discussed the Electric field due to electric dipoles at axial and equatorial positions.

In this section, we will discuss the electric potential at axial and equatorial positions.

Be assured! This is much easier than finding an Electric field as electric potential is a scalar quantity, unlike an electric field which is a vector quantity.

So we can do an algebraic sum in this case, rather than doing vector addition as in the case of electric fields due to dipoles.

Axial position: Suppose we have an electric dipole with charge ±q with separation ‘ 2a’. We wish to find the electric potential at P due to dipole which is an axial position

Potential at P will be an algebraic sum due to potential due to +q and -q charges which are at distance (r+a) and (r-a) respectively.

V at P =  V(+q) (potential due to +q) +  V(-q)(potential due to -q) = V(+q)+V(-q)

For r >> a, we can neglect ‘a’ in the denominator.

Then V is proportional to the square of the distance between them.

Equatorial Position:

Now suppose we wish to find the electric potential at the equatorial position due to the electric dipole.

Again Potential at P will be an algebraic sum due to potential due to +q and -q charges which are at distance (r+a) and (r-a) respectively.

V at P =  V(+q) (potential due to +q) +  V(-q)(potential due to -q) = V(+q)+V(-q)

Now we know that Electric potential is directly proportional to the magnitude of charge and inversely proportional to distance.

If you closely look at point P you will notice that the distance of P from either charge is the same and also the magnitude of the charges in the dipole is the same.

We can conclude from above that magnitude of potential due to

+q and -q is the same at P, but since one will be positive and the other to be negative so they will cancel each other out. The net electric potential due to the electric dipole at the equatorial position is zero.

Equipotential Surface.

• The surface at which the potential has a constant value is called the equipotential surface.
• The work done in moving a charge on an equipotential surface is zero because the potential at every point of the equipotential surface is the same as work done = charge * Potential difference, Since the potential difference is zero at the equipotential surface W=0
• For any charge configuration, the equipotential surface through a point is normal to the electric field at that point. In other words, we can say that electric field lines are perpendicular to the equipotential surface.

• The equipotential surfaces of a single point charge are concentric spherical surfaces centered at the charge
• Equipotential surfaces due to the uniform electric field are planes normal to the direction of the electric field.

• Equipotential surfaces due to electric dipole are also spheres perpendicular to electric field lines as shown in the figure below.

You will notice that the equipotential surfaces are neat to each other in between the charges and are farther on the other side.

This is because there are more electric field lines are more between the charges.

• Equipotential surface due to two similar charges ( say both positive) will also be a concentric sphere but in this case, equipotential surfaces are far from each other in between the charges and near to end on the other side. The reason is the same again as the electric field in between similar charges is less in between the similar charges.
• Equipotential surface due to two similar charges ( say both positive) will also be a concentric sphere but in this case, equipotential surfaces are far from each other in between the charges and near to end on the other side. The reason is the same again as the electric field in between similar charges is less in between the similar charges.

Relation between Electric field and electric potential

The concept of Electric field and Electric potential is introduced to understand and visualize how charges which are kept at a distance exert force on each other without any physical contact. But Electric field is a vector quantity and the electric potential is a scalar quantity and hence much simpler to deal with.

So there must be some connection between electric field and electric potential.

When V is given and E has to find out we will use the following relation.

When E is given and a potential difference is to be found out, we will use the following relation.

## 1. Electrostatic Potential

Chapter 2: Electrostatic Potential and Capacitance

Electrostatic Potential

We define potential energy of a test charge q in terms of the work done on the charge q. This work is obviously proportional to q, since the force at any point is qE, where E is the electric field at that point due to the given charge configuration. It is, therefore, convenient to divide the work by the amount of charge q, so that the resulting quantity is independent of q. In other words, work done per unit test charge is characteristic of the electric field associated with the charge configuration
POTENTIAL DUE TO A POINT CHARGE
Take Q to be positive. We wish to determine the potential at any point P with position vector r from the origin. For that we must calculate the work done in bringing a unit positive test charge from infinity to the point P. For Q > 0, the work done against the repulsive force on the test charge is positive.

Potential due to an electric dipole
What is the electric potential due to an electric dipole at an equatorial point? Zero, as potential on equatorial point, due to charges of electric dipole, are equal in magnitude but opposite in nature and hence their resultant is zero.

Potential due to system charges
derive an expression for the electric field at a point due to a system of n point charges. When there is a group of point charges say q1, q2, q3,….qn is kept at a distance r1, r2, r3,……rn, we can get the electrostatic potential at any particular point.

Equipotential surface.

If the points in an electric field are all at the same electric potential, then they are known as the equipotential points. If these points are connected by a line or a curve, it is known as an equipotential line. If such points lie on a surface, it is called an equipotential surface.
Work Done in Equipotential Surface
The work done in moving a charge between two points in an equipotential surface is zero. Then the work done in moving the charge is given by
W = q0(VA –VB)
As VA – VB is equal to zero, the total work done is W = 0.

## 2. Coulomb's Law

Coulomb’s Law

Suppose there is a charge at a point in space if we take another charge near that previous charge. These two charges will experience a force on each other. This force is consistent with Newton's third law which says that there is equal and opposite reaction for every action.

The magnitude of the force is given by Coulomb's law. If you look at Coulomb's law you will notice that it is very similar to the universal law of gravitation. Let's see what Coulomb's law states.

If you have two charges  q1 and q2 placed at a distance r  from each other then they will experience a force due to one another.

Coulomb's law states that the magnitude of this force is directly proportional to the square of these charges and inversely proportional to the square of the distance between them.

|F| = ( k q1*q2 ) / r^2

Where |F| is the magnitude of the force between two charges,

r is the distance between the charges

q1, q2 are the magnitude of the charges.

Here K is a proportionality constant whose value is  1/(4*Πϵo) whose value is 9 * 10 power 9 N.

Here ϵ0= permittivity of free space whose value is 8.85*10power(-12) .

Coulomb’s force is along the straight line connecting the points of location of the charges and hence coulomb force is central and spherically symmetric.

This force is attractive when the two charges are of different signs and the force is repulsive when two charges are of the same signs

Definition of 1 coulomb:

If  q1= q2 = 1C   and  r = 1 m   Then F= K = 9 *10 power 9 N

So one coulomb Is the charge that when placed at a distance of 1 meter from another charge of the same magnitude in vacuum experience and electric force of repulsion of magnitude 9 *10 power9 N.

The analogy between Gravitational force and Coulomb’s force

The electrostatic force is the force of attraction or repulsion between two charges at rest while the gravitational force is a force of attraction between two bodies by virtue of their masses.

Similarities:

• Both follow Inverse Square Law i.e, F is proportional to 1/r^2
• Both forces are proportional to the product of masses or charges.
• Both are Central forces in the app along the line joining the centers of two bodies.
•  Both are conservative forces; the work done against these forces does not depend upon the path followed.
•  Both forces can operate in a vacuum.
•  The range of both forces is infinite.

Dissimilarities :

• Gravitational force is attractive while electrostatic force may be attractive or repulsive.
• Gravitational force does not depend on the nature of the medium why electrostatic force depends on the nature of the medium between the two charges.
• Electrostatic forces are much stronger than gravitational force.

How much is the electrostatic force stronger than the gravitational force

From Coulomb’s law if we wish to calculate the force between an electron and a Proton separated by a distance r.

Fe=(  k q1*q2)/r^2 , here q1 and q2 are the charges of electron and proton (-e) and (e) respectively.

Fe= k(-e)(e)/ r^2 = -ke^2/r^2

Negative signs show that this force is of attractive kind.

now let's see the gravitational force between an electron and a Proton separated by a distance r.

Fg =G*me*mp/r^2

If you take the ratio of electrostatic force and the gravitational force between electron and proton ( Fe/Fg), you will see that the value of this fraction is of the order of 10power39.

Some examples show how electric forces are enormously stronger than the gravitational forces.

• If you stand on the earth there will be a gravitational force that is pulling you towards its center, but you do not move towards the center.So there must be some other force that is balancing the gravitational force upon you. That force is actually the electrostatic force. So you see that the electrostatic force between the ground and feet is enough to balance the gravitational force due to the entire earth on you.
• When we hold a book against gravity in our hand, the electrostatic force between hand and book is enough to balance the gravitational force on the book due to the whole earth.

The direction of coulomb’s Force

• If we have two charges whose signs are the same either both positive or both negative, then the force will be repulsive and they will try to push Each Other away. As electrostatic force is consistent with Newton's third law the force on the first charge due to the second  ( F12 ) is equal to the force on the second charge due to the first (F21). but they are in opposite directions.

•  If you have two opposite charges then they will attract each other, they will try to pull each other.

In the figure given below positive charge is indicated as blue ball and negative charge is indicated as a red ball.You can see that when both charges are positive the forces on one due to other is such that they are pushing the other charge away.

In the second place when one charge is positive and other is negative they try to pull each other as shown.

The magnitude of electrostatic force in both the cases is given by Coulomb’s law and |F21|= |F12| but their directions are opposite.

A fun thing to try

I am providing you the link to a simulation related to electric charges and coulomb’s force.

You can choose the two charges and their sign and it will show you the magnitude of the Force with its direction.

Electric force due to Multiple charges: Principle of Superposition.

With Coulomb's law, we can find the force between two charges, But what we will do when we have to calculate force on a charge due to multiple charges, the answer to this lies in a fantastic principle called Principle of superposition.

The principle of superposition states that the force on a charged particle due to multiple charges is actually the vector sum of the forces exerted on it due to all other charges. The force between two charges is not affected by the presence of other charges.

Let me explain this further.

suppose we have five charges ( Q1, Q2, Q3, Q4, Q5)  at a distance (d1, d2, d3, d4,d5) from a point L, A charge Q is placed at L.

So Total force on charge Q will be the vector sum of force exerted by all charges .

Force on Q  (FQ)=   F(QQ1) + F(QQ2) + F(QQ3)+F(QQ4)+ F(QQ5)

Please note that this is actually a vector sum, we need to add these vectors keeping in mind their directions too.

Eectric Field

We have already discussed the coulomb’s law and the principle of superposition. These two concepts allow us to calculate force one charges due to other charges. The electrostatic force acts between two bodies without even direct contact between them so to visualize how the electrostatic force is experienced by a charge we have introduced the concept of electric fields.

According to this theory when we place a charge Q in space it creates a field around it, specifically the electric field around it which is stretched is space around that charge,  and when another charge q is placed in the field of the previous charge it will experience a force on it due to Charge Q.

The magnitude of electric force will be the product of charge “q” and electric field due to charge Q at the position of “q”.

F= q *E due to Q.

Electrostatic force = Charge * Electric field

The Electric field or Electric field intensity for electric field strength E  at a point is defined as the force experienced by a unit positive test charge placed at that point without disturbing the position of the source charge.

E= F/q

S.I unit of  E= N/C

Direction of Electric field

The electric field is a vector quantity.  The direction of the electric field will be same as electric Force for a positive charge but it will be the opposite to electric force for a negative charge

Electric field lines:

Electric field lines are the imaginary lines that are drawn to help in visualizing electric fields.  By convention, we take the direction of electric field lines away from the positive charge and toward the negative charge.

In the above figure electric field lines are shown for positive charge, negative charge and neutral object.  There are no electric field lines around the neutral object.

Properties of Electric field lines.

•   The line of forces is a continuous smooth curve without any breaks.
• The line of forces starts from positive charge and ends at negative charges.
• They cannot form closed loops
• If there is a single charge then the line of forces will start or end at infinity.
• The tangent to a line of forces at any point will give the direction of the electric field at that point.
• The line of forces can never intersect each other because we would then have two directions of electric field at the point of intersection.

Which is not possible.

• The relative closeness of the lines gives the measure of the strength of the electric field in any region. They are closer together in a strong field and far apart in a weak field.
• Parallel and equispaced lines represent a uniform electric field.
• The line of forces has the tendency to contract lengthwise.
• This explains attraction between two unlike charges.
•  The line of forces has a tendency to expand laterally. This explains repulsion between two similar charges.

Electric field line due to dipole

Electric dipole is two equal and opposite charges separated by a small distance.   The electric field lines start at positive charge and negative charge and they seem to contract lengthwise as if the two charges are being pulled together this explains the attraction between two unlike charges.  The figure given below shows the electric field lines due to a dipole.

A fun thing to try and learn.

Below is the link to simulations of electric fields due to charges.

Electric field lines due to Similar charges.

In the figure given below, we have electric field lines due to two positively charged spheres. The field lines start from positive charge and expand laterally as if the two charges are being pushed away from each other. This explains repulsion between two like charges.

Area Vector.

Suppose we have a disc of radius “r”,  So there must be some surface area of this disc. This is actually the magnitude of the area of this plane disc. But we can also define its direction. Area vector is actually perpendicular to the surface.

The direction of a planer area vector is specified as the normal to the plane.

Electric flux.

The term flux implies some kind of flow,  This is a property of every vector. Electric flux is a property of the electric field. The electric flux through a closed surface may be thought of as a number of electric lines of Forces that intersect a given area perpendicularly.

The electric field lines directed into a closed surface are considered negative and those directed out of the closed surface are positive.

If there is no net charge enclosed within the closed surface; Electric field lines going into the closed surface are equal to the electric field lines coming out of the surface. Then electric flux through that closed surface will be zero.

As you can see in the diagram above the formula of electric flux; you will see that electric flux depends on Three things.

• The strength of Electric field  (E)
• Area of the surface (A)
• The relative orientation of E and area vector, in other words, electric flux is proportional to cosine of angle between E and Area vector.

Electric dipole.

Equal and opposite charges (±q) separated by some finite small distance are called an electric dipole.

Electric dipole moment  “P” = q*d    (either charge* distance between them)

The direction of P (electric dipole moment ) is from -q to +q.

All the distances in the electric dipole are measured from the center of the electric dipole.

The electric field at the axial and equatorial position is due to electric dipole.

Axial position:

The position in the line of axis of electric dipole is axial position.

Suppose we have a point C along with the axial position of the electric dipole and we wish to find E at point C.

In the figure above if you see the direction of E due to +q it will be away from the charge ( toward right)  and direction of E due to -q will be toward the -q charge (  toward left), since the Electric field due to +q and -q points in opposite directions, To find Net field we need to subtract them.  E due to +q is greater than E due to -q in the above case as point C is closer to +q  and we know that  E is inversely proportional to square of distance.

Enet=  E(+q) - E(-q)   as  E(+q)>E(-q)  and they are in opposite directions.

Equatorial position.

The line which is perpendicular to the axis of the dipole and passes through the middle of it is called the Equatorial position.

When we wish to find the Electric field due to the electric dipole at equatorial position  P, we first draw the direction of E due to +q and -q at that point P. You can refer to the figure below. Since this E is neither in horizontal direction and nor in vertical, so we must resolve them in horizontal ( cosine components)  and vertical directions ( sine components)

Now you can see that the Sine components of  +q and -q are pointing in opposite directions and hence cancel out. And Cosine components of both the charges are pointing toward left ( actually opposite to the direction of electric dipole moment P).

So we conclude that Net E=  2E cosθ.

For reference, you can follow the derivation given below.

Torque on Electric dipole placed in Uniform ELectric field.

Suppose there is a uniform electric field and an electric dipole is placed at some angle in the uniform field as shown in figure. If we see the direction of force on either charge of the electric dipole we will see that the force on +q and -q will be equal and in opposite direction; Net force on the electric dipole is Zero.

Force = charge * Electric field.

When charge is positive, Force is along the direction of electric field E and when the charge is negative, force is opposite to the direction of E. Since it is the case of uniform Electric field. The value of E at position of (+q) and (-q) will be the same and hence the magnitude of force.

So from above we conclude that Net force on electric dipole is zero.

Torque :

As we have seen, a couple of forces are acting on the electric dipole. Also, there is a separation between these forces, so there must be torque acting on the electric dipole.

Torque =  Either force* perpendicular distance between them.

= ( q*E) * (L*sinθ)

Where L= length of the electric dipole

Note that  P= qL  ( electric dipole moment).

For derivation please refer to the figure below.

Torque= P*E*sin(θ)

τ= p ×E

Continuous charge distribution

Whatever we have discussed so far, we took the example of point charges or discrete charges. But this is not true every time. The charge can be spread on a line,  surface or volume in a continuous manner also.

Different types of continuous charge

1. Line charge: when charges are spread in a continuous manner on a line ( 1 - Dimension)  we call it line charge distribution.

We define line charge density λ at any point on that line as the charge per unit length of the line at that point. S.I. unit of linear charge density = C/m.

1. Surface charge: When charges are spread in a continuous manner on a surface ( 2-dimension). We call it surface charge distribution.  We define surface charge density at any point of the surface as the charge per unit area at that point. S.I.  unit of surface charge density is  C/m^2.
2. Volume charge density: when charges are spread in a continuous manner in a volume ( 3-dimension), we call it volume charge distribution.
3. The volume charge density at any point is given by the charges per unit volume at that point. S.I. unit of volume charge density is  C/m^3.

## 2. Coulomb's Law

Coulomb’s Law

The force of attraction or repulsion acting along a straight line between two electric charges is directly proportional to the product of the charges and inversely to the square of the distance between them.

Forces Between Multiple Charges

• When our synthetic clothing or sweater is removed from our bodies, especially in dry weather, a spark or crackling sound appears. With females’ clothing like a polyester saree, this is almost unavoidable.
•  It does not help to calculate the force on a charge where there are not one but several charges around.  It is have been proved via an experiment that force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time.

Principle of Superposition of Electrostatic Forces This principle states that the net electric force experienced by a given charge particle q0 due to a system of charged particles is equal to the vector sum of the forces exerted on it due to all the other charged particles of the system.

Some basic Properties of Electric Field Lines

• Two-line never cross each other
• These electric field lines start on the positive charge and end in the negative charge
• Electrostatic field lines do not form any closed loops

Electric Flux

The total number of electric field lines passing a given area in a unit time is defined as the electric flux.

• Since electric field is uniform, it is created by a source very far from the closed surface. Or there is no charge enclosed within the closed surface. Hence, net flux through it is zero.
• This argument does not hold true for an open surface as an open surface can be arbitrarily extended to  a closed surface enclosing a non-zero charge in which case the electric flux through the surface may become non-zero
• Electric flux is defined as the measure of count of number of electric field lines crossing an area. Electric flux ϕ=EAcosθ SI unit of electric flux is Nm2/C.

### Physical significance of dipoles

In most molecules, the centres of positive charges and of negative charges lie at the same place. Therefore, their dipole moment is zero. CO2 and CH4 are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules, H2O,is an example of this type. Various materials give rise to interesting properties and important applications in the presence or absence of electric field.

### Electric dipole

An electric dipole is a pair of equal and opposite point charges -q and q, separated by a distance 2a. The direction from q to -q is said to be the direction of the dipole.
p=q×2a
where p is the electric dipole moment
pointing from the negative charge to the positive charge.

EXAMPLE

Force on electric dipole

Dipoles in an External Electric Field

Consider an electric dipole placed in an external electric field. The electric dipole will experience some force and is known as the torque. The torque is the force exerted on the dipoles placed in an external electric field and is

given by,τ = P x E = PE Sin θ ………(1)

Where,

P - The dipole moment

E - The applied external field

Significance of Electric Dipole and Moment

The concept of an electric dipole is not only having importance in physics but it is an equally valid and prominent topic in chemistry as well. We know that most of the matter made up of atoms and molecules will be electrically neutral. Depending upon the behaviour of the pair of charges, the molecules are subdivided into two types

• Polar molecules: If the centre of mass of positive charge doesn’t coincide with the centre of mass of negative charge then it is known as a polar molecule.
• Non-Polar molecules: If the center of mass of positive charge coincides with the center,  charges, s of negative charge then it is known as a Non-Polar molecule.

## 3. Gauss's Law

Gauss’s Law

When we have to find an electric field due to extended charge distribution, it involves the integration of charge elements and is sometimes very tedious to do. But it could be done easily in case of certain symmetry.

• Spherical symmetry- Uniform volume charge spread in the volume of a sphere.
• Cylindrical symmetry - Uniform volume charge in volume of a cylinder.
• Infinite line charge - Uniform line charge spread in an infinitely long wire.
• Infinite surface charge - Uniform charge distributed on an infinite large plane sheet.

In case of any one of the above symmetry, the calculation of electric field can be done using Gauss’s law.

Gauss’s law statement :

Gauss’s law states that electric flux through any surface enclosing charges is equal to 1/ϵo times the total charge enclosed by the surface. Mathematically,

Gaussian surface: A hypothetical closed surface enclosing a charge is called the gaussian surface of that charge.

By clever choice of Gaussian surface, we can easily find the electric field produced by certain symmetry charge configurations which are otherwise difficult to evaluate using Coulomb force and principle of superposition.

Two things that must be taken care while using gauss’s law is that

• The charge must be enclosed.
• We take only those charges which are inside the Gaussian surface

Great Significance of Gauss Law:

We already have discussed some of them but let's just gather them all under a single heading.

•  Gauss law is true for any closed surface no matter what the shape or size.
•  The term q on the right side of gauss’s law includes the sum of all the charges enclosed by the surface. The charges may be located anywhere inside the surface.
• In any situation, there may be some charge inside and outside the chosen surface. On the left side of Gauss’s law,  the electric field is due to all the charges both inside and outside the surface. The term q  on the right side however represents only the total charge inside the surface.
• Gauss’s Law Is based on the Inverse-square dependence on distance contained in  Coulomb's law.
• Gauss’s Law is often useful to word a much easier calculation of electrostatic field when the system has some symmetries as discussed above. This is facilitated by the choice of Gaussian surface

Note:

Gauss’s law is always valid, but it is useful only when there is any one of the above-mentioned symmetry and we can take out E outside the integral due to symmetry and so only surface integration is left over the chosen gaussian surface.

Application of gauss law

As we have discussed the importance of gauss law and also discussed in which symmetry situation this will be useful.

Note for the students

I will focus less on derivation in this part as anyone can get it in every book.

My focus in this section would be to give my readers a good understanding of the concept and its applications. I will directly give the Electric field of some symmetry cases without derivation but will focus on giving you a good conceptual understanding of the results and how we will use it in problem-solving.

Electric flux due to charge in a cube

Case -1  Charge is placed inside the cube.

Suppose we have a cube and a charge is placed inside the cube. Since the cube is a closed surface, we can directly use Gauss's law. electric flux coming out of the cube will be q/ϵ0

from the six faces of the cube. If we wish to calculate electric flux through one face of this cube, we will divide the whole flux by the number of faces of the cube ( here 6 ).

Electric flux through one the face when the charge is placed inside the cube is= q/6ϵo

Case -2  Charge placed on one of the faces of the cube.

Now we can have another case where the charge is kept on one of the faces of the cube.  This will not be a closed surface as the charge is not closed completely. But we can close it by putting one more cube over the charge so now the charge will be fully enclosed inside two cubes.

So the total electric flux through the close surface ( two cubes here) =q/ϵ0

The flux through one cube will be total flux divided by no of cubes. So in this case =  q/2ϵ0

Now If you imagine 2 cubes joined together, there will be 10 faces of the cubes outside.

So if we wish to calculate electric flux through each face then we will divide the total flux by no of faces on the outer side = q/10ϵ0

Case -3 when the charge is at the edge of the cube.

In this case, we would require 4 cubes to enclose the charge.

Total flux through the closed surface (4 cubes in this case) will be = q/ϵo

Electric flux through one cube is total flux divided by no of cubes = q/2ϵ0

there will be a total of 12 faces of the cube through which flux will be coming out in this case.

Electric flux passing through one face will be Total flux / no of faces toward the outer region = q/12ϵo

Case -4  Charge placed on one of the vertices.

No of cubes required to enclose this charge = 8

Electric flux coming out of 8 cubes ( enclosed surface) = q/ϵ0

Electric flux coming out of 1 cube = q/8ϵ0

No of faces on the outer side of these 8 cubes joined together = 24

Electric flux coming through one of the flux is=q/24ϵ0

Field due to an infinitely long straight uniformly charged wire.

Consider an infinitely long straight wire having charges uniformly distributed over it. The linear charge density of this wire is λ.

• Suppose this is a positive charge distribution then from previous knowledge of  Electric  field lines we can say that the direction of electric field will be radially outward from the wire as shown in figure below
• As Electric field due to this wire has cylindrical symmetry; the right choice of the gaussian surface would be a cylinder.
• Draw a cylinder of saying length L and radius ‘r’ round the wire. You will notice that this cylinder would have three surfaces, Two circles at the top and bottom and one curved surface around the wire.
• The next step would be to draw the area vector for all three surfaces. From our previous discussion, we must know that the area vector must be perpendicular to the surface. Reader may refer to the direction of Area vector for each surface  ( denoted by n )and also take a look at direction of electric field lines coming out of three surfaces

• Now you will notice that for the circular area, the area vector and Electric field are perpendicular (makes 90 degree) and for the curve surface, area vector and electric field is parallel ( angle 0).
• Now In the formula for Gauss law in the integral we have  Eds, Which means Edscosθ and we also know that cos 0=1 and cos 90=0. So if we break this integral for 3 surfaces ( two circular surfaces and one curved surface), the integral of the circular surface would vanish and we will be left with only one integral with curve surface integration.
• Now that we have simplified the left side, let's talk about the right side now.  The right side of gauss’s law is actually q/ϵ0. Here q is the charge enclosed by the gaussian surface. Here since the length of the gaussian cylinder is “L” and we have linear charge density λ. Then the total charge enclosed by the cylinder  q=λL

Eds around curved surface = λL/ ϵo

• Since the curve surface of the gaussian area is at “r” distance from the wire and so E at all the points on the curve surface is constant . We can now take constant E outside the integral and can only integrate over the area of the curve surface.
• Note that curve surface area of cylinder of length L and radius “r” is  =2ΠrL

Ecurve surfaceds = E * 2ΠrL= λL/ϵ0

E=λL/2Πϵ0*rL

E= λ/2Πϵ0*r…. Electric field due to the infinitely long wire of uniform linear charge density

The above result is for the electric field for an infinitely long wire having uniform linear charge density. Let us analyze this result.

1.  The electric field is proportional to linear charge density.
2. The field is inversely proportional to the distance from the wire. The Further you go from the wire, the less will be the value of the electric field.

Electric field due to an infinitely long plane sheet having uniform surface charge density.

Consider an infinitely long plane sheet having uniform surface charge density σ

Suppose this is a positive charge distribution than from previous knowledge of Electric field lines we can say that the direction of

• Electric field will be radially outward from the plane sheet in both directions as shown in figure below.
• As Electric field due to this plane sheet has cylindrical symmetry; the right choice of gaussian surface would be a cylinder. Consider a gaussian surface of length L and cross -section area “A” as shown in figure.
• Just like in the previous case, we have discussed that this cylinder has three surfaces and draw the area vector for each surface.
• Then we will see the direction of E and the area vector through each surface. In this case you will notice that for the curve surface E and area vector is perpendicular  (angle=90) and for two circular cross-sections, E and area vector are parallel ( angle 0).

• So in this case when we break the closed surface integral on LHS, we will get three integrals; two for cross-section area ( will contribute as angle =0) and one for curved surface area ( this will vanish as angle=90).
• Now for the RHS of Gauss’s law where q represents the charge enclosed inside the gaussian surface. As you can see from the figure, the area of the plane sheet which is equal to the cross-section area of the cylinder is enclosed inside the gaussian surface.
• So Left-hand side of Gauss’s equation will be

Eds = Eclosed surfaceds= 2Ecross-sectionds= 2EA

• Since we have uniform surface charge density σ and area enclosed is A, then the charge enclosed q=σA. So right side of equation q/ϵ0  = σA/ϵ0.

When we will equate the LHS and RHS we will get 2EA=σA/ϵ0

E= σ/2ϵ0……………( Electric field due to infinitely long plane sheet)

The above relation is the expression of Electric field due to an infinitely long plane sheet having uniform surface charge density.

Now Let's analyze it.

1.  Like the previous case, Electric field is proportional to surface charge density.
2. But the thing to be noticed here is actually that Electric field due to the infinitely plane sheet is actually independent of distance.
3.  We can conclude from the above discussion that Electric field due to the infinite plane sheet is constant in magnitude.
4. The direction of Electric field will be pointing away from the sheet from both sides of the plane in case of positive surface charge density.
5. The direction of Electric field will be pointing towards the sheet from both sides of the plane in case of negative surface charge density.

Application of Electric field due to Infinitely long plane sheet.

Electric field due to two parallel infinite plane sheet having equal and opposite surface charge density ±σ

When we have two parallel infinite plane sheets. The whole space is divided into three regions

• First region at the left of both the plates  ( say P)
• Second region is in between the plates. ( say Q)
• Third region on the right of both the plates. ( say R)

1. Since from our analysis we know that Electric field due to the infinite plane sheet is independent of the distance from the sheet, then we can take away very necessary information from this fact.The magnitude of electric field at any of these regions due two plane sheets will be equal.
2.  Now Lets talk about direction of E due to Plate A and Plate B , in these three regions. Referring to points 4 and 5 of our analysis above. We can conclude that direction of EA will be away from the plates and direction of EB will be toward the plates in all three regions. Refer above figure.
3. From the figure above you can say that at regions P and R, the direction of EA and EB is opposite and their magnitudes are equal; they would cancel each other out.
4. Above discussion concludes that Electric fields in region P and R are zero.
5. In region between the plate, EA and EB are in same directions and hence they will add to give E= σ/ϵo  between the plates.

The above result is very important, always remember that the electric field between two parallel plates of opposite surface charge density is E=σ/ϵ0

And E=0 everywhere except in between the plates.

Now you might be thinking about what would be the case if both the plates will have charges of the same sign and have the same surface charge density like each plate has   +σ  charge density.

• The magnitude of electric field in every region due to any of the two plates will be the same as it doesn’t depend on distance from the plates.
• The direction of the field between the plates due to two plates is pointing in opposite directions and hence cancel each other. So the net field in between the plates in region 2 is zero if the plates have the same charge densities.
• If we look at regions 1 and 3, we will see that the direction of the electric field due to plates in these regions are in the same direction and hence they will add. The value of electric field in region 1 and 3 will be E=σ/ϵ0.

There can be one more possibility, what if the charge densities on the plates are not the same? The answer is very simple in that case they will neither completely cancel in the region where they are in the opposite direction, nor they will double in the region where they are in the same direction.

Just simply add and subtract them with their magnitude.

## 3. Gauss's Law

Gauss's Law

As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radius r, which encloses a point charge q at its centre. Divide the sphere into small area elements

ϕ ϵ0

The Gauss law formula is expressed by;

ϕ = Q/ϵ0

Where,

Q = total charge within the given surface,

ε0 = the electric constant.

Electric Field due to Infinitely Long Straight Wire

To calculate the electric field, imagine a cylindrical Gaussian surface. Since the field is radial everywhere, flux along the two faces of the cylinders is zero. On the cylindrical part of the surface, E is perpendicular to the surface at each point and its magnitude is constant since it depends only on r. The surface area of the curved part is   where l is the length of the cylinder.

Flux through the surface = flux through the curved part of the cylinder

= E x

The surface encloses a charge equal to

Field due to a uniformly charged thin spherical shell

Electric Field Outside The Shell

Consider a point P, placed outside the spherical shell. Here, OP=r. As shown in the figure below, the Gaussian surface as a sphere is assumed to have radius r. The electric field intensity, E¯ is said to be the same at every point of a Gaussian surface directed outwards.

## 2. Potential energy

Electric potential energy is due to the system of charges.

When we assemble a charge distribution, we need to do some work in assembling the charges. The work done by the external force to assemble the charge is stored in the system of charges as Potential energy of the system of charges.

• Suppose we have to make an assembly of a few charges. We can do this by putting charges at their respective position one by one.
• The work done in bringing the first charge Q1 from infinity and placing it at its respective position is zero as there is no other charge present to exert any force on the first charge. Work done for placing the first charge (W1=0) is zero.

W1=0

• Now when we bring the second charge from infinity in the presence of the first charge, the external force has to do some work against the electrostatic repulsive force of the first charge.

Therefore, W2( work done in placing 2nd charge )= U12(   potential energy  of first and second charge) = KQ1*Q2/ r12

W2= KQ1*Q2/r12, where r12 is the distance between r1 and r2.

• Now if we wish to bring the third charge Q3 from infinity to its respective position, an external force has to do work against the electrostatic repulsion of both Q1 and Q2.

Therefore work done by the external force in bringing the charge

Q3= W3= U13+U23 = KQ1*Q3/r13 + KQ2*Q3/r23

• So the Potential energy stored in the system of charge is the total work done in assembling three charges that will be

Potential energy   W= W1+ W2+ W3

= 0 + K*Q1*Q2/r12 + K*Q1*Q3/r13+ K*Q2*Q3/r23

• Now for a system of four charges, the total potential energy will be the work done to bring all four charges together.

In the above example, I have shown the potential energy for a system of four charges placed at the corner of the square of the side “a” and diagonal “D”.

Electric Potential energy in the external Electric field.

Suppose we have two charges q1  and q2 placed in an external electric field at positions r1 and r2 with respect to some origin. These charges are separated by distance ‘rd’

•  Now suppose that the electric potential at the position of q1  is V(r1) due to external field E. So work done in placing the charge q1 at r1  in the presence of external field E will be q1(Vr1)
• The electric potential at the position of q2 is V(r2) due to external field E. So the work done in placing the charge at r2 in the presence of external electric field E is  q2(Vr2)
• But in the case of the second charge, there will be influence due to charge q1 also. We need to bring q2 in the presence of both E and q1. So work done by the external force in bringing the charge q2 in presence of q1  at a separation ‘rd’ is = Kq1q2/rd

Therefore potential energy due to charges placed in the external electric field will be as shown below.

The potential energy of a dipole in an external electric field.

When we place an electric dipole in an external electric field, the charges will experience an electrostatic force qE. The forces on either charge of the electric dipole are equal in magnitude but opposite in direction.

The direction of the force will be in the direction of E for the positive charge and opposite to E for the negative charge.

Since their magnitudes are the same, these forces cancel each other. So there will be no translational motion in the electric dipole when placed in a uniform external field.

But they do experience a torque. As torque is the moment of force.

We define torque= as force* perpendicular distance between the forces.

When we try to rotate the electric dipole in an external field, it requires work done by some external force, and this work gets converted into the potential energy of the dipole of dipole moment P placed in an external uniform electric field.

Electrostatic of conductors

The conductor contains mobile charge carriers. In metallic conductors these charge carriers are electrons. The electrons from the valence shell of the atom of the metal behave like free electrons inside the metal and therefore conduct electricity. These free charge carriers form a kind of ‘gas’, they collide with each other and with the ions and move randomly in all directions. In the presence of an external field, they move and drift opposite to the direction of the field. The positive ions are made up of nuclei and bound electrons and remain held at fixed positions.

Properties regarding electrostatics of the conductors:

1. Inside the conductor, an electrostatic field is zero.

The conductor has free charges. In the presence of electric fields, these charges experience force and drift. But in the static condition, free charges are distributed in such a way that the Electric field inside the conductor is always zero.

2. At the surface of a charged conductor, the electrostatic field must be normal to the surface at every point.

If the electric field would not normal to the surface, the charges will move due to its tangential component and charges will no longer remain in static condition.

3. The interior of a conductor can have no excess charge in a static situation.

A conductor has an equal number of positive and negative charges inside it in any volume element. Any excess charge must reside on its surface and there is no net charge at any volume of it.

4. The electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on its surface.

We have seen that E=0 inside the volume of the conductor, but does it mean that V should also be zero, certainly not!. As the derivative of constant is also zero. The conductor thus has a constant and same value of electrostatic potential inside and on the surface of it. And its value is different outside the conductor.

5. Electric field at the surface of a charged conductor  E=σ/ϵ0 and is normal to the surface.

If the conductor has charge density σ  on it. And we know that the electric field inside is zero. The electric field just outside the conductor will be normal to the surface and will have a value which is equal to σ/ϵ0. If σ>0, the direction of E will be outward normal and if σ<0, the direction of E will be inward normal.

6. Electrostatic shielding.

Consider a cavity of a conductor with no charge inside the cavity. The electric field inside the conductor is zero, whatever be the size or shape of the conductor or whatever be the charge on the conductor on the electric field outside the conductor.  This gives rise to a phenomenon called electrostatic shielding. If we place anything inside a conductor it will be shielded from any external field.

Dielectrics and polarization

Dielectrics are non conducting material, which has no charge carriers.

When we put a conductor in an external electric field, the charge inside it moves and arranges in such a way that the electric field induced inside it starts opposing the external field. This will happen till the internal induced field completely cancels the external field and the net field inside the conductor becomes zero.

But in the case of dielectric, in which free movement of charges is not possible  when put in an external field, the induced moment is  only due to stretching and re-orientation of the molecules of the dielectrics

Here also the induced field will try to oppose the external field but it cannot completely cancel it but can only reduce it.

Enet inside the dielectric = External field Eo - Induced field E.

Types of dielectric materials

We have two kinds of dielectric materials, one is polar and the other is non-polar

In polar molecules, there is a dipole moment in individual molecules but in the absence of electricity, these molecules are randomly oriented in such a way that the net dipole moment is zero.

But when these molecules are placed in external E, these individual dipoles are aligned in the direction of the Electric field, basically re-orientation of the molecules happens and thus becomes polarized in the presence of the external electric field.

In nonpolar molecules, in the absence of an external electric field, there is no polarization. But when we apply an external electric field the molecule gets stretched and there happens a separation between positive and negative charges. So the polarization of the dipole happens.

The figure given below  it is shown how a dielectric is polarized in presence of an external electric field, note that all the charges shown are actually bound charges and not the free charges. The charges in the bulk of the dielectric neutralize each other and there is a net surface charge on the boundary of the dielectric, the surface gets a surface-bound charge of ±σb (  b represents bound charge here) . Thus creating an internal induced electric field within the dielectric material and therefore reducing the net field inside it.

## 2. Potential energy

Potential energy

Potential energy of a system charge
Electric Potential Energy of a System of Charges Electric potential energy of a system of charges is equal to amount of work done in forming the system of charges by bringing them at their particular positions from infinity without any acceleration and against the electrostatic force. It is denoted by U.
U=W=qV(r)

Potential energy in an external field

consider a system of two charges q1 and qlocated at a distance r1 and r2 from the origin. Let these charges be placed in an external field of magnitude E. Let the work done in bringing the charge q1 from infinity to r1 be given as q1V(r1)and the work done in bringing the charge q2 from infinity to r2­ against the external field can be given as q2V(r2).
Potential energy of a dipole in an external field
Consider a dipole with charges q1 = +q and q2 = –q placed in a uniform electric field E, in a uniform electric field, the dipole experiences no net force; but experiences a torque τ given by      τ = p×E which will tend to rotate it (unless p is parallel or antiparallel to E).
ELECTROSTATICS OF CONDUCTORS
1. Inside a conductor, electrostatic field is zero
There may also be an external electrostatic field. In the static situation, when there is no current inside or on the surface of the conductor, the electric field is zero everywhere inside the conductor. This fact can be taken as the defining property of a conductor.
2. At the surface of a charged conductor, electrostatic field must be normal to the surface at every point
We can say that, if the electric field lines were not normal at the surface, a component of the electric field would have been present along the surface of a conductor in static conditions. Thus, free charges moving on the surface would also have experienced some force leading to their motion, which does not happen. Since there are no tangential components, the forces have to be normal to the surface.
3. The interior of a conductor can have no excess charge in the static situation
A neutral conductor has equal amounts of positive and negative charges In every small volume or surface element. When the conductor is charged, The excess charge can reside only on the surface in the static situation. This follows from the Gauss’s law.
4. Constant electrostatic potential throughout the volume of the conductor:
The electrostatic potential at any point throughout the volume of the conductor is always constant and the value of the electrostatic potential at the surface is equal to that at any point inside the volume.
5. Electric field at the surface of a charged conductor
Here σ is the surface charge density and nˆ is a unit vector normal to the surface in the outward direction. To derive the result, choose a pill box (a short cylinder) as the Gaussian surface about any point P on the surface. The pill box is partly inside and partly outside the surface of the conductor. It has a small area of cross-section δ S and negligible height.

DIELECTRICS AND POLARISATION
The Dielectric Constant is the ratio of the applied electric field strength to the strength of the decreased value of the electric field capacitor when a dielectric slab is placed between the parallel plates. The formula is as follows:
εr = E0 / E
where E0 is the applied electric field, E is the net field, & εr is the dielectric constant.The greater the dielectric constant, the greater the amount of charge that can be held. The capacitance of a capacitor is increased by a factor of the dielectric constant when the gap between the plates is completely filled with a dielectric. C = εr C0, where C0 is the capacitance between the plates with no dielectric.

## 3. Capacitors and capacitance

Conductor and Capacitors

A capacitor is a two-conductor system separated by an insulator. The conductors have charges say Q1 and Q2 and potential V1 and V2. In practice, we have two conductors having charge Q and -Q and the potential difference between them are V= V1-V2.

Note: Even a single conductor can be considered as a capacitor assuming the other at infinity.

Conductors may be charged by connecting them to the terminals of a battery. Q is the charge of one conductor, though the total charge of the capacitor is zero.

• The electric field in the region between the conductors is proportional to the charge Q

E  α Q  …….(1)

Now we also know that V= E*d  ………….(2)

where V= potential difference between the conductors.

E= electric field in the region in between the conductors of capacitor

d= distance between two conductors of the capacitors

So we get 1 and 2 equations that

Q α V    charge on the capacitor is directly proportional to the potential difference applied to the two conductors.

To remove the proportionality, you need to put some proportionality constant.

Q= C *V     ………… (3)

where  C= proportionality constant is called the capacitance of the capacitor, it basically tells us about the charge holding capacity of a particular capacitor for a given voltage difference across it.

• Capacitance is independent of Q and V
• C is only dependent on the size, shape and separation of the system of conductors
• S.I. unit of capacitance C is Farad.  1 F= 1 C/V
• For large C, V is small for given Q and Q is very large for given V.  Hence large charges can be stored by providing a small potential difference.
• For small C, V is large for given Q  and Q is small for given V.  we would need a large potential difference to get more charges on the capacitor.

Types of capacitors and their capacitances

Capacitors come in different shapes, sizes and separations between the conductors and hence differ in their capacitances.

1. Spherical conductor: It is just a single spherical conductor that can store charge on its surface and is assumed to have another surface at infinity. If the potential of the spherical conductor is V and Q is the charge stored then capacitance can be found as below.

2. Spherical capacitor:  It is basically two concentric hollow spheres of inner radius a and outer radii ‘b’. Then capacitance C is given by the following formula.

3. Cylindrical capacitor: It is made of two concentric cylindrical conductors, having length ‘l; and inner and outer radius as ‘a’ and ‘b’ respectively.

4. Parallel Plate capacitor

In a parallel plate capacitor, we have two parallel plates of surface area ‘A’  and the distance between plates is ‘d’. The capacitance of parallel is given by the following formula.

This is how we can find the capacitance of the parallel plate capacitor.

Below are some equations used in the case of parallel plate capacitors.

Effect of dielectric on capacitance

Suppose we don't have air or vacuum between the conductors of the capacitor but the space between them is filled by an insulator or a dielectric of dielectric constant ‘ K’ or relative permittivity ‘ϵr

Suppose we have a parallel plate capacitor filled with dielectric in all space between the plates. Then the new capacitance becomes K time the previous value of capacitance with air/vacuum between the plates.

This is true not only for parallel plate capacitors but valid with every kind of capacitor spherical and cylindrical too. When we filled the space between the conductor their capacitance becomes K or ϵtimes the previous values.

When we fill dielectric of thickness ‘t’ in between the plates of parallel plate capacitors such that t<d.

It means the space between the plates is filled with air in thickness (d-t) and filled with dielectric in thickness ‘t’.

Above the formula, we use to find the capacitance if a dielectric of thickness t< d is introduced between the plates in a parallel plate capacitor.

Some definitions :

Dielectric constant

We know that when we insert dielectric inside the capacitor, capacitance becomes C= K* Co

So K= C/Co

The dielectric constant may be defined as the ratio of the capacitance of the capacitor when it is filled with a dielectric to the capacitance when air/ vacuum is in between.

Combination of capacitors

We can combine several capacitors and capacitance C1, C2… Cn to obtain a system with some effective capacitance C. The effective capacitance depends on the way individual capacitors are combined.

The two simplest ways are

• Capacitors in series .
• Capacitors in parallel.

Capacitors in series

When the second plate of one capacitor is directly connected with the first plate of another capacitor, this is called a series combination of the capacitors.

In series, the combination of the capacitor charge is the same for all the capacitors and the voltage of the battery is the sum of the voltage drop of the individual capacitor.

V= V1 + V2+ V3  ; Q1= Q2= Q3= Q

We can conclude that the effective capacitance of a capacitor in series is the ‘n’ number of capacitors connected in series.

Capacitor in parallel

In the figure given below, capacitors are connected in parallel combinations. In parallel combination we have the following:

• The voltage difference is the same across every capacity

​​​​​​​V1= V2= V3= V

• A charge is different across each capacitor and the total charge in the circuit is actually the sum of the charge on the individual capacitor

Q= Q1+ Q2+ Q3

Q= Q1 + Q2+ Q3

Ceq*V= C1*V+ C2*V+ C3*V

Ceq= C1 + C2 + C3

If we have ‘n’ capacitors in parallel then the effective capacitance of the parallel combination of capacitors is given by following

Energy stored in a capacitor

In the process of charging the capacitor, some energy is stored in the capacitor. If you are wondering where this energy comes from. Imagine the process of charging the capacitor.

In the circuit given below, the positive terminal of the battery is connected with the upper plate and the negative terminal is connected with the lower plate. So the upper plate is at a higher potential than the lower.

Initially, we assume that both the plates are uncharged and then after connecting it with the battery, there starts a transfer of charge from the lower conductor to the upper conductor bit by bit, so at the end, the upper plate gets ‘Q’ charge while the lower plate gets ‘-Q’  charge when fully charged.

Since the charge is being transferred from lower potential to higher potential, work will be done externally and this external work is stored in the form of energy in the capacitor. The various formula for energy stored in the capacitor in terms of charge, capacitance and the potential difference between the plates is given below

The energy density in a parallel plate capacitor = Energy/volume

A fun thing to do.

Below is the link to the simulation of the capacitor. In this simulation, we have a battery connected with a parallel plate capacitor.

What you have to do -

• The potential of the battery can be changed by moving the slider up and down given on the battery.
• You can increase the plate area by dragging the arrow diagonally as shown in the simulator screen
• And you can also increase or decrease the distance between the plates by dragging the arrow up and down
• Take out the potentiometer by dragging it and connect it with two plates of the capacitor and measure the potential difference on the plates

What you can get from here.

For a given combination of battery potential, separation of plates and plate area you can see the value of the following quantities by clicking on the checkbox before it.

• Capacitance
• Charge on plates
• Energy stored in the capacitor

Also, you can see the visually

• Charges on the plated
• Electric field
• Current direction
• And bar graphs

simulation of capacitor

## 3. Capacitors and capacitance

Capacitors and capacitance

CAPACITORS AND CAPACITANCE
A capacitor is a two-terminal electrical device that possesses the ability to store energy in the form of an electric charge. It consists of two electrical conductors that are separated by a distance.  The space between the conductors may be filled by vacuum or with an insulating material known as a dielectric. The ability of the capacitor to store charges is known as capacitance.  C =Q/v

THE PARALLEL PLATE CAPACITOR

A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance We first take the intervening medium between the plates to be vacuum. The effect of a dielectric medium between the plates is discussed in the next section.

COMBINATION OF CAPACITORS
Capacitance in Series
Figure 1a shows a series connection of three capacitors with a voltage applied. As for any capacitor, the capacitance of the combination is related to charge and voltage by C= Q/V
Note in Figure 1 that opposite charges of magnitude Q flow to either side of the originally uncharged combination of capacitors when the voltage V is applied. Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances.

dividual capacitors Solving C=Q/V for V gives V=Q/C. The voltages across the individual capacitors are thus

V1=Q/C1,V2=Q/C2, and V3=Q/C3.

The total voltage is the sum of the individual voltages:

V1 + V2 +V3.

Now, calling the total capacitance CS for series capacitance, consider that

V=Q/Cs =V1+V2+V3.

Entering the expressions for V1V2, and V3, we get

Cancelling the Qs, we obtain the equation for the total capacitance in series CS to be

where “…” indicates that the expression is valid for any number of capacitors connected in series. An expression of this form always results in a total capacitance CS that is less than any of the individual capacitances C1C2, …, as Example 1 illustrates.
Total Capacitance in Series, Cs

Total capacitance in series:

1/CS  = 1/C1 + 1/C2 … ..
Capacitors in parallel
Two capacitors arranged in parallel. In this case, the same potential difference is applied across both the capacitors. But the plate charges (±Q1) on capacitor 1 and the plate charges (±Q2) on the capacitor 2 are not necessarily the same:
Q1 = C1V, Q2 = C2V
The equivalent capacitor is one with charge
Q = Q1 + Q2
and potential difference V.
Q = CV = C1V + C2V
The effective capacitance C is,   C = C1 + C2
VAN DE GRAAFF GENERATOR
A Van de Graaff generator is an electrostatic generator, invented by Robert J. Van de Graaff. It uses a moving belt that accumulates charge on a hollow metal structure designed like a globe, placed on the top of a column that is insulating in nature and thus, creating a very high electric potential in the order of a few million volts.  This results in a very large electric field that is used to accelerate charged particles.
Working principle of Van de Graaff Generator

Let us consider a large spherical shell of radius R. If we place a charge of magnitude Q on such a sphere, the charge will spread uniformly over the surface of the sphere and the electric field inside the sphere will be equal to zero, and that outside the sphere will be due to the charge Q placed at the centre of the sphere.

At the surface of the small sphere:

At the large spherical shell of radius R:

If we consider the total charges in the system, that is, q and Q, then the total potential energy due to the system of charges can be given as,

## 1. Electric current

Introduction

In the previous chapters, we studied the static properties of the electric charges which we call electrostatics. We have seen that charge at rest has various properties like it exerts forces on other charges, produces an electric field around it, and many more.

In this section, we will study the transient properties of the electric charges. What will happen when the charges move?

Moving charges constitute an electric current. Some currents occur naturally in many situations. Lightning is one such phenomenon although it's not steady and it can be dangerous many times when charges flow from the clouds to earth through the atmosphere.

In today's world electricity has become an inseparable part of our lives as most appliances we use run on electricity. But these appliances need a steady source of electric current to function properly. In this section, we will discuss some basic laws concerning the steady electric current.

Electric current

The flow of electric charges constitutes an electric current. Quantitatively, electric current in a conductor across an area held perpendicular to the direction of flow of charge is defined as the amount of charge flowing across that area per unit of time

Now let's understand how we get the above formula.

In the figure given below is a conductor in which charges are flowing

And we imagine a small area held normal to the direction of flow of charge like shown in dark purple color in the diagram given below.

Now through this area suppose both positive and negative charge is flowing in the forward direction across the area.

Let q+ be the positive charge flowing forward through the area

And q- be the negative charge flowing forward through the area.

Then the Net charge crossing the chosen small area is q= q+ - q-

This net charge flowing through a small cross-section is actually proportional to ‘t’ for a steady current

The above proportionality can be understood properly with a suitable analogy. Suppose on a national highway, there is a toll plaza. Imagine that the vehicles cross that toll plaza at a steady rate. If you count the number of vehicles crossing the toll plaza for one hour you will get say 100, If you keep counting the vehicles for 10 hours, you will get 1000 vehicles crossing that toll plaza. The more time has elapsed, the number of vehicles passing that toll plaza will be more.

But currents are not always steady and hence more generally, we can define current as  I=   lim Δt →0  ΔQ/Δt

Or simply by  I= ΔQ/Δt

Here we have considered ΔQ to be the net charge flowing across the cross-section in Δt time.

Units of Electric current

S.I. units of electric current is Ampere.

Definition of one ampere

I = Q / t      ; when  Q=1 C and t= 1s then  I = 1 A.

If one coulomb of charge passes through a cross-section in one second, then the current through that area is one ampere (A).

Conventional and electronic current

At the time when this phenomenon of electric current was discovered, electrons had not yet been discovered so it was thought that electric current is due to the flow of positive charges. So the conventional direction of electric current is taken in the direction of the flow of positive charges.

However, a negative charge moving in one direction is equivalent to a positive charge moving in the opposite direction. Later on, electrons were discovered by  J. J Thomson and it was later established that electric current in conductors is due to the motion of free electrons. The direction of flow of electrons in a conductor or we can say that electronic current is opposite to the direction of flow of positive charges and hence also opposite to the direction of conventional current.

Electric current : Scalar / vector ?

What do you think?  An electric current should be a scalar or vector. Many students can say as the electric current has both direction and magnitude it must be a vector. But this is not correct. Actually, electric current is a scalar quantity.

Even though it has direction, it does not follow vector addition laws. Electric current is a scalar quantity because the laws of ordinary algebra are used to add electric currents and the law of vector addition is not applicable here

For example, in the figure shown below, we have three wires and three currents.  IA= 5 A, IB= 3 A  and both are incoming currents toward point P. Ic must be an outgoing current with magnitude = ( IA+ IB = 8 A).  current in wire C is calculated via scalar addition and not vector addition laws. So electric current is a scalar quantity.

Electric current in the conductor.

When a charged particle is placed in an electric field, it will experience an electric force on it and will begin to move and this motion of charge will contribute to electric current. Now the question is every matter is made of atoms and in every atom, we have electrons and protons, so does it mean that every matter should conduct electricity when placed in an electric field?

The answer is no!

Let me tell you more precisely that electric current is due to the motion of free charges and not the bound charges. In most matters the electrons and protons in the matter are in a bound state, such matter cannot conduct electricity and are called insulators.

In other materials, notably metals, some of the electrons are practically free to move within the bulk material. These materials are generally called conductors. In solid metals, electricity is conducted by free electrons ( due to the flow of negative charge only). There are some other types of conductors like electrolytes in which conduction is due to ions both positive and negative charged ions.

In this text, we will focus only on metals for conduction.

Now another question arises at any point in time, any metals have a very large number of free charges. So is there always an electric current in the metals? Again the answer would be No! Because if it happens then you could no longer touch any metals. You will get an electric shock if you do so.

The main question is why does it happen?

At any time free electrons inside the conductors are in motion due to thermal agitation ( in other words due to thermal energy). In this motion, they collide with each other and also with the fixed ions. Electrons colliding with ions emerge at the same speed as an elastic collision.

But due to the random motion of the free charges inside the conductor, there is no preferred direction of motion. The number of electrons traveling in any direction will be equal to the number of electrons traveling in the opposite direction. So there will be no net electric current

Let us see now what will happen to such a piece of the conductor when we apply an electric field to it. Imagine we have a cylindrical conductor with a uniform circular cross-section. When we apply an electric field at the cross-section. The electrons will experience a force in a direction opposite to the direction of the applied electric field.

The electrons are accelerated toward the positive terminal, even now it collides with other electrons and ions in its path, but there is a preferred direction of drift of electrons called ‘Vd’ which is opposite to the direction of the applied electric field and current. This is all about Electric current and conductors.

## 1. Electric current

Chapter 3: Current Electricity

Current Electricity
Electric current is the flow of electrons through a complete circuit of conductors. It is used to power everything from our lights to our trains.In these activities, students will explore different kinds of circuits and investigate what is required to make a complete circuit.
Types of Current
There are two types of current
1. Direct Current (DC)
2. Alternating Current (AC)
Direct Current
The current electricity whose direction remains the same is known as direct current. Direct current is defined by the constant flow of electrons from a region of high electron density to a region of low electron density.
Alternating Current
The current electricity that is bidirectional and keeps changing the direction of the charge flow is known as alternating current. The electrical outlets at our home and industries are supplied with alternating current.

## 1.Magnetic Force

Chapter 4: Moving charges and magnetism

Magnetic Force

What Is Magnetic Force?

If we place a point charge q in the presence of both a magnitude field given by magnitude B(r) and an electric field given by a magnitude E(r), then the total force on the electric charge q can be written as the sum of the electric force and the magnetic force acting on the object (Felectric + Fmagnetic ).

F = q E = q Q rˆ / (4πε0) r 2

Magnetic Field, Lorentz Force

Let us suppose that there is a point charge q (moving With a velocity v and, located at r at a given time t) in Presence of both the electric field E  and the magnetic Field B .The force on an electric charge q due to both of Them can be written as

F = q [ E  + v × B ] ≡ Felectric + Fmagnetic ).

MOTION IN A MAGNETIC FIELD
The force on a charged particle due to an electric field is directed parallel to the electric field vector in the case of a positive charge, and anti-parallel in the case of a negative charge. It does not depend on the velocity of the particle.
In contrast, the magnetic force on a charge particle is orthogonal to the magnetic field vector, and depends on the velocity of the particle. The right hand rule can be used to determine the direction of the force.
An electric field may do work on a charged particle, while a magnetic field does no work.
The Lorentz force is the combination of the electric and magnetic force, which are often considered together for practical applications.

MOTION IN COMBINED ELECTRIC AND MAGNETIC FIELDS
Lorentz Force
If the magnitudes of electric field strength and magnetic field strength are adjusted such that the magnitudes of the two forces are equal, then the net force acting on the charged particle is zero.
F = F(electric) + F(Magnetic) = q (E = v x B)
The below figure shows the representation of the electric field and the magnetic field along with the motion of charge when they are perpendicular to each other.

F(electric) = F(Magnetic)

In the figure, we can clearly observe that the magnetic forces and electric forces are in opposite directions to each other.

Cyclotron
A cyclotron is a machine used to accelerate charged particles or ions to high energies.
To enhance the energies of charged particles, cyclotron uses magnetic as well as electric fields. It is called crossed fields since the magnetic and electric fields are perpendicular to each other.

MAGNETIC FIELD DUE TO A CURRENT ELEMENT BIOT-SAVART LAW

Assume that a conductor of a very large length L is carrying current I through it. The magnetic field due to the current, B is perpendicular to the plane of the conductor. Further, let us assume that a section of this conductor, say dL is producing a section of the magnetic field dB at point r away from it in the same plane. Let the angle between dL and dB in the direction of r be Θ.

MAGNETIC FIELD ON THE AXIS OF A CIRCULAR CURRENT LOOP

## 1.Magnetic Force

INTRODUCTION

Concepts of both electricity and magnetism was known for almost 2000 years. These two phenomena were considered as independent phenomena. But in 1820  a Danish physicist noticed that a current-carrying wire caused a noticeable deflection in the magnetic compass needle placed near that current-carrying wire. So he concluded that moving charges or currents produces magnetic fields.

Later another scientist  Oersted did some experiments with a magnet and a coil and found that the motion of the magnet near the coil is producing current in the coil. Later Faraday made laws of electromagnetic Induction based on the Oersted experiment’s observation. He concluded that change in a magnetic field produces current.

In 1864, the laws obeyed by electricity and magnetism were unified and formulated. James Maxwell then realized that light is an electromagnetic wave.  After the unification of electricity, magnetism and electromagnetic waves as a single unit called electrodynamics, remarkable scientific and technological progress took place in the 20th century.

What will we learn in this chapter?

In this chapter, we will study magnetostatics. It is a phenomenon associated with steady currents. When a steady current flows through a wire, it produces a constant magnetic field around it. We shall see how particles can be accelerated to very high energies in a cyclotron.

We will see how currents and voltages are detected by a galvanometer.

Magnetic force

Let’s first recapitulate whatever we have learned so far. First, we studied about the static property of the charges in “Electrostatics”, we understood how charges exert force on each other, we discussed electric fields, electric flux, and electric potentials. Then in the later chapter “ current electricity’’ we studied the phenomena related to charges in motion. We have discussed currents, mobility, drift velocity, resistivity and many more.

1.  Magnetic forces: Sources and fields

In this unit, we will study another property of moving charges.

Moving charges produce magnetic fields around, this property is called the magnetic effect of current.

• Also when a current carrying wire is placed in a magnetic field, it will experience some magnetic force on it.
• Just like static charges produce an electric field, moving charges produce magnetic fields. Magnetic field is again a vector field.
• The basic property of a magnetic field is very similar to an electrical field.
• Magnetic fields are also found to obey the principle of superposition.
• A charge particle moving in magnetic field also experience magnetic force
1. Magnetic field and Lorentz force

Suppose we have a charge ‘q’  in an electric field ‘E’ so it will experience an electric force on it

Electric force  Fe= q E   ….(1)

Now suppose we have switched on the magnetic field also and the charge is moving with a speed ‘v’ in this magnetic field ‘B’

So it will experience magnetic force on it.

Magnetic force Fm= q ( v ×B)    ….(2)

So total force on the charged particle when it moves in the electric and magnetic force is given by the following expression

Lorentz force F= F electric + F magnetic

F= q E + q ( v×E) = q ( E+ v×B)

The above force F is the total force on the charged particle and is called Lorentz force.

Features of magnetic force   Fm= q ( v ×B)

• It depends on q, v and B.  Magnetic force on a negative charge is opposite to magnetic force on a positive charge.
• It involves the cross-product of velocity and magnetic field. so if the velocity and magnetic field are parallel to each other, the magnetic force will be zero.  magnetic force will be maximum when velocity is perpendicular to field B.
• If charge is at rest so V=0 then magnetic force will be zero. Only a moving charge feels magnetic force
• Magnetic force is a conservative force, It do not change the energy of the system
• Magnetic force only changes direction of motion of charges and does not change the speed of the charges. S.I unit of Magnetic force is Tesla.
1. Magnetic force on current carrying conductor

We can extend our analysis for force due to the magnetic field on a single moving charge to a straight rod carrying current. Inside a current-carrying conductor, charges are moving inside the conductor and when this current-carrying conductor is placed in magnetic force, moving charge inside the conductor experience magnetic force as discussed above  Fm= q ( v ×B), and so the whole conductor also feels that magnetic force. Derivation of the magnetic force on the current carrying charge.

Fm= q ( v ×B) This formula is for a single charge ‘q’ but when in conductor we have a large no of such charges.

Suppose we have ‘n’ no of charges per unit volume of the conductor of length ‘l’ and cross-section area ‘A’.

total charges inside the conductor = nlA

Force on 1 charge is  Fm= q ( v ×B)

Force on  total charges inside the conductor = nlA * e ( vd×B) ..(1)

Here ‘vd’ = drift velocity of charges inside the conductor, q= e, a charge of carrier inside the conductor.

Now we know that current  I= neAvd  …(2)

So using 2 in 1 we get  F= I (l×B)

The above formula is the formula for force on a current carrying conductor of length ‘L” and having current ‘I’  placed in magnetic field B.

Here  l= a vector of magnitude l and direction is that of current.

Motion in a magnetic field

Magnetic forces do not do any work on the charged particle or you can say that work done by magnetic forces on the charged particle moving in the magnetic field is zero.

You have learned in mechanics that  W= F ds = F ds cosθ

So if force and the displacement is perpendicular to each other then work done= 0

In the case of the magnetic motion of a charged particle in a magnetic field, The magnetic field and the velocity vector are always perpendicular to each other as shown. So no work is done.

Also  from work-energy theorem, we have  work done = change in Kinetic Energy

So If the work done=0, then the change in kinetic energy of the charged particle would also be equal to zero. This means that the Kinetic energy of the charged particle moving in the magnetic field is constant. Also, the magnitude of the velocity of the charged particle moving in the magnetic field is constant. But the direction of velocity can be changed.

Motion of a charged particle in a Uniform Magnetic field

Case 1:  Motion of charged particle is perpendicular to Magnetic field.

Suppose a charged particle of charge ‘q’ is moving with a speed  ‘v ‘ perpendicular to the magnetic field. The magnetic force experienced by the charged particle would be.

Fm= q( v×B)= qvBsinθ= qvBsin90= qvB

This force will provide a centripetal force to the charged particle to move in a uniform circular motion.

centripetal force Fc= mv^(2 )/r

Fc= Fm , This will give    mv^2/r= qvB

So, r= mv/qB   This is the formula of the radius of the circular path Time period = Circumference/ velocity

T= 2Πr/v = 2Πmv/qBv = 2Πm/qB

So we conclude that when a charged particle moves perpendicular to the magnetic field, the magnetic force on the charged particle acts like centripetal force and produces a circular motion perpendicular to the magnetic field.

The particle would describe a circle if ‘v’ and ‘B’ are perpendicular to each other.

Radius of circular path = r=mv/qB

Time period of motion = T=2Πm/qB

Frequency of the motion = f=1/T= qB/2Πm

Case 2: Velocity is parallel to the magnetic field

Suppose we have a charged particle of charge ‘q’ moving with a velocity ‘v’ parallel to magnetic field B. The magnetic force on the charged particle will be

Fm= q(v×B)= qvBsinθ=qvBsin0= 0

So we can conclude that when a charged particle moves parallel to the magnetic field then it will not experience any force on it and will continue its motion without any change in its motion.

Case 3: when the velocity vector makes an arbitrary angle with the magnetic field Suppose we have a charged particle with charge ‘q’ . It moves in magnetic field B  with velocity ‘v’ making an angle ‘Θ'  with the Magnetic field

Since the velocity is in an arbitrary direction we can resolve the velocity in a direction parallel to B and perpendicular to B.

v||= v cosΘ  and v⊥=v sin�

So we have two components of velocity: the perpendicular component will follow case 1 and the parallel component will follow case 2 described above.

Due to v, the charged particle will try to move in a circular path in a direction perpendicular to B and due to v||, the charged particle will continue with the same speed and in the same direction.

So the resultant of the two motions will be helical motion.

The radius of the helix is determined by v  and pitch ( horizontal distance traveled in one complete circle ) is determined by v||.

r= mv⊥/qB  =mv cosΘ/qB

Time period  T=2Πm/qB, frequency f=qB/2Πm

l=v||*T= 2Πmv||/qB= 2Πm v sinΘ/qB

Motion in combined Electric and Magnetic field

When a charged particle of charge ‘q’ moves with velocity ‘v’ in the electric and magnetic fields, Lorentz force acts on it which is due to both electric and magnetic force.

F= Fe + Fm

F= q( E+ v×B)

For simplicity let's assume that the velocity, electric field and magnetic field are mutually perpendicular to each other.

Here E= Ej , B= Bk , v=vi

Fe= q*Ej=qE j, so electric force is along y-direction for positive charge

Fm= q( vi × Bk)= qvB(-j)     as   i ×k= -j

so magnetic force is in -y-direction for positive charge.

We can see that electric and magnetic forces are in opposite directions so if we adjust E and B such that magnitude of the two forces are equal. Then total force on the charge ‘q’ will be zero and the particle will go undeflected. This happens when

Fe= Fm    ; qE= qvB ;    v= E/B

This condition can be used to select charged particles of particular velocity out of the beam containing charges with different speeds, therefore serving as a velocity filter.

Magnetic field due to the current element: Biot- Savart law

We know from the above discussion that the current carrying wire produces a magnetic field around it. This phenomenon is called the magnetic effect of current.

The relationship between current and magnetic field produced due to the current element is given by Biot-Savart Law.

According to Biot-Savart law, the magnitude of magnetic field dB due to current element dl

• Is proportional to the length of the current element ‘dl’.
• Is proportional to the current in the wire.
• Is proportional to sinΘ, where Θ is the angle between dl vector and r vector ( position vector of point of observation P)
• Is inversely proportional to the square of the distance between ‘r’ of the point P from the current element.

To remove the proportionality sign we need to put a constant.

Vector form of Biot-Savart law.

Direction of dB

Right-hand thumb rule: If we put our thumb of the right hand in the direction of the current, then the direction of the curl of our fingers will give the direction of the magnetic field.

Magnetic field on the axis of the current carrying loop.

Suppose we have a current carrying loop, carrying current ‘I’ and radius R.

We need to find the magnetic at the axis of the coil at a distance ‘x’ from the center.

When we resolve dB along in the direction parallel to it and perpendicular to it, we will see that perpendicular components of dB due to half loop will be canceled by the other half of the current loop.

And parallel component of dB is integrated for the whole circular loop

Magnetic field due to  current carrying loop using Biot-Savart law

Suppose we have a currency-carrying loop of radius ‘r’ and carrying current ‘I’ and we have to find the magnetic field at the center of the circular loop at O.

From Biot-Savart law we have

Here sinθ=1 as θ=90

So to get B we have to integrate dB along the whole loop

## 1.Magnetism and Gauss’s Law

Chapter 5: Magnetism and Matter

THE BAR MAGNET
A bar magnet is a rectangular piece of an object, made up of iron, steel or any other ferromagnetic substance or ferromagnetic composite, that shows permanent magnetic properties. It has two poles, a north and a south pole such that when suspended freely, the magnet aligns itself so that the northern pole points towards the magnetic north pole of the earth.

### Types of Bar Magnet

There are two types of bar magnet:

• Cylindrical bar magnet: A cylindrical rod is also known as rod magnets that have a thickness equal to larger than the diameter enabling high magnetism property.
• Rectangular bar magnet: Rectangular bar magnets find applications in manufacturing and engineering industries as they have magnetic strength and field greater than the other magnets.

Bar magnet as an equivalent solenoid
A solenoid is a coil with a length greater than its diameter and is a type of electromagnet to produce controlled magnetic fields by passing an electric current through it.

The dipole in a uniform magnetic field
Take a compass with known value of magnetic moment m and moment of Intertia I . Allow the needle to oscillate in a magnetic field of value B.
The torque on the needle is given by

MAGNETISM AND GAUSS’S LAW

Gauss' Law for magnetism applies to the magnetic flux through a closed surface. In this case the area vector points out from the surface.

Because magnetic field lines are continuous loops, all closed surfaces have as many magnetic field lines going in as coming out. Hence, the net magnetic flux through a closed surface is zero.

Net flux = ∫ B • dA = 0

∫EdA=Q/ε0

Where,

• E is the electric field vector
• Q is the enclosed electric charge
• ε0 is the electric permittivity of free space
• A is the outward pointing normal area vector

## 1.Magnetism and Gauss’s Law

Introduction

Magnetic phenomena are universal in nature. In the last chapter, we learned that moving charges or electric currents produce magnetic fields. In this chapter, we will take a look at magnetism as a subject on its own.

The word Magnet is derived from the name of an island in Greece called Magnesia where magnetic ores deposits were found. The directional properties of the magnets have been known since ancient times. A freely suspended magnet points itself in a north-south direction.

Magnetism

Some of the commonly known ideas regarding magnetism are:

• The earth behaves like a magnet with the magnetic field pointing approximately from the geographical south to the north.
•  Directive Property: When a bar magnet is freely suspended. It points in the north-south direction. The tip which points toward the geographical north is called the north pole of the bar magnet. And the tip which points toward the geographical south is called the south pole of the bar magnet.

•  Like poles of the magnet repel each other and unlike poles attract each other.
• Attractive property: A magnet attracts small pieces of iron, cobalt and nickel etc.
• Magnetic poles always exist in pairs: Magnetic monopoles do not exist. We cannot isolate the north or south pole of the magnet. If we try to isolate two poles by breaking the magnet in the middle, each broken part is found to be a magnet with N and S poles.

• Magnetic Induction: It is possible to make magnets out of iron and its alloys.

The Bar magnet

A bar magnet is a bar of circular or rectangular cross-section magnet.

Let’s first discuss some important definitions related to magnetism:

•  Magnetic field:  The space around the magnet within which its influence can be experienced is called the magnetic field.

• Uniform magnetic field: A magnetic field in a region is said to be uniform if it has the same magnitude and direction at all points of that region.

• Magnetic poles: There are regions of apparently concentrated magnetic strength in a magnet where the attraction is maximum.
• Magnetic axis: The line passing through the poles of a magnet is called the magnetic axis of the magnet.

• Magnetic equator: The line passing through the center of the magnet and at right angles to the magnetic axis is called the magnetic equator of the magnet.
• Magnetic length: The distance between the two poles of a magnet is called the magnetic length of the magnet. It is slightly less than the geometrical length of the magnet.

Magnetic dipole and magnetic dipole moment

In electrostatics, we had electric dipoles and electric dipole moments.  In the same way in Magnetostatics we have magnetic dipoles and magnetic dipole moments.

An arrangement of two equal and opposite magnetic poles separated by a small distance is called a magnetic dipole.

Every bar magnet is a magnetic dipole. A current-carrying loop behaves as a magnetic dipole. Even an atom acts as a magnetic dipole due to the circulatory motion of electrons around its nucleus.

Magnetic dipole moment: The magnetic dipole moment of a magnetic dipole is defined as the product of its pole strength and magnetic length. It is a vector quantity, directed from the south pole to the north pole.

Magnetic dipole moment= pole strength * distance between them

M= m * 2l

Where m= pole strength.

2L is the length of a magnet.

M= magnetic dipole moment

S.I. Unit of M= Am2  or joul�� per tesla ( JT-1)

Magnetic Field Lines

Michael Faraday introduces a hypothetical concept of magnetic field lines to represent a magnetic field visually.

Magnetic field lines may be defined as the curve the tangent to which at any point gives the direction of the magnetic field at that point.

Properties of magnetic field

1. Magnetic lines of force are closed curves that start in the air from the N-pole to the S-pole and return to the N-pole through the interior of the magnet.

1. The lines of forces never cross each other. If they do so, that would mean there are two directions of the magnetic field at that point of intersection which is impossible.

1. The lines of forces have a tendency to contract lengthwise and expand sidewise. This explains attraction between unlike poles and repulsion between like poles.

1. The relative closeness of the lines of forces gives the measure of the strength of the magnetic field which is maximum at the poles.

Bar magnet as an equivalent solenoid

When a current is passed through a solenoid, it behaves like a bar magnet. Some observations of similar behavior are as follows :

1. A current-carrying solenoid suspended freely always comes to rest in a north-south direction.
2. Two current-carrying solenoids exhibit mutual attraction and repulsion when brought closer to one another.

1. The pattern of lines of forces of a bar magnet and current-carrying solenoid is exactly similar. Thus a bar magnet and current-carrying solenoid produce a similar magnetic field.

Gauss’s law in Magnetism

Gauss’s Law for magnetism is applied to the magnetic flux through a closed surface. In this case, the area vector points out from the surface. Because magnetic field lines are continuous loops, all closed surfaces have as many magnetic field lines going in as coming out and hence net magnetic flux through a closed surface is zero.

Gauss’s Law in magnetism states that the surface integral of a magnetic field over a closed surface is always zero. The net                           magnetic flux through a closed surface is zero.

Mathematically it is written in integral form as:

Differential form of Gauss’s law in Magnetism

Consequences of Gauss’s Law :

1. Gauss’s law indicated that there are no sources or the sink of the magnetic field inside a closed surface. Hence isolated magnetic poles called monopoles do not exist.
2. The most elementary magnetic element is a magnetic dipole or a current loop. All magnetic phenomena can be explained in terms of an arrangement of magnetic dipoles or current loops.
3. If a number of field lines enter a closed surface then an equal number of lines of forces must leave that surface.

The Earth’s Magnetism

Earth is a powerful natural magnet. Its magnetic field is present everywhere near the earth’s surface. The branch of physics that deals with the study of earth’s magnetism is called terrestrial magnetism or geomagnetism.

Evidence in support of earth’s magnetism:

1. A freely suspended magnetic needle comes to rest roughly in a north-south direction. This suggests that the earth behaves as a large magnet with its south pole lying somewhere near the geographic north pole and its north pole lying somewhere near the geographical south pole.
2. An iron bar buried in the earth becomes a weak magnet. Magnetism is induced by the earth’s magnetic field.
3. Existence of neutral points near a bar magnet indicated the presence of the earth’s magnetic field.

Origin of the earth’s magnetism

If earth behaves like a natural magnet. The question will arise: what is the origin of earth’s magnetism?. Many scientists have given theories about the cause of the earth’s magnetism.

1. William Gilbert suggested that magnetism is due to the presence of magnetic material at its center, which could be a permanent magnet.
2. Prof Blackett suggests that the earth’s magnetism is due to the rotation of the earth about its own axis.
3. Sir E. Bullard said that there are large deposits of ferromagnetic materials like iron, cobalt etc in the core of the earth in molten form. When the earth rotates about its axis, the circulating ions in the highly conducting liquid region of the earth’s core form current loops and hence produce a magnetic field.

Some terms related to earth’s magnetism

Geographica axis:  The straight line passing through the geographical north and south directions of the earth. It is the axis of rotation of the earth.

Magnetic axis: The straight line passing through the magnetic north and south poles of the earth is called its magnetic axis.

Magnetic equator: It is the great circle on the earth perpendicular to the magnetic axis.

Magnetic meridian: The vertical plane passing through the magnetic axis of a freely suspended small magnet. The earth’s magnetic field acts in the direction of the magnetic meridian.

Geographical meridian:  The vertical plane passing through the geographical north and south pole.

Element of Earth’s magnetic field

The earth’s magnetic field at any point can be explained completely by three parameters which are called elements of earth’s magnetism. These elements are Magnetic declination, magnetic inclination and the horizontal component of the earth’s magnetic field.

1. Magnetic declination: The angle between the geographical meridian and the magnetic meridian at a place is called the magnetic declination (α) at that point.

It arises because the magnetic axis of the earth does not coincide with the geographical axis.

1. Magnetic inclination or angle of dip: The angle made by the earth’s total magnetic field B with the horizontal direction in the magnetic meridian is called the angle of dip (δ) at any place.

The angle of dip is different at different places on earth. It is zero at the magnetic equator and 90 degrees at the poles. At all other places, it varies between 0 to 90.

1. Horizontal component of the earth’s magnetic field. It is the component of the earth’s total magnetic field B in the horizontal direction in the magnetic meridian.

It is given by the formula  BH=B cos δ

At magnetic equator  δ=0  , BH= B Cos 0= B

At magnetic poles δ=90, BH=0

## 2. Ohm's Law

Introduction :

Ohm’s law is the basic law regarding the flow of current. It was discovered by G.S. Ohm in 1828, long before the physical mechanism responsible for the flow of current - ‘electrons

The electrons were discovered by J.J. Thomson in 1897.

Ohm’s Law

Imagine a conductor through which a current is flowing

Let the V be the potential drop across the conductor. Then Ohm's  Law states that V α I

The potential drop across the conductor is directly proportional to the current.

To remove the sign of proportionality we need to put some constant so. V= R I

Here. R= resistance of the conductor.

S.I. unit of resistance = ohm Ω

When current flows through the conductor it feels a resistance to its flow. The resistance of the conductor is independent of voltage or current.

There cannot be a better explanation of ohm’s law than the cartoon given below in the figure.

The figure describes how voltage is trying hard to push the current. And the resistance is trying to constrain the path of the current. We can draw two conclusions from it.

• More the potential difference we apply on the ends of the conductor, the more will be the current passing through that conductor. V α I
• More the resistance of the conductor, the lesser will be the value of current passing through the conductor. Current is inversely proportional to resistance.

Similarly, you can conclude that V is that factor that supports the current and R is that factor that opposed it

Resistance

The resistance of the conductor depends on

• Length of the conductor
• Area of cross-section
• Nature of material of the conductor.

Now let's try to understand what caused the conductor to offer resistance to the flow of current and why it depends on the factors above. Let's try to get these answers first.

What causes the conductor to offer resistance to the flow of current?

In a conductor there are large numbers of electrons and ions, free electrons moving in the conductor suffer many collisions with the bound electrons, ions and other free electrons. These frequent collisions hinder the flow of electrons and thus provide resistance to the flow of charges and also provide resistance to the flow of electric current.

Why does resistance of conductor depend on the factors mentioned above?

I want to answer this with the help of an analogy. Imagine a very busy road where collisions with other vehicles are very frequent ( just a hypothetical situation ) with hundreds of vehicles on it. The longer that road is, the chance of more collisions. The same analogy is with a conductor, longer the conductor, more will be the number of collisions it would suffer and hence will offer greater resistance to the flow of current.

Resistance of the conductor is directly proportional to the length of the conductor  R  α L

Now imagine two roads, one is a narrow road and the other will be wider. Suppose the same number of vehicles are traveling on both roads. Now tell me on which road the collisions between vehicles should be less narrow or wider? Obviously, the answer would be that collisions will be lesser on wider roads. You can answer it based on your daily experiences. In the same way, a conductor with a wider cross-section offers less frequent collisions than a conductor with a narrow cross-section

Hence resistance is inversely proportional to the cross-section area.

R α 1/A

The third dependence of resistance is on the nature of the material of the conductor.  Resistance depends on the charge density of the conductor and also some other factors like mean free path, relaxation time, and mass of the constituent atoms or molecules. These parameters are different for different metals. We account for all these factors in a single quantity called resistivity ρ of the conductor.

Therefore resistance is directly proportional to the resistivity of the conductor.

R α ρ

If we combine all three we will get.

Resistivity:  Resistivity or specific resistance of a material may be defined as the resistance of a conductor of that material having a unit length and unit cross-section area. It is denoted by ρ, S.I. unit of resistivity is ohm meter  (Ω m). Resistivity depends on the nature of the material of the conductor and the physical conditions like temperature and pressure but it is independent of the shape and size

I- V characteristic and the resistance

I-V characteristic and the resistance are closely related, we can find the resistance of a conductor by using its I-V characteristic.

The slope of the I-V characteristic gives  1/R.

V= I*R     ;   I = (1/R) * V

If we plot current along the y-axis and voltage on X-axis then, Comparing this with the equation of straight line y = m x, we can conclude that the slope of the I-V graph gives ‘ slope  m=1/R’, we can find resistance  R= 1/slope. refer to the graph above. The greater the slope of the I-V characteristics lesser is the value of resistance.

Now if we have a V-I graph, voltage is represented along the y axis and current along the ‘x-axis. Then on comparing the V= R I with  y= m x, we can conclude that the slope of the V-I graph is equal to the resistance. Greater the slope of V-I characteristics, the greater the value of resistance.

Current density, conductance and conductivity:

Let us quickly go through these definitions

Current Density: The current density at any point inside a conductor is defined as the amount of charge flowing per second through a unit area held normal to the direction of the flow of charge at that point. It is a vector quantity. Current density j= current/ area perpendicular to the direction of the current.

J= I / A

If the area is not perpendicular to area A but makes an angle θ  with the direction of current then.

Component of Area normal to the current  ‘An’= A cosθ

J= I/ Acosθ

S.I. Unit = Ampere per square meter. ( A/m^2)

Conductance: The conductance of a conductor is the ease with which electric charges flow through it. It is equal to the reciprocal of resistance. It is denoted by  G

Conductance = 1 / resistance

G= 1/ R

S.I unit is  mho or siemens (S)

Conductivity:  The reciprocal resistivity of a material is called conductivity. It is denoted by σ.

Conductivity = 1 / resistivity

σ = 1/ρ

S.I. Unit is siemens per meter. (S/m)

Vector form of ohm’s law.

If E= magnitude of the electric field in a conductor

l= length of the conductor, A= Cross-section area of conductor

V= potential difference, ρ= resistivity, σ= conductivity

R= Resistance, J= current density

Then

The above expression J=  σ E is called the vector form of ohm’s law.

Limitations of Ohm’s Law

•  This law is not applicable to unilateral networks. Unilateral networks allow the current to flow in one direction. Such types of networks allow the current to flow in only one direction.

For example, transistors, diodes etc.

• This law is not applicable for non-linear electrical elements with parameters like capacitance, resistance etc. voltage in such circuits won't be constant with respect.
• Ohm’s law is only applicable for metallic conductors and not valid for non-metallic conductors.

I-V characteristic of various electrical elements.

Ohmic and Non-ohmic conductors

In ohmic conductors, the graph between Voltage and current is linear. The graph starts at the origin and is a straight line. These conductors follow ohm’s law. I-V characteristic graphs of non-ohmic conductors are non-linear graphs, usually curved ones.

Note: A straight-line graph with an intercept ( not starting from the origin ) is also a non-linear graph.

Drift velocity, mobility and relaxation time for the conductors

When we apply electric field ‘E’ across the end of the conductors. Free electrons experience a force on it  ‘F= -eE’ and thus also experience an acceleration. F= ma= -eE so a = -eE/m, This is the acceleration experienced by the charge in the presence of an electric field.

The electron in the conductor will start to drift in a direction opposite to the direction of the Electric field. But the electrons would still suffer multiple collisions but now there will be a preferred direction of motion of electrons. The average time between two successive collisions of an electron is called relaxation time 'τThis relaxation time is of the order of 10^(-14)s.

The velocity gained during this time is called drift velocity ‘Vd’ Vd= a* τ = -eEτ/m

Drift velocity may be defined as the average velocity gained by the free electrons of a conductor in the opposite direction of the externally applied electric field.

Mobility of charge carriers

The conductivity of any material is due to its mobile charge carriers. Greater the mobility of a charge carrier will be its conductivity.

The mobility of a charge carrier is the drift velocity acquired by it in a unit electric field. I

Mobility is denoted by μ, and  μ= Vd/E  = eτ/m

Expression of current in terms of drift velocity and mobility

We can express an electric current in terms of drift velocity and mobility also.

I= neA *Vd = neA* μ E

Where  I= electric current

n= free charge density of  the conductor

A= Area of a cross-section of conductor

Vd= drift velocity

μ= mobility of free electron

E= applied electric field.

Expression of resistivity in terms of electron density and relaxation time

We can get the resistivity in terms of electron density and relaxation time. We would start with the formula of drift velocity and current in terms of drift velocity. We will then rearrange the terms and use ohm’s law.

The above expression is the formula for resistivity

It depends on two factors :

1.  Number of free electrons per unit volume or electron density of the conductor
2. The relaxation time  τ

The resistivity of different materials.

The materials are classified as conductors, insulators and semiconductors.

Conductors: The materials which conduct electric currents are conductors. The resistivity of the conductors is very low. It ranges from 10^(-8) Ω m to 10^(-6) Ω m.

Copper and aluminum have the lowest resistivity. Nichrome has a resistivity of about 60 times that of copper so nichrome is used in electric heaters and electric iron. Metals have a positive coefficient of resistivity. With the increase in the temperature of metals their resistivity increases.

Insulators: The materials which do not conduct electric currents are insulators. They have high resistivity, more than 10^4 Ω m. Insulators like glass, mica, and bakelite have very high resistivity in the range of 10^(14) Ω m to 10^(16) Ω m. Semiconductors: These are the materials whose conductivity lies in between conductors and insulators. They usually conduct at room temperature and are insulators at very low temperatures. The range of resistivity of semiconductors is very wide and ranges from 10^(-6)  Ω m to 10^(4) Ω m.

Semiconductors have a negative temperature coefficient of resistivity. The resistivity of semiconductors decreases with an increase in temperature.

Temperature dependence of the resistivity

If we see the formula for resistivity we do not see temperature explicitly present there. Then why does the resistivity depend on temperature?

The answer to this question is that resistivity depends inversely on relaxation time. The relaxation time is the average time elapsed between two successive collisions. This relaxation time actually depends on temperature.

As the temperature increases, there will be more frequent collisions and thus reducing the relaxation time. If you follow the formula for resistivity, you can tell that resistivity is inversely proportional to relaxation time. If the relaxation time would decrease with increase in temperature. Resistivity would increase with increase in temperature.

So. Yes! There is no explicit dependence on temperature in the formula of resistivity, but it is implicitly present during the relaxation time.

In the above text, I have explained why the temperature affects the resistivity of a conductor. Now the next question will be How much does the temperature affect the resistivity of any conductor?

Resistance and resistivity both increase with the increase in temperature in exactly the same manner. As they are Resistance is directly proportional to resistivity. In the figure given below, there is given the formula.

If we consider the dependence of resistance on temperature

Here R= resistance at temperature T

R0= resistance at temperature To

To= reference temperature is usually 20 degrees Celcius but sometimes it is 0 C.

T= given temperature

α= Temperature coefficient of resistivity.

Temperature coefficient of resistivity

We can find the temperature coefficient of resistivity by just rearranging the terms. Consider,

Note: the value of α is positive for conductors and negative for semiconductors.

For conductors, the value of resistivity and hence resistance would increase with the increase in temperature. As shown in the figure below. At higher temperatures, the slope of the V-I graph will be more.

Electrical power and energy

To maintain the steady flow of current through the conductor in a circuit an external force is required which must supply the power. In a simple circuit with a cell, It is the chemical energy of the cell which supplies this power. Moreover, Inside the conductor when free charges are drifting inside the conductor under the action of the electric field, their kinetic energy would increase as they move, However, we have got that charges do not move with acceleration but move with steady drift velocity.

This is because of the collision of the ions and the atoms during transit. The kinetic energy gained by the charges is shared by the atoms during collisions and atoms start vibrating vigorously, and the conductor heats up. Thus an amount of energy is dissipated as heat in the conductor. The energy dissipated per unit time is called power dissipated. The formula for the power dissipated is given below.

Where R= resistance of the conductor, V= potential difference and I= current through the conductor. If you are someone who always struggles between energy and power. Let me try to help you with that.

Please look at the diagram above, It relates energy and power with water flowing through a pipe and collecting in the bucket.

Power is actually the rate of doing work or here it can be related to the rate of flow of water and energy is the amount of work done in some time ‘t’ and here it is related to the amount of water collected in the bucket in time ‘t’. We can conclude the discussion above about electrical energy and power in a table.

The above diagram can help you to remember all these formulas.

A fun thing to do: Below is the link to the simulation of ohm’s law, you can play with it.

simulations on ohm's law

Explanation of the simulation: In this simulation, there is a circuit with some battery and resistor and you can see the values of V, I and R.

What you do in this

• You can change the value of voltage by sliding the voltage up and down can note how current is changing
• You can change the value of resistance in the circuit for a fixed value of V and see how current will be varying.

## 2. Ohm's Law

OHM’S LAW

A basic law regarding flow of currents was discovered by G.S. Ohm in.1828,Ohm’s law states that the current through a conductor between two points is directly proportional to the voltage across the two points.

V = IR

LIMITATIONS OF OHM’S LAW

Although Ohm’s law has been found valid over a large class of materials, there do exist materials and devices used in electric circuits where the proportionality of V and I does not hold.

• Ohm’s law is also not applicable to non – linear elements. Non-linear elements are those which do not have current exactly proportional to the applied voltage that means the resistance value of those elements changes for different values of voltage and current. Examples of non – linear elements are the thyristor.
• The relation between V and I depends on the sign of V. In other words, if I is the current for a certain V, then reversing the direction of V keeping its magnitude fixed, does not produce a current of the same magnitude as I in the opposite direction. This happens for example in the case of a diode.

RESISTIVITY OF VARIOUS MATERIALS

Electrical Conductivity is an intrinsic property of a material which is defined as the measure of the amount of electrical current a material can carry. Electrical conductivity is also known as specific conductance, and the SI unit is Siemens per meter (S/m). It is also defined as the ratio of the current density to the electric field strength.  It is represented by the Greek letter σ.
TEMPERATURE DEPENDENCE
The resistivity of a material is found to be dependent on The temperature. Different materials do not exhibit the Same dependence on temperatures. Over a limited range Of temperatures, that is not too large, the resistivity of a Metallic conductor is approximately given by,
ρT = ρ0 [1 + α (T–T0)]
ELECTRICAL ENERGY, POWER
A cell has two terminals – a negative and a positive terminal. The negative terminal has the excess of electrons whereas the positive terminal has a deficiency of electrons. Let us take the positive terminal as A and the electrical potential at A is given by V(A). Similarly, the negative terminal is B and the electrical potential at B is given by V(B). Electric current flows from A to B, and thus V(A) > V (B).The potential difference between A and B is given by
V = V(A) – V(B) > 0

## 3. Combination of resistors

Combination of resistor: Series and parallel

Like we have studied the series and parallel combination of capacitance in electrostatics. Here we will see how we can combine resistors of different resistances in series and parallel and get a value equivalent to resistance in both the cases.

We will be using ohm’s law throughout the topic.

Series combination of Resistors:

When resistors are joined end to end like in the figure above. We call this combination a series combination of resistors.

In a series combination of the resistances

1. The same current passes through all the resistances.

I1=I2=I3=I

1.  The voltage of the battery gets divided between the resistors, such that the potential difference of the battery V is equal to the sum of individual potential drops of each resistor.

V= V1+ V2+ V3

If we apply ohm’s law we will get,

let Req be the equivalent resistance of the series combination of the resistors

‘I’ be the current in the circuit.

V= potential difference of the terminals of the battery

V1, V2, V3 be the potential drops  of R1, R2, R3 respectively

Then from ohm’s Law.

V= V1+ V2+ V3

I * Req= I* R1 + I*R2+ I*R3

Req= R1 + R2+ R3

If we have ‘n’ number of resistors in series then equivalent resistance will be given as:

In the circuit given below if we need to find the potential drops through each resistor in series

V1 = Potential drop of R1 =  I* R1

Similarly, V2= I*R2  and V3= I*R3

And power dissipated in each resistor,

P1= power dissipated in resistor R1= V1*I = V1^2/R1= I^2*R1

Similarly, we can find the power dissipated by R2 and R3.

Parallel combination of resistors

Below is the figure of the parallel combination of resistors.

Two or more resistors are said to be in parallel if one end of all the resistors are joined together and similarly the other ends join together.

In parallel combination of the resistors

1. Voltage is the same across all the resistors connected in parallel and is equal to the potential difference of the battery.

V1= V2= V3=V

1. In a parallel combination of resistors, the current is divided between the resistors such that the sum of currents through all the resistors are equal to the total current in the circuit.

I= I1+I2+ I3

Let Req= equivalent resistance of the circuit.

I= total current in the circuit

V= potential difference of the terminals of the battery

I1. I2 and I3 are the currents through R1, R2, and R3 respectively.

By using ohm's law we have

I1= V/ R1 , I2= V/R2 , I3= V/R3  and I= V/Req

If we put above values in I= I1 + I2 + I3

We will get 1/Req = 1/R1 + 1/R2 + 1/R3, This is the formula for finding equivalent resistance for the parallel combination of the resistors.

If we have two resistors in parallel shown below

Then equivalent resistance of the circuit will be given by

If we have ‘n’ resistors connected in parallel the equivalent resistance will be

Cells, EMF, and Internal resistance

In the circuit given below a positive charge flows spontaneously in a conductor from higher potential to lower potential in the direction of the electric field. To maintain the current through the conductor, some external devices must do some work at a steady rate to take the positive charge from lower potential to higher potential. Such a device is the source of EMF ( electromotive force). These devices can be a battery, cell, or electrolytic solution.

Electromotive force may be defined as the work done by the source in taking a unit positive charge from lower potential to higher potential. The EMF of a source is equal to the maximum potential difference between its terminals when it is in an open circuit. In other words, the emf of a source may be defined as the energy supplied by the source in taking a unit positive charge once round the complete circuit.

The term EMF- electromotive force is actually a misnomer. The emf is not a force at all. It is a special case of potential difference in which the circuit is open.

V= W/q, therefore emf has also the nature of work done per unit charge.

S.I.  unit of EMF is Volt.

Before going ahead with the topic, Let's have a pause and first try to understand the difference between the EMF of a cell and the potential difference of the cell.

Relation between  EMF, Internal resistance and terminal potential difference of the cell

In the circuit, above we have a cell with EMF ‘E’ and internal resistance ‘r’ and it is connected with a load resistance ‘R’in series.

V= potential difference across the load resistance

Total resistance in the circuit= R+ r

Current I= E/(R+r)

So V (terminal potential difference )  = I*R = ER/(R+r)

Also by simplifying we get    V= E- Ir

E= V+ I*r = I*R+ I*r = I( R+r)

Special Case:

• When the cell is open I=0  we have  V open= E

Thus the potential difference across the terminals of the cell is equal to its emf when no current is being drawn from the cell.

• A real cell has always some internal resistance ‘r’, so when the current is being drawn from the cell, we have

V closed < E

Thus  the potential difference across the terminal of a cell in a closed circuit is always less than its EMF

Combination of cells in series and parallel

Like capacitors and resistors, we can also group cells together in series and parallel combinations

Series combination of cells

The above combination is the series combination of the cell, where the positive terminal of one cell is connected with the negative terminal of the other cell.

Then the total emf of the cell will be given by some of the emf of all the individual cells.

Suppose we have practical cells with internal resistance as shown in the figure below.

Then equivalent  EMF ‘Eeq’ = E1 + E2 +E3

And total internal resistance   req= r1+ r2+ r3

Here all the cells are identical having the same EMF and internal resistance

Eeq= 3E   and req= 3r

Therefore, for a combination of ‘n’ identical cells in series

Eeq= nE, equivalent emf will be ‘n’ times the emf of single-cell

req= nr, and equivalent internal resistance would be ‘n’ times the internal resistance of a single cell.

Also, take care of one more concept. If any one of the cells is connected in reverse, then we will subtract the emf of that cell.

Combination of cells in parallel.

Below is the parallel connection of cells

Here all the positive terminals of cells are joined together and similarly, the negative terminals are joined together.

Suppose we have two cells with emf   ε1   and  ε2  and internal resistance r1 and r2.

/r = ε1/r1 + ε2/r2

1/r= 1/r1+ 1/r2

For parallel combinations of ‘n’ cells, equivalent emf and internal resistance is given by following relations.

ε/r= ε1/r1 + ε2/r2 +ε3/r3 +.... εn/rn

1/r= 1/r1+ 1/r2+1/r3 .....+ 1/rn

A fun thing to do:

Below is the link to a simulation with which you can design your own circuits using various electrical elements like wire, batter, light bulb, resistor of different values,  switch, fuse etc. Just as many circuits as you want and learn with fun.

DC circuit

What can you get from this?

• It can show the direction of conventional and electronic current
• Using voltmeter and ammeter in your circuit you can get the value of voltage and current.
• There is some advanced tools also that can change  the wire resistivity and also the battery internal resistance

## 3. Combination of resistors

COMBINATION OF RESISTORS

SERIES AND PARALLEL
The connection is in such a manner that the current flowing through the 1st register has to then flow further through the 2nd register and then through 3rd. Therefore, a common current is flowing in connection with a resistor in series. At all point in the circuit, the current amoung the resistors is same. For example,
I1 = I2 = I3 = It = 2ma
All the resistors in series that is R1, R2, R3 have current I1, I2, I3 respectively and the current of the circuit is It.
Resistor in Parallel
Unlike, series connection, in parallel connection, current can have multiple paths to flow through the circuit, hence parallel connection is also current dividers. Common voltage drop is across the parallelly connected circuits/networks. At the terminals of the circuit, the voltage drop is always the same. For example
VR1=VR2=VR3=VRT=14V

The voltage across R1 is equal to the voltage across R2 and similarly, equal to R3 and hence the total voltage drop is equal to the voltage across the circuit. Reciprocal of individual resistance of each resistor and the sum of all the reciprocated resistance of resistor will us the total resistance of the circuit.

CELLS, EMF, INTERNAL RESISTANCE
What is an Electromotive Force (EMF) of a Cell?
The electrolyte has the same potential (emf) throughout the cell. The condition of no current flowing through a cell is also known as an open circuit. An open circuit result in a potential (emf) of the cell is equal to the difference of potentials (emf) of the electrodes. Anode has a positive potential (V+) whereas Cathode has a negative potential (-V). This potential difference is known as the Electromotive Force (EMF).An electric battery is a device made up of two or more cells that make use of the chemical energy stored in the chemicals and converts it into electrical energy.

## 4. Law's and Bridge

INTRODUCTION:

In 1942, a German Physicist Kirchhoff extended Ohm’s law to complicated circuits and gave two laws, which enable us to determine the current in any part of a complicated circuit easily in an organized way. But before jumping to the topic, let me first introduce some terms that may be used in later topics.

1.  Electrical network: The term electric network is used for a complicated system of electrical conductors. The above circuit is an electrical circuit.
2. Junction: Any point in an electrical circuit where two or more conductors are joined together is a junction. In above circuit b, e are junctions.
3. Loop and Mesh: Any closed conducting path in an electric network is called a loop or mesh. In the above circuit, we have two loops, loop abefa and ebcde.
4. Branch: A branch is any part of a network that lies between two junctions.  In the above circuit  ab, bc, cd etc are branches

Kirchhoff’s Law - When we have complicated circuits and use of ohm’s law and formulae for series and parallel connection of resistors to solve the circuit are not very helpful. Kirchoff’s law became the savior for such circuits.

There are two laws of Kirchoff's

1.  Kirchhoff's Voltage Law (KVL)  also called the loop law
2.  Kirchhoff’s Current Law ( KCL ) also called junction law

Kirchhoff’s Current law ( KCL) - Kirchhoff’s current law states that in an electric circuit, the algebraic sum of current at any junction is zero.

The Sum of currents entering a junction is equal to the sum of currents leaving that junction.

Mathematically, this law can be explained by  Σ I= 0

Here you can see that currents in blue colors‘ i1 ‘ and ‘i2’ are incoming currents and currents in the red colors ‘i3’ and ‘i4’ are outgoing

Currents.

So according to KCL,  i1 + i2= i3+ i4

Let's see some other examples and try to learn from them.

here  , I1+ I4= I2 + I3+ I5

I hope with the above examples you have got a better understanding of Kirchhoff's Current Law.

You can also imagine a pipe in which water is flowing, Just like at any place inside the pipe the amount of water coming will be the amount of water going and there is not anything like pilling of water happening inside it. It's just flowing.

In the same way, when charges move inside a conductor, we expect the same ‘ no pilling of charge happens at any point in the circuit’. This is actually the cause of Kirchhoff's current law.

Kirchhoff’s Voltage Law ( KVL) - Kirchhoff’s Voltage Law states that around any closed loop of a network. The algebraic sum of the change in the potential must be zero.

In another word, The Algebraic sum of the emfs in any loop of a circuit is equal to the product of currents and resistances in it.

Mathematically, Σ Δ��=0    and  Σ ε=Σ IR

The sign convention used in Loop Law  ( KVL)

We have to assign a loop current for every loop, we can take any direction as the direction of traversal.

Like in the circuit given below, we have three loops and hence three-loop currents. The direction of all the loops is taken clockwise in the figure given below.  The direction of the loop can be anything

1.  All loops in a clockwise direction
2. All loop in an anticlockwise direction
3. Some in clockwise and some in an anticlockwise direction.

Like in the circuit given below, we have three loops and hence three-loop currents. The direction of all the loops is taken clockwise in the figure given below.

In the figure given below, the direction of the loop in loop 1 is clockwise and the direction of the loop is anticlockwise in the second loop.

• The emf of the cell is taken to be positive if the direction of traversal is from the negative to positive terminal ( rise in potential ).
•  Emf would be taken negatively if the direction of traversal is from positive to negative.

• The current-resistance product ( IR) is taken to be positive if the resistor is traversed in the same direction of the assumed current. ( loop current)
•  IR would be negative if the resistor is traversed in the opposite direction of the assumed current.

In the above figure if the direction of traversal is the same as the current ( Left to right )  then V= IR would be positive and if the direction of traversal is opposite to assumed current “I” then V= -IR.

Example - Let's take an example of an electric circuit with two loops. We will apply the above information and try to form the two-loop equations.

In the figure given below, we have two loops. Let's assume that the direction of both loops is clockwise with loop current I1 and I2 respectively.

Let's start to move in loop one abefa in the direction of the loop with loop current I1,

•  In the branch, ‘ab’  potential drop across R1 would be written as  + I1 *R1 as we are moving along the assumed current I1.
• In the branch ‘be’’, we encounter resistance R2. The potential drop across R2 = + I2*R2   as we are moving along the assumed current I2 in this branch.
• Now in the branch ‘ef’, we encounter an emf source and we are moving from its negative terminal to a positive terminal ( rise in potential so emf would be taken positive( + V1). In branch ‘fa’ there is nothing so leave it.

Now if we use all the above information in the equation Σ ε= IR

Then we will have   + V1= +I1*R1 + I2*R2   as the equation of first loop   + 24 = 3* I1+ 3* I2   ………(1)

Let's move to the second loop ‘bcdec’, notice that the current in-branch bc, cd, de is I3  and current in-branch be is I2.

•  As we move in-branch bc, voltage drop through R3 will be   +I3*R3  and in-branch cd, Voltage drop through R4 will be + I3*R4, as in both the branches we are moving along the assumed current I3.

So the equation for branch bcd will be  I3*R3 + I3*R4.

• Now in branch ‘de’ we encounter a battery V2, and while moving from d to e, we are going from the positive terminal of the battery to the negative terminal ( drop-in potential) hence emf would be taken as negative ( -V2).
• In branch ‘eb’ we are going from  e to b, but current I2 ( b to e) is opposite to the direction of traversal, So the voltage across R2 in loop 2  will be  -I2*R2

If we now combine all information in the equation Σ ε= IR

We will get        -V2 = I3*R3 + I3*R4 -I2*R2

Or ,   -29 = I3* 3 + I3* 4 - I2*3 = I3 ( 3+ 4) -I2*3

-29= 7*I3 - 3*I2    ………..(2)

Since we have three variables  I1, I2, and I3  so we must have 3 equations to solve these variables, we have already got 2 equations.

We can get the third equation by applying KCL at junction b.

At Junction b, I1= I2 + I3   ……. (3)

So, now you can use these three equations and solve for three currents  I1, I2, I3  and then you would be able to tell the value of current, voltage drop and direction of current through each resistor.

Wheatstone Bridge - Wheatstone bridge is an application of Kirchhoff’s Law.

This bridge consists of 4 resistance  P, Q, R and S and across one pair of diagonally opposite points ( say AC),  a source  E is connected. For simplicity, we assume that the internal resistance of the source is zero. And between the other two vertices ( BD)  a galvanometer G  ( a device to detect current ) is connected.  Let G be the resistance of the galvanometer and Ig current passes through it

Current I1, I2, I3, and I4 flow through resistors P, R, Q and S respectively, and Ig is the current through the galvanometer.

We can change the resistance of S. There will be one special situation when the galvanometer will show zero deflection and current through it Ig=0. This is called the balanced condition of the galvanometer.

We can use the Kirchhoff laws and use Ig= 0 in that, when we do that we will have a special result about values of resistances P, Q, R and S.

When   P / Q = R / S  ,

The Wheatstone bridge will be balanced and Ig=0, no current will pass through the diagonal branch and the galvanometer will show no deflection.

Application of Wheatstone bridge - Wheatstone bridge and its balance condition provide a practical method of finding the unknown resistance.

Here resistance R4 is unknown, When we put known resistances R1 and R2 in their places as shown and R3 here is standard resistance whose value can be varied. We go on varying R3 till the galvanometer shows a null deflection and the Wheatstone bridge is in balanced condition.

The resistance of R3 at that time is noted and we can use the condition of a balanced Wheatstone bridge to find the resistance of R4.

R4= R3 ( R2/R1).

Meter bridge - Meter bridge works on the principle of Wheatstone bridge. But the difference is that the two resistances R1 and R2 in the Wheatstone bridge are replaced by a wire of uniform cross-section and of length one meter. That’s why it is called the Meter bridge.

This wire is stretched tight and clamped between two thick metallic strips bent at right angles.

The metallic strips have two gaps ( G1 and G2)  across which the two resistors ( P, Q)  can be connected. The endpoints ( A and B)  where the wire is clamped are connected to a cell through a key.

One end of the galvanometer is connected with the metallic strip midway in between the two gaps( point E). The other end of the galvanometer is connected with the jockey ( J)

A jockey is essentially a metallic rod whose one end is a  knife-edge that can slide over the wire of a meter bridge to maintain an electrical connection.

Suppose P is an unknown resistance and Q is known resistance. When we slide the jockey on the wire. On a particular point of the wire, the galvanometer will show null deflection and the meter bridge is in a balanced condition.

Let  ‘L1’ be the length of the wire from A to the null point on the wire.

And L2 is the length of wire from B to the null point of the wire.

We know L1 + L2= 1 m = 100 cm

So L2= 100- L1

The four resistance of the bridge in balanced conditions are P, Q, RcmL1  and Rcm(100-L1), where Rcm is the resistance of wire per unit cm.

So, according to the principle of the Wheatstone bridge

P/Q= Rcm*L1 / ( Rcm*(100-L1)

Therefore,     P/Q= L1/(100-L1)

So unknown resistance P= Q * L1/ (100-L1)

Where L1 = balanced length of wire from A in cm.

Potentiometer - This is an ideal voltmeter that does not change the original potential difference and still can measure potential difference and also doesn't need to have infinite resistance.

Construction:  It is basically a long piece of uniform wire sometimes a few meters in length across which a standard cell is connected. Usually 1 m long wires are fixed on a wooden board parallel to each other. These wires are joined in series by thick copper trips. Then ends of the wire are connected with a battery, a key K and a rheostat. This circuit is called a driving circuit that sends a constant current I through the wire AB. Thus potential difference gradually falls from A to B. A jockey can slide along the length of the wire.

The principle “ When a constant current flows through a wire of uniform cross-section area and composition, the potential drop across any length of the wire is directly proportional to that length.

V α l,    V= kl      ……1

Where K= proportionality constant called potential gradient.

By ohms law we have   V= I R = Iρl/A =(/A)*l  ….2

By comparing equations 1 and 2 we can conclude that

k=/A = constant

Potential gradient -The potential drop per unit length of a potentiometer wire is called the potential gradient.  k= V/l

S.I unit is V/m  but the practical unit is V/cm

Sensitivity of the potentiometer is its capability of measuring a very small potential difference and shows a significant change in balancing length for a small change in the potential difference is measured.

How we can increase the sensitivity of the potentiometer:

•  For a given potential difference sensitivity can be increased by increasing the length of the potentiometer wire
• For a potentiometer wire of fixed length, the potential gradient can be decreased by reducing the current in the circuit with the help of a rheostat.

Application of potentiometer

1 .  comparison of emf’s of two primary cells.

For the comparison of the emf’s of the two cells, the circuit above will be used. Here E1 and E2 are the emf’s of the two cells. Suppose when key K1 is closed and K2 is open and jockey J is moved on the wire AB, the balancing length corresponding to E1 will be ‘L1’.

E1= k*L1              …(1)

Now when K2 is closed and K1 is open, the balancing length corresponding to cell E2 is ‘L2’.

E2= k*L2            …..(2)

From 1 and 2 we have

E1/ E2= L1/L2    …(3)

If emf of one cell is known and emf of another cell is known then we can find the unknown emf of the cell.

E2= E1( L2/L1)      …(4)

2. To find the internal resistance of the primary cell

Using the above circuit we can find the internal resistance of the cell.

Close the K1 and keep the Key 2 open and move the jockey on the potentiometer wire. Suppose you get balancing length ‘L1’ corresponding to the emf of cell E

E= k * L1            ..(1)

Now close both key K1 and k2 and move the jockey on the potentiometer wire and let you get a balancing length ‘L2’ corresponding to potential drop V

V=k*L2            ..(2)

From 1 and 2

E/V= L1/L2       … (3)

From Ohm’s law, we know that   E= I( R+rand V= IR

E/V= (R+r)/R=1+r/R    …. (4)

Comparing 3 and 4 we get

1+r/R= L1/L2

Therefore   r= R( L1-L2)/L2   …(5)

Equation 5 is used to find the internal resistance of the cell using a potentiometer.

## 4. Law's and Bridge

KIRCHHOFF’S RULES

Electric circuits generally consist of a number of resistors and cells interconnected sometimes in a complicated way. The formulae we have derived earlier for series and parallel combinations of resistors are not
Mvalways sufficient to determine all the currents and potential differences in the circuit. Two rules, called Kirchhoff’s rules, are very useful for analysis of electric circuits.
V = ε + I r

WHEATSTONE BRIDGE
The Wheatstone bridge works on the principle of null deflection, i.e. the ratio of their resistances are equal and no current flows through the circuit. Under normal conditions, the bridge is in the unbalanced condition where current flows through the galvanometer.

METER BRIDGE
A meter bridge consists of a wire of length 1 m and of uniform cross-sectional area stretched taut and clamped between two thick metallic strips bent at right angles with two gaps across which resistors are to be connected. The end points where the wire is clamped are connected to a cell through a key. One end of a galvanometer is connected to the metallic strip midway between the two gaps. The other end of the galvanometer is connected to a jockey which moves along the wire to make electrical connection.

POTENTIOMETER
This is a versatile instrument. It is basically a long piece of uniform wire, Sometimes a few meters in length across which a standard cell is Connected. In actual design, the wire is sometimes cut in several pieces Placed side by side and connected at the ends by thick metal strip.

## 2. Ampere's circuital law

THE SOLENOID AND THE TOROID

The solenoid

We shall discuss a long solenoid. By long solenoid we mean that thesolenoid’s length is large compared to its radius. It consists of a long wire wound in the form of a helix where the neighbouring turns are closely spaced. So each turn can be regarded as a circular loop. The net magnetic field is the vector sum of the fields due to all the turns. Enamelled wires are used for winding so that turns are insulated from each other.

Toroid?

A toroid is shaped like a solenoid bent into a circular shape such as to close itself into a loop-like structure. The toroid is a hollow circular ring, as can be seen in the image shown below, with many turns of enameled wire, closely wound with negligible spacing between any two turns.

FORCE BETWEEN TWO PARALLEL CURRENTS, THE AMPERE

The force between two long straight and parallel conductors separated by a distance r can be found by applying what we have developed in preceding sections. Figure 1 shows the wires, their currents, the fields they create, and the subsequent forces they exert on one another. Let us consider the field produced by wire 1 and the force it exerts on wire 2 (call the force F2). The field due to I1 at a distance r is given to be

What is a Moving Coil Galvanometer?

A moving coil galvanometer is an instrument which is used to measure electric currents. It is a sensitive electromagnetic device which can measure low currents even of the order of a few microamperes.

Moving-coil galvanometers are mainly divided into two types:

• Suspended coil galvanometer
• Pivoted-coil or Weston galvanometer

Moving Coil Galvanometer Principle

A current-carrying coil when placed in an external magnetic field experiences magnetic torque. The angle through which the coil is deflected due to the effect of the magnetic torque is proportional to the magnitude of current in the coil.

## 2. Ampere's circuital law

Introduction:

Like In Electrostatics we have coulomb's law to find electric fields due to charge distribution, but that is very tedious to use in many cases. So in case of some symmetry, we had Gauss's law that can be used to find electric fields in an easier way.

In the same manner in Magnetostatics we have the Biot-Savart law that can be used to find magnetic fields due to any current distribution.

But in the case of some symmetry we have, we can use ampere circuital law that can make our life easier.

Ampere circuital law

Ampere circuital law states that the line integral of the magnetic field  B around any closed circuit is equal to μ0 ( permeability constant) times the total current ‘I', Threading or passing through this closed circuit.

Mathematically,

B dl = μo* I

Where I is the net current enclosed by the closed circuit. The closed curve is called the Amperean loop, which is a geometrical entity and not a real wire loop.

Ampere circuital law can be used to find magnetic field due to magnetic field in some symmetry case like long wire, circular loop, cylindrical conductor etc,

Above are some examples of magnetic fields in some standard cases. We will discuss Solenoid and toroid in detail

Force on a current carrying conductor in a magnetic field

When the conductor carrying current is placed in an external magnetic field, it experiences a mechanical force. The direction of the force is perpendicular to both the current and the magnetic field and it is given by Fleming’s right left-hand rule.

Cause of the force: A current is an assembly of moving charges and magnetic field exerts a force on moving charges, That is why a current-carrying conductor experiences a side-ways force as the force experienced by the moving charges( free electrons) is transmitted to the conductor as a whole.

The expression of the force will be derived as

Thus Force on the current carrying conductor placed in a magnetic field is given by F= IlBsinθ= I ( l×B)

Direction of force is given by Fleming’s left-hand rule.

If we place the fingers of our left hand in mutually perpendicular directions. If the forefinger is along the direction of the magnetic field and the middle finger points in the direction of current then the thumb will give the direction of the force on the conductor.

The force between two parallel current

As we know, the current carrying conductor produces a magnetic field around it and when any current carrying conductor is placed in a magnetic field it will experience a force on it.

So if we have one current carrying conductor it will produce a magnetic field around it. If we place another current carrying conductor near the first conductor, it will actually be a current carrying conductor placed in the magnetic field and hence will experience a force on it.

In the figure given above It is shown that two parallel current carrying conductors placed near each other exert force on each other.

• When they have parallel currents, they attract each other.
• When they have antiparallel current, they repel each other.
• The magnitudes of forces F1 and F2 are equal but they are in opposite directions which is in accordance with Newton’s third law.

The force between two parallel current carrying conductors.

Consider two conductors having parallel currents  I1, I2  are placed at a distance ‘r’ from each other. Magnetic field of one conductor at the place of the second conductor is B1=μo* I1/(2Πr)

Force on 2nd conductor having length ‘L’  due to 1st conductor= F21F21=I2*L*(B1)

Force on 1st conductor due to 2nd conductor = F12

F12= - F21

Force per unit length  on each conductor F= μo*I1*I2/ (2Πr)

Similarly, force on two parallel conductors having antiparallel currents is shown below. Conductors having ant parallel currents repel each other.

Force on two straight wires having ant parallel currents.

Torque on a current loop in a uniform magnetic field.

The figure shows a rectangular loop carrying a steady current I and placed in the uniform magnetic field B. In this discussion, we will see that the current loop placed in uniform magnetic field experiences no net force but experiences a torque.

Initially when the rectangular loop is placed such that Uniform magnetic field B is in the plane of the loop. Then Side AD and BC would be parallel to magnetic field B

F= I ( l×B)= IlBsinθ

Therefore force on side AD and BC is zero.

F4=F3=IlBsin0=0

Force on AB and CD is maximum as these sides are perpendicular to magnetic field B and also equal in magnitude.

F1=F2= IlBsin90= IlB

F1 is directed into the plane of paper and F2 is directed out of the plane of the paper. Thus the net force on the loop is zero.

Thus the net force on the loop is zero.

There is a torque on the loop due to a pair of forces F1 and F2. The torque on the loop tends to rotate it.

Torque= Force * perpendicular distance

τ=F1*(d/2)+ F2*(d/2)

τ= (IlB)*(d/2) + (IlB)*(d/2)=I(ld)B= IAB

Where A= ld is the area of a rectangular loop.

When the plane of the loop makes an angle ‘θ’ with the uniform magnetic field B.

Then Force on Side BC and DA will be equal and opposite as their currents are in opposite directions. There is no net force and torque due to force on side BC and DA as these forces are collinear along the axis and hence cancel each other.

Force on arm AB and CD are F1 and F2, They too are equal and opposite in magnitude but they are not collinear and hence constitute a torque.

F1=F2=IlB

The torque on the loop, in this case, is however less than the torque when the loop was placed in the plane of the magnetic field B.

τ= F1(a/2)sinθ+ F2(a/2)sinθ=Il aBsinθ= IABsinθ

When angle ‘θ’ tends to zero, the perpendicular distance between them also approaches zero, Thus making the force collinear and the net force and net torque zero.

For N turns τ=NIABsinθ

Also, τ=mBsinθ= m×B

Where m= NIA magnetic moment of N turns coil.

Magnetic moment is a measure of an object's tendency to align with a magnetic field. It is a vector quantity.

m= current* Area= I A

The Magnetic dipole Moment of revolving electron

In Bohr’s Model of atom, electrons revolve in circular orbits around the positively charged nucleus under electrostatic force just the way planets revolve around the sun under gravitational force.

Thus the electrons of charge (-e) perform uniform circular motion around a stationary heavy nucleus (+Ze), This constitutes a current ‘I’.

The time period of the electron around circular orbit with uniform speed ‘v’ is given by

The circulating current ‘I’ is given by

Moving coil galvanometer

Moving coil galvanometer is an instrument used for deflection and measurement of small electric currents and voltages.

Principle: It is working is based on the fact that when a current-carrying coil is placed in a magnetic field. It experiences a torque.

• Moving coil galvanometer consists of a coil with many turns, free to rotate about a fixed axis in a uniform radial magnetic field.
• There is a cylinder with a soft iron core which not only makes the field radial but also increases the strength of the magnetic field.

• When current flows through the coil, torque NIBA acts on it. The magnetic torque tends to rotate the coil by angle Φ. The spring provides counter-torque kΦ that balances the magnetic torque NIAB, resulting in angular deflection Φ.
• In equilibrium   KΦ= NIBA, where K= torsional constant of spring ( restoring torque per unit twist)
• Deflection Φ= (NAB/k) *I, Thus deflection is directly proportional to the current in the coil so it is used to detect and measure currents.

• Deflection per unit current (Φ/ I)= NAB/k is called the sensitivity of the galvanometer.
• A convenient way to increase the sensitivity of the galvanometer is to increase the number of turns.

Galvanometer as an ammeter and Voltmeter

A Galvanometer can be used as an ammeter to measure the value of the current.

• For measuring currents, the galvanometer has to be connected in series but the resistance of the galvanometer is very high, this will change the current in the circuit.

• To overcome this situation a shunt resistance ‘S’ is attached parallel to the galvanometer coil to drastically reduce the resistance of the galvanometer.
• The scale of this ammeter is calibrated and graduated to read off the current values.

The galvanometer can be used as a voltmeter to measure the voltage across a given section of the circuit.

•  For this galvanometer must be connected parallel with the section of the circuit whose potential difference is to be measured.
• It must draw a very small current from the circuit, otherwise it will disturb the original voltage.

• To ensure that the voltmeter draws negligible current from the circuit, the resistance of the voltmeter must be very high. Therefore a very high resistance is connected in series with a galvanometer to make it work like a voltmeter.

## 2.Magnetisation and Magnetic Intensity

MAGNETISATION AND MAGNETIC INTENSITY

Mathematically,

Let us take a solenoid with n turns per unit length and the current passing through it be given by I, then the magnetic field in the interior of the solenoid can be given as,

Now, if we fill the interior with the solenoid with a material of non-zero magnetization, the field inside the solenoid must be greater than before. The net magnetic field B inside the solenoid

Where Bm gives the field contributed by the core material. Here, Bm is proportional to the magnetization of the material, M

Here, µ0 is the constant of permeability of a vacuum.

Let us now discuss another concept here, the magnetic intensity of a material. The magnetic intensity of a material can be given as,

From this equation, we see that the total magnetic field can also be defined as,

Here, the magnetic field due to the external factors such as the current in the solenoid is given as H and that due to the nature of the core is given by M.

Here, the term µr is termed as the relative magnetic permeability of a material, which is analogous to the dielectric constants in the case of electrostatics. We define the magnetic permeability as,

MAGNETIC PROPERTIES OF MATERIALS

Magnetic materials are classified into three categories, based on the behaviour of materials in the magnetic field. The three types of materials are diamagnetic, paramagnetic and ferromagnetic.

Intensity of magnetisation (I)

The electrons circulating around the nucleus have a magnetic moment. When the material is not magnetised the magnetic dipole moment sum up to zero.

Coercivity

The coercivity of a material is the ability to withstand the external magnetic field without becoming demagnetised.

Retentivity

The ability of a material to retain or resist magnetization is called retentivity.

Magnetic Field (H)

The magnetic field produced only by the electric current flowing in a solenoid is called the magnetic intensity.

Magnetic susceptibility

When a material is placed in an external magnetic field, the material gets magnetised. For a small magnetising field, the intensity of magnetisation (I) acquired by the material is directly proportional to the magnetic field (H).

I H

I = χmH , χis the susceptibility of the material

PERMANENT MAGNETS AND ELECTROMAGNETS

Permanent Magnet

1. Flexible Magnets: They are utilised in refrigerator door seals. Rubber polymers, plastics, and magnetic powders can all be used to create them.
2. NdFeB (Neodymium Iron Boron Magnet): These are rare earth magnets. It’s fairly simple to oxidise. It’s a high-priced substance. It’s frequently used in jewellery making, bookbinding, and other crafts.
3. A permanent magnet’s magnetic field can only be created below a particular temperature. As a result, these magnets aren’t suitable for use in hot-device applications.
4. Hard drives, motors, vehicles, generators, TVs, phones, headphones, speakers, transducers, and sensors all require permanent magnets. A magnet’s most common purpose is to attract other magnetic things, but it also serves a variety of tasks in electrical devices.
5. The majority of speakers use a permanent magnet that interacts with a wire coil (an electromagnet, really). The audio signal travels along the cable and causes the speaker to move. The speaker creates sound by moving air.

Electromagnet​​​​​​​

1. Resistant electromagnets: This sort of magnet uses copper wires to create a magnetic field. The magnetic field is created when the copper wire is twisted around a piece of iron and an electric current is sent through the copper wire. The stronger the field, the more copper wires are twisted.
2. Hybrid electromagnets: They are a mix of the two types of electromagnets mentioned above, resistive and superconductor electromagnets.
3. Electromagnets require a constant current source. Due to numerous variables such as ohmic heating, inductive voltage spikes, core losses, coil coupling, and so on, this may impact the magnets and their field at some point in the future.

## 2.Magnetisation and Magnetic Intensity

Magnetization and magnetic intensity.

Magnetization, also called magnetic polarization, is a vector quantity that gives the measure of the density of permanent or induced dipole moment in a given magnetic material.

We have seen that electrons in an atom have a magnetic moment. In bulk, these magnetic moments add up vectorially and they give a net magnetic moment.

We define magnetization M of a sample to be equal to its net magnetic moment per unit volume.

Consider a long  current-carrying solenoid, Magnetic field inside the solenoid is given by

When the interior is filled with magnetic material with non-zero magnetization. Then field inside the solenoid  B will be greater than B0

B = BBm, Where  Bm is a field contributed by the magnetic cores. Bm is proportional to the magnetization M.

Now introducing another vector field H called the magnetic intensity, which is defined by

Magnetization M  is proportional to magnetic intensity H.

Where χ is a dimensionless quantity and called magnetic susceptibility.

Magnetic materials

Materials that are attracted by the magnet are called magnetic materials. And materials that are not attracted to a magnet are called non-magnetic materials.

Classification of Magnetic materials

Magnetic materials are classified into the following types. In this chapter, we will discuss diamagnetic, paramagnetic and ferromagnetic materials in detail.

Diamagnetism

• Materials that have a tendency to move from the stronger to the weaker part of the external magnetic field.
•  These materials are weakly repelled by the magnetic field.

Cause of Diamagnetism

Electrons revolve around the nucleus in an orbit possessing orbital angular momentum. The orbiting electrons are equivalent to the current-carrying loop and thus possess a magnetic moment. In diamagnetic materials, the resultant magnetic moment of the atom is zero.

When the magnetic field is applied, those electrons having orbital magnetic moments in the same direction slow down and those in opposite directions speed up in accordance with Lenz’s law.

Thus the substance develops a net magnetic moment in the direction opposite to the applied field and is hence repelled by the field.

Paramagnetism

• Paramagnetic substances are those which get weakly magnetized when placed in an external magnetic field.
• They have a tendency to move from a region of the weaker magnetic field to a stronger magnetic field, these are weakly attracted by the external field.

Cause of paramagnetism

The individual atoms (ions or molecules) of a paramagnetic material possess magnetic moments of their own. In the absence of external B, these magnetic moments are oriented in a random direction and hence net magnetic moment is zero. When we apply an external magnetic field, the individual magnetic dipoles orient themselves in the direction of the magnetic field and hence get weakly magnetized

Ferromagnetism

• Ferromagnetic materials are those which get strongly magnetized when placed in an external magnetic field.
• Ferromagnetic materials have a strong tendency to move from a region of the weaker magnetic field to a stronger field when placed in an external magnetic field.
• These materials got strongly attracted by the external magnetic field.

Cause of ferromagnetism

In ferromagnetic materials, individual atoms are associated with large magnetic moments. The magnetic moments of the neighboring atoms interact with each other and align themselves spontaneously in a common direction over microscopic regions called domains.

In a ferromagnetic material in the unmagnetized state, atomic dipoles in small regions called domains are aligned in the same direction. The domain exhibits a net magnetic moment even in the absence of an external magnetizing field.

However, the magnetic moment of the neighboring domains are oriented in opposite directions and they cancel out and therefore the net magnetic moment of the material is zero.

On applying an external magnetic field these domains all align themselves in the direction of the applied field. In this way, the material is strongly magnetized in a direction parallel to the magnetizing field.

Hysteresis

When a ferromagnetic sample is placed in a magnetizing field, the sample gets magnetized by induction. As the magnetizing field intensity H carries, the magnetic induction B does not vary linearly with H.

The figure given below shows the variation of magnetic induction B with magnetizing field intensity H.

• The origin represents the initial unmagnetized state of a ferromagnetic sample. As the magnetizing field intensity H increases, the magnetic induction B first gradually increases and attains a constant value.
• Now if the magnetizing field intensity H is gradually decreased to zero, B decreases but along a new path AB. The sample is not demagnetized even when the magnetizing field has been removed.
• The magnetic induction left behind in the sample after the magnetizing field has been removed is called residual magnetism or retentivity.
• To reduce the magnetism to zero, the field H id gradually increases in the reverse direction, the induction B decreases and becomes zero at a value of H= OC.
• The value of reverse magnetizing field intensity H required for the residual magnetism of a sample to become zero is called coercivity of the sample.

A study of the hysteresis loop provides us with information about retentivity, coercivity and hysteresis loss of magnetic. This helps in the proper selection of material for designing cores of transformers and electromagnets and making permanent magnets.

Soft ferromagnetic materials

These are the ferromagnetic materials in which the magnetization disappears on the removal of the external magnetization field. Such materials have narrow hysteresis loops.

Example: soft iron.

Hard ferromagnetic material

These are the ferromagnetic materials that retain magnetization even after the removal of the external magnetizing field.

Example: Steel, lodestone etc.

Permanent magnets and electromagnets

Permanent magnets

Substances which at room temperature retain their ferromagnetic property for a long period of time are called permanent magnets.

Methods of making permanent magnet :

• One can hold an iron rod in a north-south direction and hammer it repeatedly.
• One can also hold a steel rod and stroke it with one end of a bar magnet a large number of times , always in the same sense to make permanent magnets.
• An efficient way to make a permanent magnet is to place a ferromagnetic rod in a solenoid and pass a current. The magnetic field of the solenoid magnetizes the rod.

Desired property of a material to make permanent magnets

Hysteresis curve allows us to select suitable material for the permanent magnet.

• Material should have high retentivity so that the magnet is strong.
• Material should have high coercivity so that the magnetization is not erased by any stray magnetic field, temperature fluctuations etc.
• Steel is the favorable choice of material for a permanent magnet, Other suitable materials for permanent magnet are alnico, cobalt steel and ticonal.

Electromagnets

Electromagnets are the type of magnets where a magnetic field is produced by electric current. It is made up of a coil of wire which acts as a magnet when an electric current passes through it.

• An electromagnet only displays magnetic properties when an electric current is applied to it.
• The strength of an electromagnet can be adjusted by the amount of electric current allowed to flow into it.
• Cores of electromagnets are made of ferromagnetic material which have high permeability and low retentivity. Soft iron is a suitable material for electromagnets.

Application of the electromagnets

• It is used in generators and motors, which are necessary for the conversion of mechanical energy into electrical energy.
• Electric buzzers and bells.
• Headphones and loudspeakers use electromagnets.
• Data storage devices like VCR’s, tape recorders, hard discs etc.
• MRI machines also used electromagnets.

## 1.Electromagnetic Induction

Chapter 6: Electromagnetic Induction

Electromagnetic Induction
Magnetic flux
Magnetic flux is a measurement of the total magnetic field which passes through a given area. It is a useful tool for helping describe the effects of the magnetic force on something occupying a given area.
If we choose a simple flat surface with area A as our test area and there is an angle θ  between the normal to the surface and a magnetic field vector (magnitude B) then the magnetic flux is
Φ=BAcosθ

## 1.Electromagnetic Induction

Introduction

We have studied electrostatics and magnetostatics in previous chapters as two different phenomena. But later on, it was discovered that they are not two different phenomena. These are like two faces of the same coin. They are not independent but linked to each other under a wider category of interaction which is called electromagnetic interactions. Electromagnetic interactions are one of the four fundamental interactions of the universe.

In this unit, we will see that changing magnetic fields also produce emf and hence electric current.This phenomenon is called electromagnetic induction. We will discuss such phenomena in detail in this chapter.

Experiment of Faraday and Henry

Discovery of electromagnetic induction is based on a series of experiments carried out by Faraday and Henry. The theoretical explanation of the experiment was later given by Faraday and Lenz which are known as the laws of electromagnetic induction.

Note for the studentIn the following experiments along with the observation, I have given the explanation of the observation too in blue color and used the word magnetic flux many times in that explanation which I have not discussed before. Students can read about the magnetic flux in the latter topic.

Experiment 1: A coil and a magnet

Consider a loop of wire, connected with a galvanometer and a bar magnet as shown in the figure. We will keep the wire loop fixed in place and the bar magnet can be moved back and forth as shown in the figure.

• Initially, when the magnet is far from the coil, there is no deflection in the coil, This is because no magnetic flux will be linked with the coil and hence the deflection of the galvanometer is zero.
• As the coil is moved near the coil, magnetic field lines will be linked with the coil and these field lines will change as the coil continues to move and the galvanometer will show deflection.

• Suppose as in figure ‘b’, a magnet is placed inside the coil and it is kept stationary, there will be no deflection in the galvanometer.

This happens because the magnetic flux linked with the coil is constant and does not change with time

Again like in figure ‘c’, if we now try to move the magnet away from the coil, we will again see that the galvanometer shows deflection but now in the opposite direction. Here magnetic flux

• linked with the coil is also changing, it is decreasing as the magnet is moving away from it.
• Now like in the figure ‘d’, if we keep the magnet at rest outside the coil, the galvanometer will show no deflection

This happens as the magnetic flux linked with the coil is constant when there is no relative motion between the coil and the magnet

• If we now keep the magnet stationary and move the coil near the magnet, we will again observe that the galvanometer will show deflection like before.
• Deflection was found larger when we move the magnet faster near the coil keeping the coil fixed or moving the coil near the fixed magnet.

Conclusion:  The results from the above experiment show that It is the relative motion between the magnet and the coil which is responsible for the generation ( induction ) of electric current in the coil.

Experiment 2: One coil with a galvanometer and another coil connected with a battery.

We have two coils C1 and C2. Coil C1 is connected with a galvanometer and coil C2 is connected with a battery source that produces a constant current.

• Keep C1 fixed and move C2 toward C1, the galvanometer will show deflection. Now if we move coil C2 away from coil C1, again the deflection will produce in the galvanometer. Hence there will be an induced current in coil C1.
• Keep C2 fixed and move C1 toward C2, the galvanometer will show deflection. Now if we move coil C1 away from coil C2, again the deflection will produce in the coil. Hence there will be an induced current in coil C1.

Conclusion: Again it is the relative motion between the coils that induce the electric current

Experiment 3: Both coils are kept stationary

In this experiment, we have two coils C1 and C2. Coil C1 is connected with the galvanometer and C2 is connected with a battery and key k.

• When the key of the circuit is closed, there is a momentary deflection in the galvanometer

Explanation is that when the key is open, there would be no current in C2 and hence the magnetic field would also be zero. So there will be no magnetic flux linked with the coil, but when the switch is closed, there will be a sudden increase in current in Coil C2 and magnetic flux linked with C1 suddenly changes so there is momentarily deflection in the coil.

• There is no deflection in the galvanometer after the key of C2 is closed.

Explanation is that when there is a steady current in C2 and hence magnetic field produced by the C2 is also constant. So the magnetic flux linked with coil C1 will also remain the same and hence no current will be produced in C1 and hence the galvanometer shows zero deflection.

• When the key is opened again, there will be again momentarily deflection in the coil.

Explanation is that when the key is opened, the current in Coil C2 changes from a steady value to zero and hence produces a changing magnetic field. The magnetic flux linked with the coil also changes from a finite value to zero. And hence there is a momentarily induced current in the coil C1 and hence the galvanometer shows momentarily deflection.

Magnetic flux

There was a need for a mathematical tool to explain the series of experiments carried out by Faraday on electromagnetic induction.

So he defined the notion of magnetic flux ‘ ϕB’ in the same way as electric flux. It is a scalar quantity.

The magnetic flux through any surface placed in a magnetic field is the total number of magnetic lines of force crossing this surface normally.

ϕB= BA= BA cosθ

Magnetic flux would be maximum when cosθ= 1, θ=0

Magnetic flux will be zero when cosθ=0, θ =90.

Dimension of ϕB=[M1L2A-1T-2]

In S.I. The unit of magnetic flux is Weber.

One weber is the flux when a uniform magnetic field of one tesla acts normally over an area of 1m2.

S.I. unit is Weber and in C.G.S. The unit is maxwell.

1 Wb= 108 maxwell

How can magnetic flux links be changed?

Mathematically magnetic flux is equal to ϕB= BA= BA cosθ

So there are following ways to change the magnetic flux

•  By changing the magnetic field
• By changing the area of the loop or closed circuit
• By changing the orientation of the closed circuit with respect to Magnetic flux.

We can either do one of the above-mentioned ways to change the magnetic flux or we can do more than one change simultaneously.

The Faraday Law of Induction

In the experiment discussed above, we have following observation that

• When there is a relative motion between the coil and the source of the magnetic field, an electric current is induced in the coil.
• The common point in all observations is that more deflection is produced when the magnet is moved rapidly near the coil.

Now we will use the concept of magnetic flux to explain how this happens.

Faraday explained that whenever there is relative motion between the coil and the magnet or a source of magnetic field ( current-carrying coil), the magnetic flux linked with the coil changes and thus an emf is induced in the coil and hence there is an induced electric current in the coil. He also noticed that in every experiment,  emf induced is directly proportional to the rate of change of magnetic flux in the coil.

Faraday’s Law of Electromagnetic induction:

1. Whenever the magnetic flux linked with a closed circuit changes, an emf (and hence a current) is induced in it which lasts only so long as the change in flux is taking place. This phenomenon is called electromagnetic induction.
2.  The magnitude of the induced emf is equal to the rate of change of magnetic flux linked with the closed circuit.

Mathematically,

For the N turn of the coil Emf= N (Δϕ/Δt)

Thus induced current emf can be increased by increasing the number of turns N of a closed coil.

## 1. AC Voltage to resistor and circuit

Chapter 7: Alternating Current

AC VOLTAGE APPLIED TO A RESISTOR

​​Alternating Currents are used almost as a standard by electricity distribution companies. In India, 50 Hz Alternating Current is used for domestic and industrial power supply. Many of our devices are in fact nothing but resistances. These resistances cause some voltage drop but since the voltage this time is alternating, these voltage drops are dealt with differently

AC VOLTAGE APPLIED TO AN INDUCTOR

AC voltage is applied to the inductor. In order to find out the equation, we will consider the circuit as shown in the figure below. we have an inductor and an AC voltage V, which is represented by the symbol ~. The voltage produces a potential difference across its terminals that varies using a sinusoidal equation. The difference that is, the AC voltage thus can be given as,

From the equation, we deduce that vis used to signify the amplitude for the oscillating potential to denotes the differences. The angular frequency is given by ω. The current can be calculated by using the Kirchhoff’s loop rule. The equation which forms is as under,

Using the above equation in the given circuit,

The value of current as given by,

Therefore, the integration constant is zero.

Here, the amplitude of the current is given by

The quantity ωL can be said to be equivalent to the resistance of this device and is termed as the inductive resistance. We denote the inductive resistance of the device as XL.

Thus, we can say that the amplitude of current in this circuit is given as

AC Voltage Applied to a Capacitor
An AC voltage source is connected to a capacitor. The expression for the voltage from the voltage source is given by v = vmsin(ωt). A capacitor is an electrical device that stores electrical energy. It is a passive electronic component with two terminals. The effect of the capacitor is known as capacitance. A capacitor when connected to a voltage source draws current from the source so as to charge itself. Once the capacitor is charged, the potential at its plates becomes equal to the potential at the battery. At this point, the current stops flowing into the capacitor. This is called the charging of the capacitor.
At a particular time “t”, denotes the charge on the capacitor by “q”. The instantaneous voltage across the capacitor is given by,

Using the kirchhoff’s rule,

Since the current is continuously changing, to find the current. Derivative of the charge is required,

Differentiating the given equation,

i = vmωC cos(ωt)

Rearranging the above equation,

i = imsin(ωt + π/2)

Here, im = vmωC. It is the amplitude of the oscillating current. It can also be re-written as,

This equation when compared to the ohm’s law gives 1/ωC as resistance. It is called capacitive reactance and it is denoted by XC

Now, the amplitude of the current becomes,

im =

AC VOLTAGE APPLIED TO A SERIES LCR CIRCUIT
we will follow the analytical analysis of the circuit.
Analytical solution
As i

, we can write

Hence, writing the voltage equation in terms of the charge q through the circuit, we can write,

The above equation can be considered analogous to the equation of a forced, damped oscillator. In order to solve the equation, we assume a solution given by,

So,

And

Substituting these values in the voltage equation, we can write,

Here, we have substituted the value of Xc and XL by Xc = 1/ωC and XL = ω L.

As we know,

hence, substituting this value in the above equation, we get,

Now, let

So we can say,

Now, comparing the two sides of the equation, we can write,

And,

Hence, the equation for current in the circuit can be given as,

LC OSCILLATION
The Difference between the Direct and Alternating current is that the direct current (DC), travels only in one direction while the alternating current (AC) is an electric current that alternates direction on occasion and alters its amplitude continuously over time.In LC Oscillations, the charge equation is as follows:
q = qm cos(ωt)
We derive the present equation by differentiating this equation:
i = dq/dt
i = –qm ωsin(ωt)
The formula for calculating the energy stored in a capacitor is:
U= q2/2C
Substituting the equation for a given time interval t;
U= qm2/2C × (ωt)
The formula for calculating the energy stored in an inductor is:
U= 1/2 L i2
Substituting the capacitor’s equation for the same amount of time;
U= 1/2 L qmω(ωt)
Since the angular frequency, ω=1/ √LC
U= qm2/2C × (ωt)
As a result, the LC Oscillations’ total energy will be;
U = U+ UC
U = qm2/2C × (ωt)+qm2/2C × (ωt)
U = qm/ 2C

Transformer
There are usually two coils primary coil and secondary coil on the transformer core. The core laminations are joined in the form of strips. The two coils have high mutual inductance. When an alternating current pass through the primary coil it creates a varying magnetic flux. As per faraday’s law of electromagnetic induction, this change in magnetic flux induces an emf (electromotive force) in the secondary coil which is linked to the core having a primary coil. This is mutual induction.

Overall, a transformer carries the below operations:

1. Transfer of electrical energy from circuit to another
2. Transfer of electrical power through electromagnetic induction
3. Electric power transfer without any change in frequency
4. Two circuits are linked with mutual induction

1. Core

The core acts as a support to the winding in the transformer. It also provides a low reluctance path to the flow of magnetic flux. The winding is wound on the core as shown in the picture. It is made up of a laminated soft iron core in order to reduce the losses in a transformer. The factors such as operating voltage, current, power etc decide core composition. The core diameter is directly proportional to copper losses and inversely proportional to iron losses.

2. Windings

Windings are the set of copper wires wound over the transformer core. Copper wires are used due to:

• The high conductivity of copper minimizes the loss in a transformer because when the conductivity increases, resistance to current flow decreases.
• The high ductility of copper is the property of metals that allows it to be made into very thin wires.

There are mainly two types of windings. Primary windings and secondary windings.

• Primary winding: The set of turns of windings to which supply current is fed.
• Secondary winding: The set of turns of winding from which output is taken.

The primary and secondary windings are insulated from each other using insulation coating agents.

3. Insulation Agents

Insulation is necessary for transformers to separate windings from each other and to avoid short circuit. This facilitates mutual induction. Insulation agents have an influence on the durability and the stability of a transformer.

Following are used as an insulation medium in a transformer:

• Insulating oil
• Insulating tape
• Insulating paper
• Wood-based lamination

Ideal Transformer

The ideal transformer has no losses. There is no magnetic leakage flux, ohmic resistance in its windings and no iron loss in the core.

Vs = (Ns/ Np) Vp

Power at the input end is same as the power at the output end.

Therefore Pintput = Poutput

IpVp =  IsVs

## 1. AC Voltage to resistor and circuit

Introduction:

In the discussion so far we have used direct current (DC), the magnitude and the direction of DC remain the same and do not change with time. In this chapter, we will discuss the Alternating current (AC). In Alternating currents and voltages the magnitude as well as direction change periodically with time.

Most of our electrical appliances today work on AC because electrical energy produced by power companies is transmitted and distributed as Alternating current. This is because electrical energy can be transmitted more economically over long distances in the form of AC. And It can be easily and efficiently converted from one voltage to another by the means of transformers.

Alternating current

An alternating current is that current whose magnitude changes continuously with time and direction reverses periodically.

It is in contrast to a direct current whose magnitude and direction do not change with time.

• Alternating emf is given by E= Eosin(wt)

By ohm’s law  current I= E/R  = Eosin(wt)/ R= (Eo/R)*sin(wt)

Therefore alternating current of resistance is  I= Iosinwt

Where Io=Eo/R   which is called peak current or amplitude of AC.

• Time period of AC is the time taken by the alternating current to complete one cycle of its variation is called time period

T = 2Π/w

• Frequency of AC current is the number of complete cycles per second.

f=w/(2Π)

• The average value of AC over one complete cycle is zero. Therefore we defined the average value of a.c over half a cycle.

Average Value of AC

It is defined as the value of a direct current which sends the same charge in a circuit in the same time as it is sent by the given alternating current in its half-time period. It is denoted by Iav

Iav= 2Io/Π= 0.637 Io

Similarly, average value of Alternating EMF is given by

Eav=(2/Π)*Eo= 0.637 Eo

Root mean square(RMS) value of AC

It is defined as that value of direct current that produces the same heating effect in a given resistor as is produced by the given alternating current when passed for the same time. It is denoted by Irms.

Irms =(Io/2 ) =0.707*Io

Similarly, the root mean square value of Alternating emf is given by

Erms =Eo/2= 0.707 *Io

AC voltage applied to a Resistor

The figure shows a resistor connected to source E of ac voltage. We consider a source that produces sinusoidally varying potential differences across its terminal.

The potential difference is called ac voltage is given by

V=Vm( sinwt) Where Vm is the amplitude of oscillating potential difference and ‘w’ is its angular frequency.

By kirchoff’s voltage rule   Vm sin wt=IR

• So current in the circuit I=(Vm sin wt)/R=Vm/R sin wt=Im sin wt

Where Im= Vm/R  = peak current or current amplitude.

• Ohm’s law works equally well for both ac and dc voltages
• We can conclude that V and I both reach zero, minimum and maximum at the same time. Clearly voltage and current are in phase with each other in a resistor connected with ac source.

Representation of AC current and voltage by rotating vector- Phasors

We have seen that current through a resistor is the phase with the ac voltage. But this is not in the case of the inductor and capacitor or a combination of these circuits.

In order to show the phase relationship between V and I in various ac circuits, we use the representation of the rotating vectors called Phasors.

A phasor is a vector that rotates about the origin with angular speed ‘w’

Both cosine and sine functions can be represented by the phasor diagram

AC voltage applied to Inductor

Consider a circuit having a pure inductor of inductance L  connected with an alternating voltage source.

We have assumed that the inductor has zero resistance.

When an AC source that produces variable current and hence varying magnetic field, is connected with the inductor. Then according to the principle of electromagnetic induction, a back emf is produced in the coil.

When we apply kirchhoff's Voltage law in the circuit we will get

Applied emf  (e)+ back emf (-e’) =0

e+ (-e')=0

From the expression of applied alternating voltage and alternating current, it is clear that current lags behind the voltage by 90 degrees in the purely inductive circuit.

AC circuit having capacitor only

Consider a circuit having a capacitor of capacitance C connected with an alternating voltage source.

The alternating voltage is given by V=Vo sin(wt)

Consider ‘q’ to be the instantaneous charge in the circuit.

The emf across the capacitor will be  V= q/C then according to kirchhoff’s voltage law we have      V-qC= Vo sin(wt)-q/c=0

So we can write the q=CVo sin(wt)

Now we know that current i=dq/dt

Where   io= current amplitude in the capacitive circuit

Xc=(1/wcis the called capacitive reactance

Capacitive reactance is the resistance offered by the capacitor to the flow of alternating current in the circuit

From the expressions of applied alternating voltage and alternating current, we can conclude that current leads the voltage by 90 degrees in the pure capacitive circuit.

Below is the phasor diagram and the wave diagram for a purely capacitive circuit.

Average power for purely capacitive circuit

because  < sin 2wt> =0, an average value of sine over a complete cycle is zero. Hence capacitors do not consume any power from the circuit.

Now before moving to the Series LCR circuit Let’s first recap all we have learned about purely resistive, inductive and capacitive circuits.

Purely resistive circuit

Purely Inductive circuit

Purely capacitive circuit

AC Voltage is applied to a series LCR circuit.

Consider a circuit having a resistor of resistance R, a capacitor of capacitance C and an inductor of inductance L connected in series with an alternating voltage source.

The voltage of the alternating source V=Vo sin(wt)

If ‘q’ is the charge on the capacitor and ‘I’ be the current in the circuit at time ‘t’, then from Kirchoff’s law we have

L di/dt  + IR +  q/C= Vo sin(wt)

When current passes through the series LCR circuit, the voltage of the resistor VR is in phase with the current, while it is out of phase in the case of the inductor and capacitor as shown in the diagram.

• VR is in phase with the current I in the circuit.
• VL leads the current I by 90 degrees.
• VC lags the current I by 90 degrees.

We can represent the voltages across R, C and L and current with the help of phasor diagrams.

The resultant voltage of R, C and L will give the applied Alternating current

when VL > VC

Alternating current in series LCR circuit I=Io sin(wt-ϕ),

Where Io=Vo/Z   is the amplitude of the current in the series LCR circuit.

And ϕ = phase difference between the applied alternating voltage V and alternating current ‘I’ in series LCR circuit and when VL > VC is given by -

In the figure given above Current ‘I’ and alternating applied voltage V along with voltage across R, C and L are shown

When  Vc > VL  then I= Io sin(wt+ϕ) and ϕ=tan^(-1) ((Xc-XL)/R)

Resonance

In series, LCR resonance occurs when the magnitude of inductive reactance and capacitive reactance becomes equal and they are 180 degrees out of phase with each other and hence cancel each other.

Xc = XL   ;    1/wc= wl  ;   w^2=1/(LC)

• At the resonance impedance of the series, LCR is minimum.

Z= R    as  (Xc= XL)

• Current in the circuit is maximum as impedance is minimum

I= E/R

• Series LCR circuits behave like a resistive circuit at resonance as capacitive reactance and inductive reactance cancel each other.
• Current and voltage are in phase with each other in a series LCR circuit at resonance

• Resonant angular frequency is inversely proportional to square root of product of inductance and capacitance.​​​​​​

Wo =  1/LC

• Frequency of oscillation at resonance fr is given by

fr=1/(2ΠLC)

Quality factor: Quality factor is the ratio of the voltage across the capacitor or the inductor at resonance to the applied voltage. It is dimensionless and has no units.

Bandwidth of the

• It signifies the sharpness of the peak at resonance. More the value of Q will be sharper will be the resonance curve.​​​​​​​

• As Quality factor Q is inversely proportional to R, smaller value of R means the greater value of Q and hence the resonance curve will be sharper

• Bandwidth( BW) of series LCR circuits in resonance: The bandwidth of a resonant circuit is defined as the total number of cycles below and above the resonant frequency for which the current is equal to or greater than 70.7% of its resonant value.

• Larger be the Q value of the series LCR circuit, smaller be the bandwidth of the circuit.

Power in AC circuit

Power factor: The power factor is an indicator of efficient utilization of power. In an AC circuit, PF is defined as the ratio of real power flowing to the load to the apparent power in the circuit and is a dimensionless number.

Power in series LCR circuit

In pure inductive circuit and pure capacitive circuit  ϕ=90

So , Pav= Vrms*Irms*cos 90= 0

Therefore current in the inductive and capacitive circuit is called wattless current as there is no power loss across the inductor and capacitor.

LC oscillation

In this discussion we have a capacitor of capacitance C which is fully charged, then we connect an inductor of inductance L in the circuit as shown.

In this circuit, there will be oscillation between charge and current. We can visualize this as an oscillation of electrical energy and magnetic energy in the circuit such that total energy of the circuit remains the same.

• Suppose initially at t=0, the capacitor is fully charged with charge Qand current in the circuit i=0, so all the energy is of electrical nature.   q(t=0)=Q0    i(t=0)=0
• At t>0, the capacitor begins to discharge and current starts rising in the circuit, therefore electrical energy of the capacitor decreases and magnetic energy of the inductor increases by the same amount. Energy is transferred from capacitor to inductor.
• At t= T/4, the capacitor is completely discharged so electrical energy of the capacitor becomes zero and magnetic energy of the inductor becomes maximum and current flows in anticlockwise direction as shown in the figure below.
• When the capacitor is totally discharged and current is now beginning to fall and energy is being transferred from inductor to capacitor and the capacitor is being charged but now in the opposite direction.
• At time t= T/2, the capacitor is now fully charged and current in the circuit is zero.  All of the energy is in electrical form and magnetic energy is zero as the capacitor is fully charged again and current in the circuit is zero.
• After this instant, charge in the capacitor starts to fall and current is now in the clockwise direction as shown. Energy is being transferred from capacitor to inductor. Electrical energy of the capacitor starts to decrease and the magnetic energy of the inductor does increase.
•  At t=3T/4, again the capacitor is now discharged and the current in the circuit is maximum. Electrical energy of the capacitor is zero and the magnetic energy is maximum.​​​​​​​

Transformer: A transformer is a device that changes the voltage of the AC source. A transformer consists of two coils, primary and secondary. Input alternating voltage is given to the primary circuit. The number of turns and the voltage and current in primary and secondary coils is given below. Here we have considered an ideal transformer in which there is no energy loss.

Vp/Vs=Np/Ns=Is/Ip

There are two types of transformers: Step down and step-up transformers.

In a step-down transformer, the number of turns in the primary is more than the number of turns (Np> Ns) in the secondary coil. So

Here  Np > Ns      so , Vs < Vp and Is > Ip.

In the step-up transformer, the number of turns in the secondary is more than the number of turns in the primary.

Here  Ns > Np   so , Vs > Vp and Ip > Is.

Working principle of transformer.

When an alternating voltage is applied to the primary, the resulting current produces an alternating magnetic flux which links the secondary and induces an emf in it. The value of emf depends on the number of turns in the secondary.

In the Ideal transformer we have assumed that there is no energy loss in the process. But In actual transformers, there is small energy loss due to the following reasons.

1. Flux leakage: There is always some flux leakage. Not all the flux due to primary passes through the secondary due to poor design of the core or air gaps between the windings.
2. Resistance of the windings: The wires used for winding also have some resistance and some energy is lost in heating of the wire.
3. Eddy currents:  Alternating magnetic flux induces eddy current in the core and causes heating.
4. Hysteresis: the magnetization of the core is repeatedly reversed by the alternating magnetic field so there is some hysteresis loss also.

A fun thing to try

AC simulation

Click on the link above and you could actually make your own AC circuit using the desired value of R, C and L with any combination. We can see the direction of electron current and conventional current. Also, we can use an ammeter and voltmeter to track the values of the voltages and current. Make as many circuits as you like and learn from them.

Happy learning !!

## 1. Electromagnetic Waves

Chapter 8: Electromagnetic Waves

Electromagnetic Waves
DISPLACEMENT CURRENT
Displacement current is a quantity appearing in Maxwell’s equations. Displacement current definition is defined in terms of the rate of change of the electric displacement field (D).Apart from conduction current, there is another type of current called displacement current. It does not appear from the real movement of electric charge as is the case for conduction current.

ELECTROMAGNETIC WAVES

• The Magnetic field is produced by a moving charged particle. A force is exerted by this magnetic field on other moving particles. The force on these charges is always perpendicular to the direction of their velocity and therefore only changes the direction of the velocity, not the speed.
• So, the electromagnetic field is produced by an accelerating charged particle. Electromagnetic waves are nothing but electric and magnetic fields travelling through free space with the speed of light c. An accelerating charged particle is when the charged particle oscillates about an equilibrium position. If the frequency of oscillation of the charged particle is f, then it produces an electromagnetic wave with frequency f. The wavelength λ of this wave is given by λ = c/f.  Electromagnetic waves transfer energy through space.
• Electromagnetic wave equation describes the propagation of electromagnetic waves in a vacuum or through a medium.

Maxwell’s Equations of Electromagnetic Waves Maxwell’s equations are the basic laws of electricity and magnetism. These equations give complete description of ail electromagnetic interactions.
There are four Maxwell’s equations which are explained below:

Electromagnetic Spectrum The systematic sequential distribution of electromagnetic waves in ascending or descending order of frequency or wavelength is known as electromagnetic spectrum. The range varies from 10-12 m, to 104 m, i.e. from γ-rays to radio waves.

13. Elementary facts about the uses of electromagnetic waves
(i) In radio and TV communication.
(ii) In astronomical field.
Microwaves
(i) In RADAR communication.
(ii) In analysis of molecular and atomic structure.
(iii) For cooking purpose.
Infrared waves
(i) In knowing molecular structure. (ii) In remote control of TV VCR, etc.
Ultraviolet rays
(i) Used in burglar alarm. (ii ) To kill germs in minerals.
X-rays
(i) In medical diagnosis as they pass through the muscles not through the bones.
(ii) In detecting faults, cracks, etc., in metal products,
γ-rays
(i) As food preservation. (ii) In radiotherapy.

## 1. Electromagnetic Waves

Introduction

We have studied magnetostatics in the 4th chapter where we have seen the magnetic effect of current. Moving charges or current produces magnetic fields. Later in the 6th chapter, we have seen that changing magnetic fields produce current and this phenomenon is called electromagnetic induction.

What do you think the converse of electromagnetic induction exists?

Can changing electric fields produce magnetic fields?

The answer is Yes!

Maxwell proposed the converse: a changing electric field has a magnetic field associated with it. Maxwell also argued that not only electric current but also a time-varying Electric field generated magnetic field

While applying Ampere’s circuital law to find the magnetic field at a point outside a capacitor connected with a time-varying current. He noticed some inconsistencies with Ampere’s circuital law and suggested the existence of an additional current in addition to the conventional current. He called that additional current the displacement current Id to remove the inconsistency in the Ampere’s Law.

Maxwell formulated a set of four equations involving electric and magnetic fields and their sources, the charges and current sensitises. These equations are called Maxwell’s equations.The most important prediction of Maxwell’s equation is the existence of electromagnetic waves. We will discuss displacement currents and electromagnetic waves in this chapter. Below is the list of four maxwell equations.

Displacement current

I would like you to think something before actually jumping on the discussion of displacement current.

Imagine a  parallel plate capacitor that is charging through a dc battery. If there is nothing but a vacuum between the parallel plates of the capacitor then why doesn't it behave like an open circuit ?. How does the flow of charges through this vacuum happen?

As for conducting current to flow through a wire we need the circuit to be closed.  Just hold on for a while and you will get answers to all the questions above.

Let me first introduce displacement current.

Displacement current is that current that comes into existence in addition to the conduction current, whenever the electric field and hence the electric flux changes with time

Now let's try to understand what happens during the charging/discharging of capacitors.

When the capacitor is charging the charge density on the plates of the capacitor increases with time. The electric field between plates due to charges on the plates also increases with time. So due to this changing electric field between the plates of the capacitor, the displacement current comes into existence between the plates and makes the current continuous.

The displacement current between the plates is equal to the conduction current in the circuit.

When the capacitor is fully charged, the charge density on the plates of the capacitor is constant and therefore the electric field between the plates is also constant. So the displacement current is zero.

Conduction current is also zero when the capacitor is fully charged.

That is why the capacitor blocks DC when it is fully charged.

Note: Displacement current is not due to flow of charges but due to change in electric flux.

Electromagnetic waves

From our previous knowledge, we know that Charges at rest produce electric fields and Charges in motion produce electric current and electric current then produces magnetic fields. Changing magnetic fields produce electric current and this phenomenon is electromagnetic induction. So what would be the source of electromagnetic waves?

By Maxwell's theory, it was suggested that accelerated charges radiate electromagnetic waves.

In 1864, British physicist James Maxwell made the remarkable suggestion that the accelerated electric charges generate linked electric and magnetic disturbance that can travel indefinitely through space. If the charged particle oscillates periodically, the disturbance is waves whose electric and magnetic components are perpendicular to each other and also perpendicular to the direction of propagation.

Consider a charge oscillating with some frequency. An oscillating charge is an example of accelerating charge. This produces , changing electric field in space and this produces an oscillating magnetic field which in turn produces an oscillating electric field and thus oscillating magnetic field. We can conclude that oscillating electric and magnetic fields thus regenerate each other as the waves propagate through space and these waves are called electromagnetic waves.

Characteristic of electromagnetic waves:

Amplitude: The amplitude of an electromagnetic wave determines the maximum intensity of its field quantities. It is the maximum disturbance from the mean position of the waves.

Wavelength: The distance traveled by the wave in one complete cycle. It is the distance between two consecutive crests and troughs. Then S.I.  unit of wavelength is the meter.

Time periodThe time taken by the wave in one complete oscillation.Then S.I.  unit of time period is seconds

FrequencyIt is the complete oscillation of the waves in 1 second.Then S.I.  unit of  frequency is hertz.

Nature of electromagnetic waves:

• Electromagnetic waves do not require any medium to travel through space. It can travel in vacuum also.
• The speed of the electromagnetic waves in vacuum is 3*108 m/s and is an important fundamental constant ‘C’.
• The Electric field , magnetic field and the direction of propagation is in mutually perpendicular directions like shown below.

• The speed of the electromagnetic waves changes in different mediums. It depends on electric and magnetic properties of the medium.
•  Speed or electromagnetic waves are maximum in vacuum. Speed of electromagnetic waves changes when it travel from one medium to another
• The frequency of the electromagnetic wave remains the same in all mediums.
• EM waves show phenomena like dispersion, reflection, refraction and polarization just like light waves do. Light is an electromagnetic wave.
• EM waves carry energy and momentum with it, since it carries momentum, an electromagnetic wave also exerts pressure called radiation pressure.

Radiation pressure P= U ( energy density of EM wave)/ C. When the sun shines on our hand , we feel energy being absorbed as warmth from the electromagnetic wave. Do we feel pressure too? No ! We do not feel the pressure but it is present.

EM waves also transfer momentum to our hands but since ‘c= 3*108 m/s ’ and is very large so the momentum transfer is so small that we cannot feel it.

The great technological importance of electromagnetic waves stems from their capability to carry energy from one place to another. The radio signal and TV signals from broadcasting stations carry energy.  Light carries energy from the sun to the earth and thus makes life possible on the earth.

Electromagnetic spectrum

The orderly distribution of electromagnetic radiation according to their wavelength or frequency is known as the electromagnetic spectrum.

The main parts of EM waves are γ-rays , X-rays , ultraviolet rays, visible light, Infrared , microwave and radio waves

The various regions of the EM spectrum do not have sharply defined boundaries and they overlap. The classification is based roughly on how they are detected or produced.

Radiowave : These are the electromagnetic waves of longest wavelength and minimum frequency. It was discovered  by Marconi.

Source of radio wave of accelerated motion of charges in conducting wires of oscillating circuits.

Wavelength range : 600m and 0.1 m

Frequency range : 500 KHz to 1000 MHz

Use of radio wave :

1. In radiowave and television communication systems.
2. In radio astronomy.

Microwave : They are the electromagnetic waves having wavelength next smaller to radio waves. Source of the microwave is oscillating currents in special vacuum tubes like klystrons, magnetrons and Gunn diodes.

It was discovered by Marconi.

Wavelength range : 0.3 m to 10-3 m

Frequency range: 109 Hz to 1012 Hz.

Uses of microwave

• Mobile uses a microwave for communication.

• Wifi connection uses Microwave

• Microwaves are used in microwave ovens for cooking purposes.

Infrared Waves :

These radiations lie close to the low-frequency or long wavelength of the visible spectrum. Infrared waves produce heating effects, so they are also known as heat waves or thermal radiation.

Source of infrared is hot bodies.

Wavelength range- 5×10-3 m to 10-6 m.

Frequency range : 1011 Hz to 5×1014 Hz.

Uses of infrared:

Visible Light :

It is a very small part of the electromagnetic waves spectrum towards which the human retina is sensitive. The visible light emitted or reflected from bodies around us gives information about the world.

Source of visible gas discharge tubes, arcs of iron and mercury and the sun.

It was discovered by Ritter in 1800.

Wavelength Range : 3.5×10-7m to 1.5×10-7

Frequency range : 1016 Hz to 1017 Hz.

Uses of visible light

• It provides us with information about the world around us.
• It can cause chemical reactions . Example: Photosynthesis.
• Some man made applications are light bulb and lasers which have numerous applications​​​​​​​

Ultraviolet light

The region of the electromagnetic spectrum has a shorter wavelength than visible light and can be detected beyond the violet end of the solar spectrum. Source of visible gas discharge tubes, arcs of iron and mercury and the sun.

It was discovered by Ritter in 1800.

Wavelength Range : 3.5×10-7m to 1.5×10-7m.

Frequency range : 1016 Hz to 1017Hz

Uses of Ultraviolet light

• Treatment of water is done by Ultraviolet light.

• To kill bacteria in hospitals.
• Treats jaundice in newborns.

• Used in detective work to reveal the presence of substance that cannot be seen by visible light

X- Ray

These EM waves have wavelengths just shorter than ultraviolet light. As X-ray can pass through many form of matter, so they have many medical and industrial applications

Source of X-ray is sudden dece;eration of fast moving electron by a target metal. It was discovered by Rontgen in 1895.

Wavelength range : 10-8 m to 10-11m.

Frequency range: 1018 Hz to 1020  Hz

Uses of X-Ray

• In medical diagnosis because X-ray can pass through flesh but not through bones.

• In the study of crystal structure because X-ray can be reflected and diffracted by crystals.

• In engineering for detecting faults, cracks and holes in finished metal products.
• In radiotherapy to cure untraceable skin disease and malignant growth.

γ-rays

These are the e.m. wave of highest frequency range and lowest wavelength range. These are most penetrating e.m. waves.

Source of γ-ray is radioactive nuclei and nuclear reactions. It was discovered by Henry Becqurel in 1896.

Wavelength range : 10-14 m to 10-10m

Frequency range : 1018 Hz to 1022 Hz.

Uses  of γ-rays

• Gamma rays are used for treatment of tumors.

• It is used for treatment of cancerous cells in the human body without the use of surgery. Cancer cells cannot regenerate once destroyed by gamma rays.

• It is used to sterilize medical equipment.

At a glance

## Lenz’s Law

Lenz’s law is given by a physicist of Germany named Heinrich Friedrich Lenz. He described the direction of electric current relative to the magnetic flux. He deduced that the direction of an induced current in a circuit is such as to oppose the change that causes it.

Applications of Lenz’s Law

• Braking systems in trains,
• Metal detectors,
• Eddy current dynamometers,
• AC Generators,
• Microphones, etc.

MOTIONAL ELECTROMOTIVE FORCE
The process of induction occurs when a change in magnetic flux causes an emf to oppose that change. One of the main reasons for the induction process in motion. We can say, for example, that a magnet moving toward a coil generates an emf, and that a coil moving toward a magnet creates a comparable emf.
Let x be the distance between the resistance and the rod at any timer t . According to faraday’s law, emf produced in a loop due to change in magnetic flux is,
ε = – dφ/dT
φt (flux at any time t) = B . A
= B l x
d (φ)/dt = d (Blx)/dt
= Bl dx/dt
= Blv
Therefore,
ε = – dφ/dT
= Blv
Using Ohm’s law: V = IR or I = V/R
Current through the resistance R is,
I = Blv/R                                                                                                                                                                (in a clockwise direction)
Amount of charge (q) passed through the loop in time ‘t’ = Δφ/R
where Δφ = Total change in flux in time ‘t’.

ENERGY CONSIDERATION: A QUANTITATIVE STUDY

Let us consider a rectangular loop as shown in the figure above, with the sides PQ, QR, RS and SP. Here, the three sides of the loop are fixed and one of the sides, the side PQ, is free to move. Let r be the resistance of the movable arm under consideration.

loop be taken as ε, then the current in the loop can be given as,

The energy that goes into the motion of the rod in this system is dissipated in the form of heat given by,

Here we see that the value of PJ is equal to the value of P. the magnitude of the induced emf is,

However, as we know that,

And thus, from the above two equations, we can write,

EDDY CURRENTS
Eddy currents are currents which circulate in conductors like swirling eddies in a stream. They are induced by changing magnetic fields and flow in closed loops, perpendicular to the plane of the magnetic field. So far we have studied the electric currents induced in well defined paths in conductors like circular loops. Even when bulk pieces of conductors are subjected to changing magnetic flux, induced currents are produced in them. However, their flow patterns resemble swirling eddies in water. This effect was discovered by physicist Foucault (1819-1868) and these currents are called eddy currents
AC GENERATOR

The phenomenon of electromagnetic induction has been technologically exploited in many ways. An exceptionally important application is the generation of alternating currents (ac). The modern ac generator with a typical output capacity of 100 MW is a highly evolved machine. In this section, we shall describe the basic principles behind this machine. The Yugoslav inventor Nicola Tesla is credited with the development of the machine.