Capacitance
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
Capacitance is charge stored per volt: C = Q/ΔV. It depends only on the conductors’ geometry (and any dielectric). To find it: assume charge ±Q, use Gauss’s law for E, integrate to get ΔV, then divide.
1. Three standard shapes
2. The method
Spherical capacitor: between the spheres E = kQ/r², so ΔV = ∫ab kQ/r² dr = kQ(1/a − 1/b) = kQ(b − a)/(ab). Then C = Q/ΔV = ab/[k(b − a)] = 4πε0ab/(b − a).
Q cancels: capacitance doesn’t depend on how much charge is stored.
In series, each capacitor carries the same charge (charge can’t cross the gap between them). In parallel, each has the full voltage.
Double the voltage and Q doubles; C doesn’t change.
3. Example, then try it
A spherical capacitor has an inner radius of 0.10 m and an outer radius of 0.20 m. Find its capacitance.
C = ab/[k(b − a)] = (0.10)(0.20) ÷ [(9.0 × 10⁹)(0.10)] ≈ 2.2 × 10⁻¹¹ F (22 pF).
In the circuit below, a 6 µF capacitor is in series with a parallel pair of 2 µF and 4 µF, on a 12 V battery. Find the equivalent capacitance, the charge from the battery, and the charge and voltage on each capacitor.
Parallel pair: 2 + 4 = 6 µF.
Series with 6 µF: 1/C = 1/6 + 1/6, so C = 3 µF. Q = CV = (3)(12) = 36 µC.
6 µF capacitor: carries 36 µC, so ΔV = 36/6 = 6 V. The pair gets the other 6 V.
Pair: 2 µF holds (2)(6) = 12 µC; 4 µF holds (4)(6) = 24 µC. Check: 12 + 24 = 36 ✓.
Answer: 3 µF; 36 µC; 6 V across each part; 12 µC and 24 µC in the pair.
4. Common mistakes
Two identical capacitors, each C, are connected in series. What is their equivalent capacitance?
1/Ceq = 1/C + 1/C = 2/C, so Ceq = C/2. (In parallel it would be 2C.)
Ready to practice?
5 questions on Capacitance, each with a hint and an explanation for every wrong answer.