Let us consider two closely spaced equipotential surfaces A and B as shown in figure. Let the potential of A be VA = V and potential of B be VB= V−dV is decrease in potential in the direction of electric field \(\vec E\) normal to A and B.
Let dr be the perpendicular distance between the two equipotential surfaces. when 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
\(W _{BA} = - \vec E (dr)\)
This work done equals the potential difference VA−VB
∴ WBA = VA−VB
= V − (V−dV)
= dV
\(\therefore - \vec E.dr = dV\)
or, \(\vec E = - \frac {dV}{dr}\) = negative of potential gradient
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SI unit of the potential gradient is volt/metre.