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

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

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.

Electric Dipole

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

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

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

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

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.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

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 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

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.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.

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. 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. 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
Radio 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.

2. Lenz's law and energy

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,
  • Card Readers,
  • 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.
Working of an AC Generator
When the armature rotates between the poles of the magnet upon an axis perpendicular to the magnetic field, the flux linkage of the armature changes continuously. As a result, an electric current flows through the galvanometer and the slip rings and brushes. The galvanometer swings between positive and negative values. This indicates that there is an alternating current flowing through the galvanometer.
Field
The field consists of coils of conductors that receive a voltage from the source and produce magnetic flux. The magnetic flux in the field cuts the armature to produce a voltage.
Armature
The part of an AC generator in which the voltage is produced is known as an armature. This component primarily consists of coils of wire that are large enough to carry the full-load current of the generator.
Prime Mover
The component used to drive the AC generator is known as a prime mover. The prime mover could either be a diesel engine, a steam turbine, or a motor.
Rotor
The rotating component of the generator is known as a rotor. The generator’s prime mover drives the rotor.
Stator
The stator is the stationary part of an AC generator. The stator core comprises a lamination of steel alloys or magnetic iron to minimize the eddy current losses.
Slip Rings
Slip rings are electrical connections used to transfer power to and fro from the rotor of an AC generator. They are typically designed to conduct the flow of current from a stationary device to a rotating one.