Short on time? This page is the recap. For explanations, Figures 1 and 2 and the worked examples, use the full study guide.
Recap
- Electric potential energy U_E of two point charges = the work an external force does to bring them slowly from infinitely far apart to their positions.
- U_E belongs to the system, not to one charge. It is a scalar with a sign.
- Like charges repel: you push them together, so U_E > 0. Unlike charges attract: U_E < 0, and you must add energy to separate them.
- On a U_E–r graph both curves approach zero at large r and become steep at small r. Like and unlike curves are mirror images.
- For three or four charges, the total is the sum over every pair.
- If charges start and end at rest, the external work equals the change in U_E.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| U_E = k q₁ q₂ / r | k = 9.0 × 10⁹ N·m²/C², q in C (with sign), r in m, U_E in J | find the energy of one pair |
| U_E,total = U₁₂ + U₁₃ + U₂₃ (+ …) | one term per pair | find the energy of 3 or 4 charges |
| W_ext = ΔU_E | charges at rest at start and end | find the work to assemble or rearrange charges |
| U_E ∝ q₁q₂/r | ratio method | predict factor changes |
| Number of charges | 2 | 3 | 4 |
|---|---|---|---|
| Number of pairs | 1 | 3 | 6 |
Assumptions behind the numbers
- Charges are point charges (or small spheres treated as points).
- U_E = 0 when the charges are infinitely far apart.
- Charges are moved slowly, so no kinetic energy is gained.
Mistakes to avoid
- Dropping the signs of the charges. An attracting pair must give U_E < 0.
- Using 1/r². Energy goes as 1/r; force goes as 1/r².
- Missing pairs (such as square diagonals) or counting a pair twice.
- Giving U_E to one charge. It belongs to the pair.
- Reading “negative” as “impossible”. It means less energy than at infinite separation.
- Confusing U_E (J) with potential V (Topic 10.5, volts).
Quick self-check
- What is U_E for +2.0 nC and −3.0 nC that are 0.020 m apart? (−2.7 × 10⁻⁶ J)
- How many pairs do four charges make? (6)
- A pair has U_E = +0.50 J. The separation is tripled. What is the new U_E? (+0.167 J)
- Is the external work positive or negative when you pull an electron away from a proton? (Positive)
Next: practice questions.