Electric Potential

The electric potential describes electric interactions in terms of energy rather than force. Just as gravitational potential energy relates to the gravitational force, electric potential energy relates to the electric force. When a charge moves through a field, its potential energy changes; we define the electric potential $V$ as the potential energy per unit charge.

Potential Energy Landscapes

Visualizing an object “rolling” on a potential energy surface connects the force and energy pictures. Below, a mass feels either a spring-like force or simple gravity. In both cases it tends to move downhill along the 1D potential energy curve, trading potential energy for kinetic energy and back.

This downhill tendency is a general property of conservative forces. The force always points toward decreasing potential energy, captured in one dimension by

which is the slope of the potential energy function $U(x)$.

From Energy to Potential

Before the electrical case, recall gravitational potential energy near Earth’s surface, $U = mgh$.

Suppose we want to analyze the gravitational potential energy function independent of a particular mass. Define $V = gh$ with the rule $U = mV$. This $V$ (units of J/kg) tells us the potential energy per unit mass at a given height.

For the electric force we do exactly this. The electric potential $V$ is defined so that the electric potential energy $U$ is

where $q$ is some particular charge “experiencing” this potential (energy) function. The units of electric potential are J/C, which we call volts. Only changes in potential energy — and therefore changes in potential — are physical: $V$ can be shifted by any constant without changing the physics, just as elevation does not depend on our choice of sea level. Picking that constant is choosing a zero potential.

Reading a Landscape with Contour Lines

Imagine walking around a hill without climbing or descending. Your path follows a contour line: every point on it has the same elevation. A topographical map draws these lines from above, marking heights at a fixed interval.

The lines are equally spaced in height, not in horizontal distance. Closely spaced lines indicate a steep slope in that region. Electric equipotential lines work the same way, with volts replacing meters: each line joins points of equal potential, and closer lines at a fixed voltage interval indicate a stronger electric field.

To take the analogy further: just as water tends to flow toward lower elevation, the electric field lines will “flow” toward lower electric potential. If we then consider the dynamics of electric charge in this potential landscape, we can also say (positive) charge will feel a force in the steepest “downhill” direction.

Point Charge Potential

The electric potential function evidently carries a lot of information and will make electrodynamics relatively simple. But how do we determine it in the first place? One method is identical to how we found the electric field: consider all point charges and sum to find their net effect.

The electric potential due to a point charge $Q$ is

where $r$ is the distance from the charge to the location of interest and $k = 9 \times 10^9\ \tfrac{\text{N·m}^2}{\text{C}^2}$.

This choice of potential function implicitly chooses the zero potential to be infinitely far away; $V(r \to \infty) \to 0$. The important feature of the function is the $1/r$ proportionality. There are many other functions that would work perfectly well.

Although not as common, it can be fun to visualize this electric potential function in 2D with a color map.

In 3D we draw equipotential surfaces, the analog of the equipotential lines shown above. Because the electric potential energy $U = qV$, positive and negative charges behave differently in the same potential: both tend toward lower potential energy, not lower potential. When the signs get confusing, fall back on “like charges repel, opposites attract.”

This last explorer overlays the potential color map and equipotentials in the space around a collection of point charges.

The explorer uses one voltage interval throughout each view. It adapts that interval to the charges and leaves out extreme values near the charge cores to keep the drawing readable.

Notice that the equipotential lines are everywhere perpendicular to the field lines. Just as $F \approx -\tfrac{\Delta U}{\Delta x}$ in one dimension, the field is the (negative) slope of the potential: