Electromagnetism

Electric Field

Define the electric field as force per unit charge, build it from point charges, and explore superposition and field lines.

Electric Potential

Here

Move from force to energy: potential energy landscapes, the electric potential, and its relationship to the field.

Electric Current

Finish

Follow charges into motion: the Drude model of electron drift, Ohm’s law, and series–parallel circuits.

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

F \approx -\frac{\Delta U}{\Delta x},

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

Which forces have a potential energy?

The work done by a conservative force depends only on the start and end positions, not the path taken. Those are exactly the forces for which we can define a scalar potential energy. The electric, gravitational, and spring forces are conservative; friction and air resistance are not.

In one dimension, the work done over a small step is $W \approx F\,\Delta x$ . For a conservative force we call this a change in potential energy, $\Delta U = -W$ , so

F \approx -\frac{\Delta U}{\Delta x}.

In three dimensions the same idea is the gradient, which points in the direction of steepest increase of a scalar field:

\vec{\mathbf F} = -\nabla U = -\left\langle \frac{\partial U}{\partial x}, \frac{\partial U}{\partial y}, \frac{\partial U}{\partial z} \right\rangle.

Don’t worry if that notation is unfamiliar — picture a 2D energy landscape $U(x,y)$ of hills and valleys, with matter tending to flow downhill.

From Energy to Potential

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

Does U = mgh agree with our definition?

Yes. With “up” positive, the gravitational force near the surface is $F = -mg$ . The work done moving an object up a small height $\Delta h$ is $W = F\,\Delta h = -mg\,\Delta h$ , so

\Delta U = -W = mg\,\Delta h.

Choosing $U = 0$ at $h = 0$ gives $U(h) = mgh$ .

Suppose we want the potential energy independent of 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

U = qV,

where $q$ is the charge. Its units 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 of potential.”

Point Charge Potential

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

V = k\,\frac{Q}{r},

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

Where does V = kQ/r come from?

Start from $V = \tfrac{U}{q}$ , so we need the potential energy of a charge $q$ in the field of $Q$ . Coulomb’s law gives the force $F = k\dfrac{Qq}{r^2}$ . The change in potential energy bringing $q$ from infinity to $r$ is

\Delta U = -\int_{\infty}^{r} k\,\frac{Qq}{r^2}\,dr = -kQq\left[-\frac{1}{r}\right]_{\infty}^{r} = k\,\frac{Qq}{r}.

Dividing by $q$ ,

V = \frac{U}{q} = k\,\frac{Q}{r}.

We can picture $V(x,y)$ with a color map. Below, brighter colors are higher potential; it decreases as you move away from the positive charge, since $V \propto 1/r$ . Drag the charge, and toggle the equipotential rings — they are circles of constant $V$ .

In 3D we draw equipotential surfaces, the analog of these rings. Because $U = qV$ , 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 on the field arrows and lines. Build intuition for how source charges shape the potential, how a test charge moves, and how field and potential relate.

Electric Potential Explorer

Toggle the colormap, equipotentials, field arrows, and field lines; add, drag, and remove charges.

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:

E \approx -\frac{\Delta V}{\Delta x}.

Problem Solving

Potential of a Point Charge

Problem

What is the electric potential $0.25\,\text{m}$ from a $+3\,\mu\text{C}$ point charge?

Solve

V = k\,\frac{q}{r} = 9\times10^9 \cdot \frac{3\times10^{-6}}{0.25} \approx 1.1\times10^5\,\text{V}.

Work Along an Equipotential

Problem

Why is no work required to move a charge along an equipotential surface?

Reasoning

Along an equipotential, $\Delta V = 0$ . Since $\Delta U = q\,\Delta V$ , the change in potential energy is zero — and the work done by the electric force equals $-\Delta U = 0$ .

Change in Potential Energy

Problem

Find the change in electric potential energy of a $-3\,\mu\text{C}$ charge that moves from $5\,\text{V}$ to $12\,\text{V}$ .

Solve

With $\Delta V = 7\,\text{V}$ and $q = -3\times10^{-6}\,\text{C}$ ,

\Delta U = q\,\Delta V = (-3\times10^{-6})(7) = -2.1\times10^{-5}\,\text{J}.

The negative charge loses potential energy moving toward higher potential.

Electric Potential Checkpoint

Question 1 of 3

What does electric potential measure?

Choose an answer to get instant feedback.