Chapter 1: Electric Charges and Fields

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  [ M⁰ L ⁰ T ¹ A ¹]

“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.

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 .

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.

https://phet.colorado.edu/sims/html/coulombs-law/latest/coulombs-law_en.html

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.

https://phet.colorado.edu/sims/html/charges-and-fields/latest/charges-and-fields_en.htm

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.

2. 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.
3. Volume charge density: when charges are spread in a continuous manner in a volume ( 3-dimension), we call it volume charge distribution.
4. 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.

 

 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, H2​O,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. 

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  E∙ds, 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

E∙ds 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 

E∙ds = 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.

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

Q = 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.