Understanding Electric Potential and Equipotentials (AP Physics 2: Unit 2)

Electric Potential Energy

What electric potential energy is

Electric potential energy is the energy stored in a system of electric charges because of their positions relative to each other. If you have two charges in space, the electric force between them can do work as they move closer together or farther apart. That “ability to do work due to position” is what we call potential energy.

A good way to build intuition is to compare to gravity. Near Earth’s surface, lifting a mass increases gravitational potential energy because the gravitational force could later do work on the mass as it falls. Similarly, placing charges in certain arrangements can “store” energy because the electric force could later do work as charges move.

Why it matters

Electric potential energy is the bridge between forces/fields and motion/energy. In AP Physics, you often have two ways to solve problems:

  • A force approach (Coulomb’s law and Newton’s laws)
  • An energy approach (work and potential energy)

The energy approach is often simpler because energy is a scalar—no components—so you avoid vector algebra.

How it works: work and potential energy

The key relationship is that the work done by the electric force changes the potential energy of the system.

If the electric field does work on a charge, the system’s electric potential energy decreases.

The work done by the electric field is

W_{\text{field}} = -\Delta U

  • W_{\text{field}} is the work done by the electric force (or field) on the charge.
  • \Delta U = U_f - U_i is the change in electric potential energy.

If instead you move a charge slowly (so kinetic energy doesn’t change) using an external agent (like your hand or a machine), then the external work goes into changing potential energy:

W_{\text{ext}} = \Delta U

This “slow move” idea shows up a lot: you imagine moving a charge at constant speed so that the net work goes into potential energy rather than kinetic energy.

Potential energy for point charges (Coulomb potential energy)

For two point charges q and Q separated by distance r, the electric potential energy of the pair (choosing zero at infinite separation) is

U = k\frac{qQ}{r}

  • k is Coulomb’s constant.
  • r is the distance between the charges.

Important meaning:

  • If qQ > 0 (like charges), then U > 0. You must do positive external work to push them close together against repulsion.
  • If qQ < 0 (opposite charges), then U < 0. The system releases energy as they attract.

A common misconception is thinking “negative potential energy means something is wrong.” It doesn’t—negative just means your chosen zero (at infinity) is higher than the system’s current energy.

Multiple charges: superposition for potential energy

For more than two charges, total potential energy is the sum over all distinct pairs:

U_{\text{total}} = \sum_{i