Potential of Point Charges
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
A point charge creates a potential V = kq/r around it, with V = 0 infinitely far away. Potential is a scalar, so with several charges you just add numbers, keeping their signs. No components, no directions.
1. Adding potentials
2. V and E are different
V can be zero where E isn’t: at the midpoint between +q and −q, the potentials cancel but the fields add.
E can be zero where V isn’t: at the midpoint between two equal positive charges, the fields cancel but V = 2kq/r.
To bring a charge q in from far away, an outside agent does work W = qVfinal. If the result is negative, the field pulls the charge in on its own.
3. Example, then try it
Find the potential at P in the figure below.
From +3.0 nC: (9.0 × 10⁹)(3.0 × 10⁻⁹) ÷ 0.30 = +90 V. From −2.0 nC: (9.0 × 10⁹)(−2.0 × 10⁻⁹) ÷ 0.50 = −36 V.
VP = 90 − 36 = +54 V.
Using the figure below: how much work must an outside agent do to bring a +1.0 nC charge slowly from far away to P? Where on the line between the two charges is V = 0?
Work: W = qVP = (1.0 × 10⁻⁹)(54) = 5.4 × 10⁻⁸ J.
V = 0 between them: at distance x from the +3.0 nC charge, k(3.0)/x = k(2.0)/(0.40 − x).
Solve: 3.0(0.40 − x) = 2.0x → 1.2 = 5.0x → x = 0.24 m from the positive charge, closer to the smaller (negative) charge.
Answer: 5.4 × 10⁻⁸ J; V = 0 at 0.24 m from the positive charge.
4. Common mistakes
You move twice as far from a point charge. What happens to the potential there?
V = kq/r, so doubling r halves V. (The field, kq/r², drops to ¼.)
Ready to practice?
5 questions on Potential of Point Charges, each with a hint and an explanation for every wrong answer.