Spherical Capacitor

Capacitance of two concentric spherical shells of radii a (inner) and b (outer). For isolated sphere (b → ∞): C = 4πε₀a.

Setup

Inner shell radius $a$, outer shell radius $b$, charge $+Q$ on inner and $-Q$ on outer.

Field in the gap ($a < r < b$)

By Gauss's law (enclosed charge = $+Q$):

$$E = \frac{Q}{4\pi\varepsilon_0 r^2}$$

Field is zero for $r < a$ (inner shell) and $r > b$ (outer shell shields it).

Potential difference

$$V = V_a - V_b = \int_a^b E\, dr = \frac{Q}{4\pi\varepsilon_0}\int_a^b \frac{dr}{r^2} = \frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{a} - \frac{1}{b}\right) = \frac{Q(b-a)}{4\pi\varepsilon_0 ab}$$

Capacitance

$$C = \frac{Q}{V} = \frac{4\pi\varepsilon_0 ab}{b - a}$$ $$\boxed{C = 4\pi\varepsilon_0\frac{ab}{b-a}}$$

Limiting case: isolated sphere ($b \to \infty$)

$$C = 4\pi\varepsilon_0 \frac{a \cdot b}{b - a} \xrightarrow{b\to\infty} 4\pi\varepsilon_0 a$$

An isolated sphere of radius $a$ has capacitance $4\pi\varepsilon_0 a$.