Question

For any charge configuration, equipotential surface through a point is normal to the electric field. Justify :

Answer 1

For any charge configuration, equipotential surface through a point is normal to the electric field. We will prove this statement using contradiction.

So, if we assume the statement to be false, there can be a component of electric field along the equipotential surface. Now, a non-zero work is done on a test charge moved along the equipotential surface given by $dW=qEdx$. But work done on the test charge on movement through an equipotential surface is zero. Hence, the assumption was wrong. Thus, the electric field cannot have any component along the equipotential surface which implies that it has to be perpendicular to the equipotential surface.