**Class 12 Physics CHAPTER 2 ELECTRIC CHARGES AND FIELDS**

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**NCERT Notes For Class 12 Physics CHAPTER 2 ELECTRIC CHARGES AND FIELDS**

**Class 12 Physics CHAPTER 2 ELECTRIC CHARGES AND FIELDS**

**ELECTROSTATIC POTENTIAL AND CAPACITANCE**

**Potential energy difference **

- Electric potential energy difference between two points is the work required to be done by an external force in moving charge q from one point to another.

**Electrostatic potential energy **

- Potential energy of charge q at a point is the work done by the external force in bringing the charge q from infinity to that point.

**ELECTROSTATIC POTENTIAL **

**The electrostatic potential (V ) at any point is the work done in bringing a unit positive charge from infinity to that point.**

**W – work done, q – charge.**

**Also,***W*=*qV***It is a scalar quantity.****Unit is J/C or volt (V)**

## POTENTIAL DUE TO A POINT CHARGE

- The force acting on a unit positive charge (+1 C) at A , is

- Thus the work done to move a unit positive charge from A to B through a displacement dx is

- The negative sign shows that the work is done against electrostatic force.
- Thus the total work done to bring unit charge from infinity to the point P is

- Integrating

- Therefore electrostatic potential is given by

**Variation of potential V with r**

**POTENTIAL DUE TO AN ELECTRIC DIPOLE**

The potential due to the dipole at P is the sum of potentials due to the charges q and –q

- Using cosine law

- For r >> a

- Neglecting the higher order terms we get

- Similarly

- Thusand
- Using the Binomial theorem and retaining terms up to the first order in a/r,

- Thus the potential is

- Using p=q x2a, we get

**Special cases**

- Potential at point on the axial line

At the axial point θ=0, therefore

- Potential at point on the equatorial line At the equatorial line θ=90
^{0}, thus , V=0.

## POTENTIAL DUE TO A SYSTEM OF CHARGES

- By the superposition principle, the potential at a point due to a system of charges is the algebraic sum of the potentials due to the individual charges.

**Thus***V = V*……._{1 }V_{2}+*V*_{n}

**Potential due to a uniformly charged spherical shell**

- For a uniformly charged spherical shell, the electric field outside the shell is as if the entire charge is concentrated at the centre
- Thus potential at a distance r, from the shell is

- Where
*r R*≥ , radius of the shell - Inside the shell, the potential is a constant and has the same value as on its surface.

## Equipotential surface

- Surface with constant value of potential at all points on the surface.

## Properties of Equipotential surface

- Work done to move a charge on an equipotential surface is zero.
- Electric field is perpendicular to the surface.
- Two equipotential surfaces never intersect.

#### Equipotential surface of a single charge

#### Equipotential surfaces for a uniform electric field

#### Equipotential surfaces for a dipole

#### Equipotential surfaces for two identical positive charges

#### Relation between electric field and potential

- The work done to move a unit positive charge from B to A is

*Work* =* Edr*

- This work equals the potential difference V
_{A}–V_{B}. - Thus
- That is

**POTENTIAL ENERGY OF A SYSTEM OF CHARGES **

**For a system of two charges **

- As there is no external field work done in bringing q
_{1}from infinity to r_{1}is zero. - The potential due to the charge q
_{1}is

- The work done in bringing charge q
_{2}from infinity to the point r_{2}is

- This work gets stored in the form of potential energy of the system.
- Thus, the potential energy of a system of two charges q
_{1}and q_{2}is

For a system of three charges

- The work done to bring q
_{1}from infinity to the point is zero. - The work done to bring q
_{2 }is

- The charges q
_{1}and q_{2}produce a potential, which at any point P is given by

- Work done in bringing q
_{3}from infinity to the point r_{3}is

- The total work done in assembling the charges

## POTENTIAL ENERGY IN AN EXTERNAL FIELD

### Potential energy of a single charge

- Potential energy of q at r in an external field is

- where V(r) is the external potential at the point r.

### Potential energy of a system of two charges in an external field

- Work done to bring q1 is

- Work done on q2 against the external field

- Work done on q2 against the field due to q1

- Thus the total Work done in bringing q
_{2}to r_{2}

- Thus, Potential energy of the system = the total work done in assembling the configuration

Potential energy of a dipole in an external field

- The torque experienced by the dipole is

- The amount of work done by the external torque will be given by

- This work is stored as the potential energy of the system.
- Thus

- Therefore

*U*=−*pE*cosθ,

- Where p- dipole moment , E – electric field

## Properties of conductors

- Inside a conductor, electrostatic field is zero
- A conductor has free electrons
- In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside
- At the surface electric field is normal.
- If E were not normal to the surface, it would have some non-zero component along the surface.
- Free charges on the surface of the conductor would then experience force and move.
- In the static situation, therefore, E should have no tangential component.
- 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.
- Electric potential is constant throughout the volume of the conductor.
- Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface.
- That is, there is no potential difference between any two points inside or on the surface of the conductor.
- Electric field at the surface of a charged conductor

- where σ is the surface charge density and ˆn is a unit vector normal to the surface in the outward direction

**Derivation**

- 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 δS and negligible height.
- The contribution to the total flux through the pill box comes only from the outside (circular) cross-section of the pill box.
- The charge enclosed by the pill box is σδS.
- By Gauss’s law

## Electrostatic shielding

- The vanishing of electric field inside a charged conducting cavity is known as electrostatic shielding.
- The effect can be made use of in protecting sensitive instruments from outside electrical influence.
- Why it is safer to be inside a car during lightning?
- Due to Electrostatic shielding, E=0 inside the car.

## DIELECTRICS

- Dielectrics are non-conducting substances.
- They have no (or negligible number of ) charge carriers.

Conductor in an external field

- In an external field the free charge carriers in the conductor move and an electric field which is equal and opposite to the external field is induced inside the conductor.
- The two fields cancel each other and the net electrostatic field in the conductor is zero.

Dielectric in an external field

- In a dielectric, the external field induces dipole moment by stretching or reorienting molecules of the dielectric.
- Thus a net electric field is induced inside the dielectric in the opposite direction.
- The induced field does not cancel the external field.

- Dielectric substances may be made of polar or non polar molecules.

Non polar molecule in an external field

- In an external electric field, the positive and negative charges of a nonpolar molecule are displaced in opposite directions.
- The non-polar molecule thus develops an induced dipole moment.

Linear isotropic dielectrics

- When a dielectric substance is placed in an electric field, a net dipole moment is induced in it.
- When the induced dipole moment is in the direction of the field and is proportional to the field strength the substances are called linear isotropic dielectrics.

Polar molecule in external field

- In the absence of any external field, the different permanent dipoles are oriented randomly due to thermal agitation; so the total dipole moment is zero.
- When an external field is applied, the individual dipole moments tend to align with the field.
- A dielectric with polar molecules also develops a net dipole moment in an external field.

## Polarisation

- The dipole moment per unit volume is called polarisation and is denoted by P.
- For linear isotropic dielectrics,

- where χ
_{e}is the electric susceptibility of the dielectric medium.

A rectangular dielectric slab placed in a uniform external field

- The polarised dielectric is equivalent to two charged surfaces with induced surface charge densities, say σ
_{p}and –σ_{p} - That is a uniformly polarised dielectric amounts to induced surface charge density, but no volume charge density.

## Capacitor

- It is a charge storing device.
- A capacitor is a system of two conductors separated by an insulator.

- A capacitor with large capacitance can hold large amount of charge Q at a relatively small V.

## Capacitance

- The potential difference is proportional to the charge , Q.
- Thus
- The constant C is called the capacitance of the capacitor. C is independent of Q or V.
- The capacitance C depends only on the geometrical configuration (shape, size, separation) of the system of two conductors
- SI unit of capacitance is farad.
- Other units are, 1 μF = 10
^{–6}F, 1 nF = 10^{–9}F, 1 pF = 10^{–12}F, etc.

### Symbol of capacitor

Fixed capacitance Variable capacitance

Dielectric strength

- The maximum electric field that a dielectric medium can withstand without break-down is called its dielectric strength.
- The dielectric strength of air is about 3 × 10
^{6 }Vm^{–1}.

## THE PARALLEL PLATE CAPACITOR

- A parallel plate capacitor consists of two large plane parallel conducting plates separated by a small distance

Capacitance of parallel plate capacitor

- Let A be the area of each plate and d the separation between them.
- The two plates have charges Q and –Q.
- Plate 1 has surface charge density σ = Q/A and plate 2 has a surface charge density –σ.

At the region I and II, E=0

- At the inner region

- The direction of electric field is from the positive to the negative plate.
- For a uniform electric field the potential difference is

- The capacitance C of the parallel plate capacitor is then

- Thus

## Effect of dielectric on capacitance

- When dielectric medium is placed capacitance increases.
- When there is vacuum between the plates,

- The capacitance C
_{0}in this case is

- When a dielectric is introduced

- so that the potential difference across the plates is

- For linear dielectrics, σ
_{p}is proportional to E_{0}, and hence to σ. - Thus, (σ – σ
_{p}) is proportional to σ and we can write

- where K is the dielectric constant.
- Thus

- The capacitance C, with dielectric between the plates, is then

- That is
- The product ε
_{0}K is called the permittivity of the medium and is denoted by ε.

- For vacuum K = 1 and ε = ε
_{0}; ε_{0}is called the permittivity of the vacuum. - Thus the dielectric constant of the substance is

- Also

- C
_{0}– capacitance in vacuum, C- capacitance in dielectric medium.

## Combination of capacitors Capacitors in series

- In series charge is same and potential is different on each capacitors.
- The total potential drop V across the combination is

- Considering the combination as an effective capacitor with charge Q and potential difference V, we get

- Therefore, effective capacitance is

- For n capacitors in series

**Capacitors in parallel**

- In parallel the charge is different, potential is same on each capacitor.
- The charge on the equivalent capacitor is

- Thus

- Therefore

- In general , for n capacitors

## Energy stored in a capacitor

- Energy stored in a capacitor is the electric potential energy.

- Charges are transferred from conductor 2 to conductor 1 bit by bit, so that at the end, conductor 1 gets charge Q.
- Work done to move a charge dq from conductor 2 to conductor 1, is
*dW*=*Potential*×*Ch*arg*e* - That is
- Since potential at conductor 1 is , q/C.
- Thus the total work done to attain a charge Q on conductor 1, is

- On integration we get,

- This work is stored in the form of potential energy of the system.
- Thus energy stored in the capacitor is
- Also or

Alternate method

- We have the Q – V graph of a capacitor,

- Energy = area under the graph

Energy Density of a capacitor

- Energy density is the energy stored per unit volume.
- We have
- Thus we get

- Thus energy per unit volume is given by

- That is the energy density of the capacitor is

## VAN DE GRAAFF GENERATOR

Uses

- Used to generate high potential about million volts and electric field close to 3 × 10
^{6}V/m. - High potential generated is used to accelerate charged particles to high energies.

Principle

- When a small conducting sphere is kept inside a large sphere, potential is high at the small sphere.

- Potential due to small sphere of radius r carrying charge q

- Taking both charges q and Q, we have

- Thus the potential at the smaller sphere is higher than the larger sphere.
- If the two spheres are connected with a wire , charge will flow from smaller sphere to larger one.

Construction

- A large spherical conducting shell is supported at a height several meters above the ground on an insulating column.
- A long narrow endless belt insulating material, like rubber or silk, is wound around two pulleys – one at ground level, one at the centre of the shell.

Working

- The belt is kept continuously moving by a motor driving the lower pulley.
- It continuously carries positive charge, sprayed on to it by a brush at ground level, to the top.
- At the top, the belt transfers its positive charge to another conducting brush connected to the large shell.
- Thus positive charge is transferred to the shell, where it spreads out uniformly on the outer surface.
- In this way, voltage differences of as much as 6 or 8 million volts (with respect to ground) can be built up.
- The voltage produced by an open-air Van de Graaff machine is limited by corona discharge to about 5 megavolts.
- Most modern industrial machines are enclosed in a pressurized tank of insulating gas these can achieve potentials of as much as about 25 megavolts.