Question

Two closely spaced equipotential surfaces A and B with potentials V and V + $\delta$V, (where $\delta$V is the change in V), are kept $\delta l$ distance apart as shown in the figure. Deduce the relation between the electric field and the potential gradient between them. Write the two important conclusions concerning the relation between the electric field and electric potentials.

Answer 1

$\overrightarrow{E_A}=-\overrightarrow{\nabla }V$ and $\overrightarrow{E_B}=-\overrightarrow{\nabla }(V+\delta V)$
$\therefore \overrightarrow{E_B}=-\overrightarrow{\nabla }V-\overrightarrow{\nabla }\left(\delta V\right)=\overrightarrow{E_A}-\overrightarrow{\nabla }\left(\delta V\right)$
net field $=\overrightarrow{E_B}-\overrightarrow{E_A}=-\overrightarrow{\nabla }\left(\delta V\right)$
1) Electric filed is perpendicular to the equipotential surfaces.
2) Net field is also gradient of potential $\delta V$ .