Relation Between Field And Potential

Consider two closely spaced equipotential surfaces A and B Fig. with potential values V and V + δV, where δV is the change in V in the direction of the electric field E.

Let P be a point on the surface B. δl is the perpendicular distance of the surface A from P. Imagine that a unit positive charge is moved along this perpendicular from the surface B to surface A against the electric field. The work done in this process is |E|δl.

This work equals the potential difference V_A –V_B. Thus,

We thus arrive at two important conclusions concerning the relation between electric field and potential:

(i)            Electric field is in the direction in 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.