Electric Potential in Electrostatics (AP Physics C: E&M Unit 1)

Electric Potential Energy

What it is (and what it is not)

Electric potential energy is the energy associated with the configuration (positions) of electric charges. It is energy stored in the system of charges because electric forces can do work as charges move.

A crucial mindset shift: potential energy is not “contained in” a single charge by itself. It belongs to an interaction between charges (or between a charge and an externally created electric field). When you change the arrangement, you change the stored energy.

Why it matters

Potential energy is your bridge between force/field ideas and energy conservation. Many electrostatics problems are dramatically easier with energy:

  • Forces can vary with distance (like 1/r^2), but energy methods often avoid vector force components.
  • Conservation of energy connects electric potential energy to kinetic energy, letting you predict speeds without tracking acceleration.
  • It sets up the definition of electric potential V, a scalar that is often easier to compute than the electric field.

How it works: work and sign conventions

In electrostatics, the electric force is conservative (as long as fields are time-independent), so you can define a potential energy function U.

The work done by the electric field, W_{\text{field}}, is related to the change in potential energy:

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

So:

  • If the field does positive work on a charge, the system’s potential energy decreases.
  • If you move a charge “against” the field, you must do positive external work, and potential energy increases.

If an external agent moves the charge slowly (so kinetic energy doesn’t change), the external work equals the increase in potential energy:

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

Common sign intuition (for a positive test charge):

  • Moving along the electric field direction tends to decrease U.
  • Moving opposite the field tends to increase U.

Potential energy for point charges

For two point charges q_1 and q_2 separated by distance r, choosing U=0 at infinite separation gives:

U(r) = k\frac{q_1 q_2}{r}

where

k = \frac{1}{4\pi\epsilon_0}

Key implications:

  • Like charges: q_1 q_2 > 0 so U>0. You must put energy in to bring them close.
  • Opposite charges: q_1 q_2 < 0 so U

For multiple point charges, the total potential energy is the sum over all distinct pairs:

U = k\sum_{i