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In Chapters 6 and 8 (Class XI), the notion of potential energy was

introduced. When an external force does work in taking a body from a

point to another against a force like spring force or gravitational force,

that work gets stored as potential energy of the body. When the external

force is removed, the body moves, gaining kinetic energy and losing

an equal amount of poten

tial energy. The sum of kinetic and

potential energies is thus conserved. Forces of this kind are called

conservative forces. Spring force and gravitational force are examples of

conservative forces.

Coulomb force between two (stationary) charges, like the gravitational

force, is also a conservative force. This is not surprising, since both have

inverse-square dependence on distance and differ mainly in the

proportionality constants – the masses in the gravitational law are

replaced by charges in Coulomb’s law. Thus, like the potential energy of

a mass in a gravitational field, we can define electrostatic potential energy

of a charge in an electrostatic field.

Consider an electrostatic field E due to some charge configuration.

First, for simplicity, consider the field E due to a charge

*Q*placed at the

origin. Now, imagine that we bring a test charge

*q*from a point R to a

point P against the repulsive force on it due to the charge

*Q*. With reference

Chapter Two

ELECTROSTATIC

POTENTIAL AND

CAPACITANCE

Physics

52

to Fig. 2.1, this will happen if

*Q*and

*q*are both positive

or both negative. For definiteness, let us take

*Q*,

*q*> 0.

Two remarks may be made here. First, we assume

that the test charge

*q*is so small that it does not disturb

the original configuration, namely the charge

*Q*at the

origin (or else, we keep

*Q*fixed at the origin by some

unspecified force). Second, in bringing the charge

*q*from

R to P, we apply an external force Fext just enough to

counter the repulsive electric force FE (i.e, Fext= –FE).

This means there is no net force on or acceleration of

the charge

*q*when it is brought from R to P, i.e., it is

brought with infinitesimally slow constant speed. In

this situation, work done by the external force is the negative of the work

done by the electric force, and gets fully stored in the form of potential

energy of the charge

*q*. If the external force is removed on reaching P, the

electric force will take the charge away from Q – the stored energy (potential

energy) at P is used to provide kinetic energy to the charge

*q*in such a

way that the sum of the kinetic and potential energies is conserved.

Thus, work done by external forces in moving a charge

*q*from R to P is

W

RP =

P R

d

∫ F r

*ext*

=

P R

d

−∫ F r

*E*(2.1)

This work done is against electrostatic repulsive force and gets stored

as potential energy.

At every point in electric field, a particle with charge

*q*possesses a

certain electrostatic potential energy, this work done increases its potential

energy by an amount equal to potential energy difference between points

R and P.

Thus, potential energy difference

∆ = − =

*U U U W*

*P R RP*(2.2)

(Note here that this displacement is in an opposite sense to the electric

force and hence work done by electric field is negative, i.e.

*, –W*

*RP*.)

Therefore, we can define electric potential energy difference between

two points as the work required to be done by an external force in moving

(without accelerating) charge

*q*from one point to another for electric field

of any arbitrary charge configuration.

Two important comments may be made at this stage:

(i) The right side of Eq. (2.2) depends only on the initial and final positions

of the charge. It means that the work done by an electrostatic field in

moving a charge from one point to another depends only on the initial

and the final points and is independent of the path taken to go from

one point to the other. This is the fundamental characteristic of a

conservative force. The concept of the potential energy would not be

meaningful if the work depended on the path. The path-independence

of work done by an electrostatic field can be proved using the

Coulomb’s law. We omit this proof here.

FIGURE 2.1 A test charge

*q*(> 0) is

moved from the point R to the

point P against the repulsive

force on it by the charge

*Q*(> 0)

placed at the origin.

Electrostatic Potential

and Capacitance

53

(ii) Equation (2.2) defines

*potential energy difference*in terms

of the physically meaningful quantity

*work*. Clearly,

potential energy so defined is undetermined to within an

additive constant.What this means is that the actual value

of potential energy is not physically significant; it is only

the difference of potential energy that is significant. We can

always add an arbitrary constant α to potential energy at

every point, since this will not change the potential energy

difference:

( ) ( )

*U U U U*

*P R P R*+ − + = − α α

Put it differently, there is a freedom in choosing the point

where potential energy is zero. A convenient choice is to have

electrostatic potential energy zero at infinity. With this choice,

if we take the point R at infinity, we get from Eq. (2.2)

*W U U U*∞ ∞

*P P P*= − = (2.3)

Since the point P is arbitrary, Eq. (2.3) provides us with a

definition of potential energy of a charge

*q*at any point.

*Potential energy of charge q at a point*(in the presence of field

due to any charge configuration)

*is the work done by the*

*external force*(equal and opposite to the electric force)

*in*

*bringing the charge q from infinity to that point*.

2.2 ELECTROSTATIC POTENTIAL

Consider any general static charge configuration. 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

*q*E, 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. This leads to

the idea of electrostatic potential

*V*due to a given charge

configuration. From Eq. (2.1), we get:

Work done by external force in bringing a unit positive

charge from point R to P

=

*V*

*P*

*– V*

*R*

*U U*

*P R*