Topic 9.1 Notes – Electric Potential Energy

1. What Electric Potential Energy Is

Think about gravity. Lifting a mass stores gravitational potential energy because you did work against gravity. Electric potential energy is the same idea, but with electric forces between charges.

Electric potential energy $U_E$ is the work required by an external force to assemble a system of charges from infinitely far apart (where we define $U_E = 0$) to their final positions.

A few core facts:

• It comes from Coulomb’s law, an inverse-square, conservative force.
• Because the electric force is conservative:

  $$W_{\text{electric}} = -\Delta U_E$$

• Energy can move between electric potential energy and kinetic energy.
• Reference point for AP Physics C:
  • $U_E \to 0$ as $r \to \infty$

The most important mindset shift:
Potential energy belongs to the system of charges, not to a single charge.

2. Electric Potential Energy Between Two Point Charges

For two point charges, the expression is clean and must be automatic for you.

$$U_E = k \frac{q_1 q_2}{r}$$

• $k = \dfrac{1}{4\pi\varepsilon_0}$
• $q_1, q_2$ in coulombs
• $r$ = separation distance
• Units: joules

How the Sign Works

The sign tells you everything about the interaction.

• Like charges ($+/+$ or $-/-$)
  • $q_1 q_2 > 0$
  • $U_E > 0$
  • Repulsive interaction
  • You must do positive work to push them together.
• Unlike charges ($+/-$)
  • $q_1 q_2 < 0$
  • $U_E < 0$
  • Attractive interaction
  • The system releases energy as they come together.

The electric potential energy $U(r)$ behaves differently in the two cases: for like charges it is positive and decreases toward 0 as $r$ increases, and for unlike charges it is negative and increases toward 0 (from below) as $r$ increases.

Notice:

• $U_E \propto \frac{1}{r}$
• Force depends on $\frac{1}{r^2}$
  Students mix this up constantly. Energy falls off more slowly than force.

As $r \to \infty$, $U_E \to 0$ in both cases. That’s why infinity is our reference.

3. Electric Potential Energy of a System of Multiple Charges

Now extend the idea. If you have more than two charges, nothing new conceptually happens. You just account for every pair interaction.

\[ U_{E,\text{total}} = \sum_{i

What that means in practice:

1. List all unique pairs.
   • 3 charges → 3 pairs
   • 4 charges → 6 pairs
2. Compute $U_{ij} = k \dfrac{q_i q_j}{r_{ij}}$ for each.
3. Add them algebraically.

Key reminders:

• Do not double-count pairs.
• Do not include a charge interacting with itself.
• This works because electric potential energy is a scalar.

Here’s a simple three-charge setup where each pair contributes to the total energy:

Three point charges forming three pairwise interactions

Even though the forces are vectors, the potential energies just add as numbers.

On free-response questions, symmetry often makes several $r_{ij}$ equal. That’s usually your algebra simplifier.

4. Work and Energy in Charge Systems

Since the electric force is conservative:

$$W_{\text{electric}} = -\Delta U_E \qquad W_{\text{external}} = +\Delta U_E$$

This lets you switch to energy methods instead of force analysis.

Energy conservation form:

$$K_i + U_i = K_f + U_f$$

Common physical situations:

• Bring like charges closer → $U_E$ increases → external agent does positive work.
• Bring opposite charges closer → $U_E$ decreases → electric force does positive work.
• Let charges move freely → system moves toward lower $U_E$.

On tests, they love giving you a charge released from rest and asking for its speed at some distance. Skip forces. Write conservation immediately.