Relation Between Electric Field Intensity and Potential

Because electric field $\vec{E}$ and electric potential $V$ describe exactly the same electrostatic field, one as a vector and the other as a scalar, an exact mathematical relation must connect the two -- and it takes the form of a derivative in one direction and an integral in the other.

From potential to field (differentiation). Consider a small displacement $dl$ along the direction in which the field $\vec{E}$ points, moving a test charge $q_0$ through it. The work done by the field itself on the charge over this small step is $dW_{\text{field}} = q_0 E\, dl$; the work an EXTERNAL agent must supply to move the charge quasi-statically (balancing the field's force at every instant) is therefore $dW_{\text{ext}} = -q_0 E\, dl$. Dividing by $q_0$, this external work per unit charge is, by definition, the change in potential over that step, $dV$:

So the component of the electric field along any direction equals minus the rate at which potential falls off in that direction. Two consequences follow immediately: first, the electric field points in the direction of the STEEPEST DECREASE of potential (never toward increasing potential, for a positive test charge to be pushed the way it actually is); second, wherever the field is strong, potential must be changing rapidly with position, and wherever the field is weak (or zero), potential must be changing slowly (or be constant) -- exactly the picture equipotential surfaces make visual in Section 2.7.

From field to potential (integration). Running the same relation the other way, the potential difference between two points $A$ and $B$ is recovered by integrating the field along any path joining them:

Special case: a uniform field. Between the plates of a parallel plate capacitor (Section 2.13), the field $E$ is uniform and directed straight from the positive plate to the negative plate. Integrating the relation above along a straight path of length $d$ between the plates, in the direction of $\vec{E}$, gives simply …