Why are electric field lines perpendicular to equipotential surfaces?

In physics, the relationship between the electric field and electric potential is defined by the gradient of the potential. Specifically, the electric field vector $𝐄$ is equal to the negative gradient of the potential $V$, expressed as $𝐄=-\text{grad }V$.12

Because the gradient of a scalar function always points in the direction of the most rapid increase, the electric field—being the negative of this gradient—points in the direction where the potential decreases most rapidly. By definition, an equipotential surface consists of points where the potential is constant; therefore, no work is required to move a charge along such a surface.314

Since the electric field represents the force acting on a charge, and moving along an equipotential surface involves no change in potential energy, the force (and thus the electric field) must have no component tangent to the surface. Consequently, the electric field lines must be directed normal, or perpendicular, to the equipotential surfaces at every point.312