UY1: Capacitance Of Spherical Capacitor

A spherical capacitor consists of two concentric conducting shells separated by an insulating region.

1) At a glance

• Prerequisites: Gauss’s law (Gauss’s Law (Simple Version)) and potential from field (Electric Potential)
• Outcomes: derive the capacitance of two concentric spherical conductors and interpret useful limits
• Key result: C = 4πε₀ab/(b-a) (a < b)
• Common trap: integrating potential with wrong limits/signs
• Physical check: as gap shrinks, capacitance increases

Motivation / intuition

Spherical geometry is where Gauss’s law is at its cleanest: symmetry gives E(r) immediately. Once you can integrate E to get Δ V, the capacitance drops out as a pure geometry constant.

2) Setup

Inner conductor radius a carries + Q, outer conductor inner radius b carries -Q, with a < b.

Field exists only for a < r < b (ideal electrostatic case).

3) Core derivation/explanation

For a < r < b, spherical Gaussian surface gives E(4π r²) = Q/ε₀ E(r) = (1/4πε₀)Q/r²

Potential difference magnitude between shells: V_ab = V(a)-V(b) = ∫ₐ^b E dr V_ab = ∫ₐ^b(Q/4πε₀r²)dr = (Q/4πε₀)(1/a-1/b)

Therefore, C = Q/V_ab = 4πε₀ab/(b-a)

4) Worked example(s)

Let a = 0.040 m and b = 0.060 m.

C = 4πε₀ab/(b-a) = 4π(8.854 × 10⁻¹²)(0.040)(0.060)/0.020 C ≈ 1.34 × 10⁻¹¹ F = 13.4 pF

5) Practice set (with hints + answers)

1. If b → ∞, what does spherical-capacitor formula reduce to?
2. If b-a decreases, does C increase or decrease?
3. Why is field outside outer conductor ideally zero?

Hints

1. Take the limit in 4πε₀ab/(b-a).
2. Inspect denominator.
3. Use spherical symmetry plus zero net enclosed charge outside both shells.

Answers

1. C → 4πε₀a (isolated sphere).
2. Increases.
3. For r > b, spherical symmetry makes E uniform on a Gaussian sphere and Q_encl = 0, so E = 0.

6) Summary + next steps

• Gauss’s law gives E(r) in the gap, then integration gives Δ V.
• The concentric spherical capacitor has C = 4πε₀ab/(b-a)
• Limiting cases provide good checks against algebra errors.

Next: Capacitance Of A Cylindrical Capacitor Previous: Energy Stored In Capacitors Back To Electromagnetism (UY1)