Question

Why electric field is perpendicular to the equipotential surface?

Answer 1

The reason for field lines to be perpendicular to equipotential surfaces

1. A surface on which at each and every point potential is the same is called an equipotential surface.
2. For an equipotential surface, the work done to move a charge is always zero because the potential at each and every point is the same.

Work done in an electric field,

$W=q\left(V_a-V_b\right)$

Here,

$$\begin{gathered} W=Workdonetomovechargefrompointatob \\ {V}_a=Potentialata \\ {V}_b=Potentialatb \\ q=Valueofthecharge \end{gathered}$$

1. From the above formulaic potential is the same everywhere on the surface then the work done will be zero on the entire surface.
2. If the electric field here is not perpendicular to the equipotential surface then there exists a non-zero component of the field along the surface and some work must be done against this field to move a charge on the surface which is absolutely false according to the definition of the equipotential surface

Hence, the field lines are perpendicular to the equipotential surfaces.