Electrostatic Potential
and Capacitance

2.1 INTRODUCTION

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 potential 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 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 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 $F_{ext}$ just enough to counter the repulsive electric force $F_E$ (i.e, $F_{ext}$= –$F_E$). 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

$$\begin{array}{rl}{\mathrm{W}}_{\mathrm{RP}}& ={\int }_{\mathrm{R}}^{\mathrm{P}}{\mathbf{F}}_{ext}\cdot \mathrm{d}\mathbf{r}\\ & =-{\int }_{\mathrm{R}}^{\mathrm{P}}{\mathbf{F}}_E\cdot \mathrm{d}\mathbf{r}\end{array}$$ (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

$$\mathrm{\Delta }U=U_P-U_R=W_{RP}$$ (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:

1. 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.
2. 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 $\alpha$ to potential energy at every point, since this will not change the potential energy difference: $$(U_P+\alpha )-(U_R+\alpha )=U_P-U_R$$

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)

$$W_{\mathrm{\infty }P}=U_P-U_{\mathrm{\infty }}=U_P$$ (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.