2.2 ELECTROSTATIC POTENTIAL

2.2

Work done on a test charge \( q \) by the electrostatic field due to any given charge configuration is independent of the path, and depends only on its initial and final positions.

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 \( qE \), 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 \( P \) \[ =V_{P}-V_{R}\left(=\frac{U_{P}-U_{R}}{q}\right) \] (2.4)

where \( V_{P} \) and \( V_{R} \) are the electrostatic potentials at P and R, respectively. Note, as before, that it is not the actual value of potential but the potential difference that is physically significant. If, as before, we choose the potential to be zero at infinity, Eq. (2.4) implies:
Work done by an external force in bringing a unit positive charge from infinity to a point = electrostatic potential (\( V \)) at that point.

In other words, the electrostatic potential (\( V \)) at any point in a region with electrostatic field is the work done in bringing a unit positive charge (without acceleration) from infinity to that point.

The qualifying remarks made earlier regarding potential energy also apply to the definition of potential. To obtain the work done per unit test charge, we should take an infinitesimal test charge \( \delta q \), obtain the work done \( \delta W \) in bringing it from infinity to the point and determine the ratio \( \delta W/\delta q \). Also, the external force at every point of the path is to be equal and opposite to the electrostatic force on the test charge at that point.