Electric Force is Gradient of Electric Potential Field

Theorem
Let $R$ be a region of space in which there exists an electric potential field $F$.

The electrostatic force experienced within $R$ is the negative of the gradient of $F$:


 * $\mathbf V = -\grad F$

Proof
The electrostatic force at a point of $R$ is in the direction of the greatest rate of decrease of electric potential.

That is, it is normal to the equipotential surfaces.

It also has a magnitude equal to the rate of decrease.

Hence the result.