Class 12- Chapter 2 : Electric Potential and Capacitance.

 

Chapter 2: 

Electrostatic Potential and Capacitance  

Table of Contents

ELECTROSTATIC POTENTIAL AND CAPACITANCE
Potential energy difference
ELECTROSTATIC POTENTIAL
POTENTIAL DUE TO A POINT CHARGE
POTENTIAL DUE TO A SYSTEM OF CHARGES
Equipotential surface
Properties of Equipotential surface 

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
POTENTIAL ENERGY OF A SYSTEM OF CHARGES
POTENTIAL ENERGY IN AN EXTERNAL FIELD

Potential energy of a single charge
Potential energy of a system of two charges in an external field
Properties of conductors
Electrostatic shielding
DIELECTRICS
Polarisation
Capacitor
Capacitance
Symbol of capacitor
THE PARALLEL PLATE CAPACITOR
Effect of dielectric on capacitance
Combination of capacitors Capacitors in series
Energy stored in a capacitor
VAN DE GRAAFF GENERATOR

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

What is electrostatic potential energy?

To move a positive charge from lower potential to higher potential region, an external agent needs to work. This work done is stored as the potential energy in the electric field. This is called the electrostatic potential energy. So, the work done to move a positive charge against an electric field is the electric potential energy of the electric charge.

• 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.

Units of electrostatic potential energy

The units of electric potential energy are similar to that of the energy we know. So, its SI unit is Joule (J) and CGS unit erg.

Dimensional formula of electric potential energy

The dimensional formula for electric potential energy is the same as that of the normal energy we know. Thus, it has the dimension of [ML2T-2].

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)

Expression for minimum velocity of a charge q to cross a potential difference V

Let an electric charge q is to cross a potential difference V with a velocity u. Now, kinetic energy of the charge will be used to do the electrostatic work to overcome the potential difference. This satisfies the law of conservation of energy.

So, $\color{Blue}\frac{1}{2}mu^2= qV$

or, $\color{Blue}u=\sqrt{\frac{2qV}{m}}$ …………..(9)

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

• Thus

• and

• 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, So Cos θ = 1, therefore

• Potential at point on the equatorial line At the equatorial line θ=900, 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 = V1 V2 + …….Vn

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

• An Equipotential Surface is a surface with constant or the  same  value of potential at each point 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 VA–VB.
• Thus
• That is

What is electrostatic potential energy?

To move a positive charge from lower potential to higher potential region, an external agent needs to work. This work done is stored as the potential energy in the electric field. This is called the electrostatic potential energy. So, the work done to move a positive charge against an electric field is the electric potential energy of the electric charge.

Electric potential energy formula

diagram to find electrostatic potential energy

The work done by the external agent to move a charge through a potential difference is stored as the electrostatic potential energy in the electric field.

Now, electric potential V is the work done per unit charge. So, the work done in moving a charge q through a potential difference V is equal to qV.

Thus, the formula for electrostatic potential energy, W = qV………..(1)

Now, If VA and VB be the electric potentials at the points A and B respectively, then the potential difference between these points is VAB = (VA-VB). Then electrostatic energy required to move q charge from point-A to point-B is,

       W = qVAB

or,   W = q(VA-VB) ……………(2)

     Again, the formula for electric potential at r distance from a point charge Q is $\color{Blue}V=\small \frac{Q}{4\pi \epsilon _{0}r}$.

Then from equation- (1), the electrostatic potential energy to bring a charge q from infinity to r distance of a source charge Q is,

$\small {\color{Blue}W= \frac{qQ}{4\pi \epsilon_{0}r}}$ ………….(3)

This is the formula for electrostatic potential energy. 

Units of electrostatic potential energy

The units of electric potential energy are similar to that of the energy we know. So, its SI unit is Joule (J) and CGS unit erg.

Dimensional formula of electric potential energy

The dimensional formula for electric potential energy is the same as that of the normal energy we know. Thus, it has the dimension of [ML2T-2].

POTENTIAL ENERGY OF A SYSTEM OF CHARGES

For a system of two charges

• As there is no external field work done in bringing q1 from infinity to r1 is zero.
• The potential due to the charge q1 is

• The work done in bringing charge q2 from infinity to the point r2 is

• This work gets stored in the form of potential energy of the system.
• Thus, the potential energy of a system of two charges q1 and q2 is

For a system of three charges

• The work done to bring q1 from infinity to the point is zero.
• The work done to bring q2 is

• The charges q1 and q2 produce a potential, which at any point P is given by

• Work done in bringing q3 from infinity to the point r3 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 q2 to r2

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=−pEcosθ,

• 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 χe 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 σp and –σp
• 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 × 106 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 C0 in this case is

• When a dielectric is introduced

• so that the potential difference across the plates is

• For linear dielectrics, σp is proportional to E0, and hence to σ.
• Thus, (σ – σp) 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 ε0K is called the permittivity of the medium and is denoted by ε.

• For vacuum K = 1 and ε = ε0; ε0 is called the permittivity of the vacuum.
• Thus the dielectric constant of the substance is

• Also

• C0 – 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

The application of voltage produces electric field and this electric field forces the charges to move from one plate to another plate. This rises a voltage across the capacitor which is different from the voltage of the battery. Electrons will continue to move from one plate to another plate until the voltage across the capacitor becomes equal to the voltage of the battery. As the voltage across the capacitor develops an energy starts to be stored in the capacitor.

Here, we are going to derive the formula of energy in a capacitor. This equation for the capacitor energy is very important to study the character of a capacitor.

The energy stored by the capacitor is the electrostatic potential energy. Such type of energy appears due to the storage of electric charges in the electric field. All type of capacitors like parallel plate capacitor, spherical capacitor, cylindrical capacitor, etc. stores same type of energy inside them. The unit of the energy stored in the capacitor is same as the unit of energy we know. It is Joule in SI system and erg in CGS.

Energy stored in capacitor formula

If Q, V and C be the charge, voltage and capacitance of a capacitor, then the formula for energy stored in the capacitor is, $\small {\color{Blue} U=\frac{1}{2}CV^{2}}$. ……………….(1)

Again, Q=CV. 

So, we can re-write the equation in two different ways as, $\small {\color{Blue} U=\frac{1}{2}QV}$ ……( 2)

And, $\small {\color{Blue} U=\frac{1}{2}\frac{Q^{2}}{C}}$ ……………..(3)

The above three equations give the formula for the energy stored by a capacitor.

Derivation of formula for energy stored in capacitor

As the charges shifted from one plate to another plate of a capacitor, a voltage develops in the capacitor. This voltage opposes further shifting of electric charges. Now, to give more charges to the capacitor a work is to be done against the voltage drop. This work stores as the electrostatic potential energy in the capacitor.

Let at any instant the electric charge on the capacitor is q and the voltage is v. Now, to give another dq amount charge to the capacitor, the work done against the developed voltage is, $\small {\color{Blue} dW=v.dq}$

Now, q = C.v or, v = q/C.

Then, $\small {\color{Blue} dW=\frac{q}{C}dq}$

Now, let we want to charge the capacitor up to Q amount from zero value. Then total work done in charging the capacitor by Q is, $\small {\color{Blue} W=\int_{0}^{Q}\frac{q.dq}{C}}$

or, $\small {\color{Blue} W=\frac{1}{2}CV^{2}}$ …………………..(4)

This work done is stored as the electrostatic potential energy (U) of the capacitor and this is the equation for energy stored in a capacitor.

Now, using Q=CV formula one can re-write this equation in other two forms.

Explanation:

• 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 ×Charge
• 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

 

How to increase the potential energy of a capacitor?

Parallel plate Capacitor with circular plate

There are two ways to increase the energy in capacitor.

1. By increasing the voltage of the capacitor without changing the parameters and properties of the capacitor.
2. In the above formulae, one can see that electrostatic potential energy of the capacitor will increase if the capacitance increases when voltage remains same. So, one can increase the energy stored in a parallel plate capacitor by inserting dielectric medium or slab between the plates at the time of charging the capacitor.

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 × 106 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.

Problems on Capacitance :

1. A parallel plate capacitor is fully charged by 2 coulomb and thereby its potential increases to 3 volts. What will be the maximum energy stored in the parallel plate capacitor?

Answer: 

Here, the maximum charge of the parallel plate capacitor is 2 C and the corresponding voltage is 3 volts. Then using equation-2 we get,

Energy stored = 1/2 (QV) = (2×3)÷2 = 3 Joule.

2. A parallel plate capacitor has its capacitance of 2 micro-farad. Now, if you place a dielectric medium (K=2) between the plates keeping a battery of 10 voltage on. What will be the ratio of potential energy of the capacitor before and after placing the dielectric medium?

Answer: 

Here, the battery is always on. So, the voltage is constant all time. given, V=10 volts, C= 2 micro-farad. After placing the dielectric medium or slab, the capacitance becomes C2=KC=4 micro-farad.

Now, from equation-1, $\small U=\frac{1}{2}CV^{2}$.

So, $\small \frac{U_{1}}{U_{2}}=\frac{\frac{CV^{2}}{2}}{\frac{4CV^{2}}{2}}$

or, $\small \frac{U_{1}}{U_{2}}=\frac{1}{4}$

or, U1 : U2 = 1 : 4

Thus, the final energy in capacitor increases and becomes four times the initial value of energy.

Revision at a glance

1. Electrostatic Potential The electrostatic potential at any point in an electric field is equal to the amount of work done per unit positive test charge or in bringing the unit positive test charge from infinite to that point, against the electrostatic force without acceleration.

NOTE: Electrostatic potential is a state dependent function as electrostatic forces are conservative forces.
2. Electrostatic Potential Difference The electrostatic potential difference between two points in an electric field is defined as the amount of work done in moving a unit positive test charge from one point to the other point against of electrostatic force without any acceleration (i.e. the difference of electrostatic potentials of the two points in the electric field).

where, is work done in taking charge q0 from A to B against of electrostatic force.
Also, the line integral of electric field from initial position A to final position B along any path is termed as potential difference between two points in an electric field, i.e.

NOTE: As, work done on a test charge by the electrostatic field due to any given charge configuration is independent of the path, hence potential difference is also same for any path.
For the diagram given as below, potential difference between points A and B will be same for any path.

3. Electrostatic potential due to a point charge q at any point P lying at a distance r from it is given by

4. The potential at a point due to a positive charge is positive while due to negative charge, it is negative.
5. When a positive charge is placed in an electric field, it experiences a force which drives it from points of higher potential to the points of lower potential. On the other hand, a negative charge experiences a force driving it from lower potential to higher.
6. Electrostatic potential due to an electric dipole at any point P whose position vector is r w.r.t. mid-point of dipole is given by----

7. The electrostatic potential on the perpendicular bisector due to an electric dipole is zero.
8. Electrostatic potential at any point P due to a system of n point charges q1, q2, ……………, qn whose position vectors are r1,r2,…,rn respectively, is given by----

where, r is the position vector of point P w.r.t. the origin.
9. Electrostatic potential due to a thin charged spherical shell carrying charge q and radius R respectively, at any point P lying-----

10. Graphical representation of variation of electric potential due to a charged shell at a distance r from centre of shell is given as ----

11 Equipotential Surface A surface which have same electrostatic potential at every point on it, is known as equipotential surface.
The shape of equipotential surface due to
(i) line charge is cylindrical.
(ii) point charge is spherical as shown along side:
(a) Equipotential surfaces do not intersect each other as it gives two directions of electric field E at intersecting point which is not possible.
(b) Equipotential surfaces are closely spaced in the region of strong electric field and vice-versa.
(c) Electric field is always normal to equipotential surface at every point of it and directed from one equipotential surface at higher potential to the equipotential surface at lower potential.
(d) Work done in moving a test charge from one point of equipotential surface to other is zero.

12. Relationship between electric field and potential gradient

where, negative sign indicates that the direction of electric field is from higher potential to lower potential, i.e. in the direction of decreasing potential.
NOTE: (i) Electric field is in the direction of which the potential decreases steepest.
(ii) Its magnitude is given by the change in the magnitude of potential per unit displacement normal to the equipotential surface at the point.
13. Electrostatic Potential Energy The work done against electrostatic force gets stored as potential energy. This is called electrostatic potential energy.
∆U = UB-UA =WAB
14. The work done in moving a unit positive test charge over a closed path in an electric field is zero. Thus, electrostatic forces are conservative in nature.
15. Electrostatic potential energy of a system of two point charges is given by

16. Electrostatic potential of a system of n point charges is given by

17. Potential Energy in an External Field
(i) Potential Energy of a single charge in external field Potential energy of a single charge q at a point with position vector r, in an external field is qV(r),
where V(r) is the potential at the point due to external electric field E.
(ii) Potential Energy of a system of two charges in an external field

18. Potential energy of a dipole in a uniform electric field E is given by
Potential energy = -p .E
19. Electrostatic Shielding The process which involves the making of a region free from any electric field is known as electrostatic shielding.

It happens due to the fact that no electric field exist inside a charged hollow conductor. Potential inside a shell is constant. In this way we can also conclude that the field inside the shell (hollow conductor) will be zero.

 The S.I. unit of electric potential and a potential difference is volt.

→ 1 V = 1 J C-1.

→ Electric potential due to a + ve source charge is + ve and – ve due to a – ve charge.

→ The change in potential per unit distance is called a potential gradient.

→ The electric potential at a point on the equatorial line of an electric dipole is zero.

→ Potential is the same at every point of the equipotential surface.

→ The electric potential of the earth is arbitrarily assumed to be zero.

→ Electric potential is a scalar quantity.

→ The electric potential inside the charged conductor is the same as that on its surface. This is true irrespective of the shape of the conductor.

→ The surface of a charged conductor is equipotential irrespective of its shape.

→ The potential of a conductor varies directly as the charge on it. i.e., V ∝ lA

→ Potential varies inversely as the area of the charged conductor i.e.
→ S.I. unit of capacitance is Farad (F).
→ The aspherical capacitor consists of two concentric spheres.
→ A cylindrical capacitor consists of two co-axial cylinders.
→ Series combination is useful when a single capacitor is not able to tolerate a high potential drop.
→ Work done in moving a test charge around a closed path is always zero.
→

→ The parallel combination is useful when we require large capacitance and a large charge is accumulated on the combination.

→ If two charged conductors are connected to each other, then energy is lost due to sharing of charges, unless initially, both the conductors are at the same potentials.

→ The capacitance of the capacitor increases with the dielectric constant of the medium between the plates.

→ The charge on each capacitor remains the same but the potential difference is different when the capacitors are connected in series.

→ P. D. across each capacitor remains the same but the charge stored across each is different during the parallel combination of capacitors.

→ P.E. of the electric dipole is minimum when θ = 0 and maximum when θ = 180°

→ θ = 0° corresponds to the position of stable equilibrium and θ = π to the position of unstable equilibrium.

→ The energy supplied by a battery to a capacitor is CE2 but energy stored

→ A suitable material for use as a dielectric in a capacitor must have a high dielectric constant and high dielectric strength.

→ Van-de Graaff generator works on the principle of electrostatic. induction and action of sharp points on a charged conductor.

→ The potential difference between the two points is said to be 1 V if 1 J of work is done in moving 1 C test charge from one point to the another.

→ The electric potential at a point in E→: It is defined as the amount of work done in moving a unit positive test charge front infinity to that point.

→ Electric potential energy: It is defined as the amount of work is done in bringing the charges constituting a system from infinity to their respective locations.

→ 1 Farad: The capacitance of a capacitor is said to be 1 Farad if 1 C charge given to it raises its potential by 1 V

→ Dielectric: It is defined as an insulator that doesn’t conduct electricity but the induced charges are produced on its faces when placed in a uniform electric field.

→ Dielectric Constant: It is defined as the ratio of the capacitance of the capacitor with a medium between the plates to its capacitance with air between the plates

→ Polarisation: It is defined as the induced dipole moment per unit volume of the dielectric slab.

→ The energy density of the parallel plate capacitor is defined as the energy per unit volume of the capacitor.

→ Electrical Capacitance: It is defined as the ability of the conductor to store electric charge.

Important Formulae

→ Electric potential at a point A is 

→ Electric field is related to potential gradient as:

→Electric potential at point on the axial line of an electric dipole is:

→ Electric P.E. of a system of point charges is given

→ V due to a charged circular ring on its axis is given by:

→ V at the centre of ring of radius R is given by

→ The work done in moving a test large from one point A to another point B having positions vectors rA→ and rA→ respectively w.r.t. q is given by

→ Line integral of electric field between points A and B is given by.

→ Electric potential energy of an electric dipole is

→ Capacitance of the capacitor is given by

→ P.E. of a charged capacitor is:

→ C of a parallel plate capacitor with air between the plates is:

→ C of a parallel plate capacitor with a dielectric medium between the plates is:

→ Common potential as

→ loss of electrical energy = 12(C1C2C1+C2)(V1−V2)

→ Energy supplied by battery is CE2 and energy stored in the capacitor is 12 CE2.

→ The equivalent capacitance of series combination of three capacitor is given by
1Cs=1C1+1C2+1C3

→ The equivalent capacitance of parallel grouping of three capacitors is

→ Capacitance of spherical capacitor is

→ Capacitance of a cylindrical capacitor is given by:

→ Capacitance of a capacitor in presence of conducting slab between the plates is .

→Capacitances of a capacitor with a dielectric medium between the plates is given by

→ Reduced value of electric field in a dielectric slab is given by

→ Capacitance of an isolated sphere is given by