Short on time? This page is the recap. For the derivation, Figure 1 and the worked examples, use the full study guide. The calculus derivation is part of the calculus-based course.
Recap
- The electric potential energy of two point charges is the work an external force does to bring them slowly from infinitely far apart to their present separation.
- U belongs to the system of charges, not to one charge.
- The electric force is conservative, so U depends only on the separation, not on the path.
- Zero of U: the charges infinitely far apart.
- For uniformly charged spheres that do not overlap, use the centre-to-centre distance.
- k = 8.99 × 10⁹ N·m²/C²; e = 1.60 × 10⁻¹⁹ C.
Key relationships
| Idea | Relationship |
|---|---|
| Pair of point charges | U = q₁q₂/(4πε₀r) = kq₁q₂/r |
| Derivation | U(R) = ∫_∞^R (−kq₁q₂/r²) dr |
| Force from energy | F_r = −dU/dr |
| System of N charges | U_total = Σ kq_iq_j/r_ij over the N(N − 1)/2 pairs |
| Work, start and end at rest | W_ext = ΔU; W_electric = −ΔU |
| Scaling | U ∝ 1/r; F ∝ 1/r² |
Assumptions
- Point charges, or spherically symmetric charges treated from their centres.
- Charges moved slowly, so no kinetic energy is gained (changes in kinetic energy are Topic 9.3).
- No other forces store energy in the system.
Mistakes to avoid
- Leaving out signs. Substitute q₁ and q₂ with their signs.
- Using 1/r². That is the force; energy goes as 1/r.
- Double-counting pairs. List each pair once.
- Reading force from height. Force is the negative slope of U(r).
- Mixing up external and electric work. They have opposite signs.
Quick self-check
- How many pairs do 5 charges make? (10)
- Find U for +1.0 nC and −1.0 nC at 0.10 m. (−8.99 × 10⁻⁸ J)
- A pair’s separation doubles. What happens to U and to F? (U halves; F falls to a quarter)
- Like charges are released from rest. Does U rise or fall? (It falls towards zero as they move apart)
- Three equal charges +q sit at the corners of an equilateral triangle of side a. What is U_total? (3kq²/a, one term for each of the 3 pairs)
- Removing one charge from a system: which pair terms change? (Only the pairs that include the charge being removed)
Next: practice questions.