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.

$\blacksquare$

Sources

• 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text {IV}$: The Operator $\nabla$ and its Uses: $2$. The Gradient of a Scalar Field
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): field: 2. (field of force, force field)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): field: 2. (field of force, force field)