Electric Potential Energy
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 6 practice questions.
Two charges have potential energy U = kq1q2/r (zero when they’re infinitely far apart). For a group of charges, add the energy of every pair. U is the work needed to assemble the charges; it turns into kinetic energy if they’re released.
1. Adding up pairs
2. What the sign means
U > 0: like charges. You had to push them together; released, they fly apart and gain kinetic energy.
U < 0: unlike charges. They attract; you’d have to do work to pull them apart.
A charge accelerated through ΔV gains K = |q|ΔV. An alpha particle (+2e) through 1000 V gains 2000 eV.
Count each pair once. Three charges have three pairs, four have six.
3. Example, then try it
Find the total potential energy of the three charges in the figure below (triangle side 0.30 m).
Positive pair: (9.0 × 10⁹)(2.0 × 10⁻⁶)(2.0 × 10⁻⁶) ÷ 0.30 = +0.12 J.
Each +/− pair: (9.0 × 10⁹)(2.0 × 10⁻⁶)(−3.0 × 10⁻⁶) ÷ 0.30 = −0.18 J. Total: 0.12 − 0.36 = −0.24 J.
Two protons are released from rest 1.0 × 10⁻¹⁰ m apart. Find each proton’s kinetic energy and speed once they’re far apart. (e = 1.6 × 10⁻¹⁹ C, mp = 1.67 × 10⁻²⁷ kg)
Starting energy: U = ke²/r = (9.0 × 10⁹)(1.6 × 10⁻¹⁹)² ÷ (1.0 × 10⁻¹⁰) ≈ 2.3 × 10⁻¹⁸ J.
Share it: far apart, U → 0. By symmetry each proton gets half: K ≈ 1.15 × 10⁻¹⁸ J (about 7.2 eV).
Speed: v = √(2K/m) = √(2 × 1.15 × 10⁻¹⁸ ÷ 1.67 × 10⁻²⁷) ≈ 3.7 × 10⁴ m/s.
Answer: about 1.15 × 10⁻¹⁸ J (7.2 eV) and 3.7 × 10⁴ m/s each.
4. Common mistakes
A positive charge is moved closer to another fixed positive charge. What happens to the system’s potential energy?
U = kq1q2/r is positive for like charges, and it grows as r shrinks. You have to do work to push them together.
Ready to practice?
6 questions on Electric Potential Energy, each with a hint and an explanation for every wrong answer.