Calculating the Capacitance — Question 2

Question 2

A spherical capacitor approaches the self-capacitance of one sphere

Concentric conducting spheres have radii a and b>a and charges +Q and -Q. The gap is vacuum; $k=1/(4\pi\epsilon_0)$.

1. Derive their capacitance from the radial field.

2. Take b to infinity and then compare the exact capacitance with that of a narrow gap b=a+d.

Question 2 - Solution

A spherical capacitor approaches the self-capacitance of one sphere

(a) Method and calculation

$$V_a-V_b=\int_a^b\frac{Q}{4\pi\epsilon_0r^2}dr=\frac{Q}{4\pi\epsilon_0}(1/a-1/b).$$$$C=4\pi\epsilon_0\frac{ab}{b-a}.$$

(b) Interpretation and further calculation

$$b\to\infty:\ C\to4\pi\epsilon_0a.$$$$b=a+d:\ C=\frac{4\pi\epsilon_0a^2}{d}(1+d/a),$$with planar leading term $\epsilon_0(4\pi a^2)/d$ for d much smaller than a.

Physical check

The isolated-sphere limit uses infinity as the second terminal; the small-gap limit uses the inner sphere area.