General capacitance method, spherical capacitor

The capacitance of a capacitor is C = q/V, with q the magnitude of the charge on each plate and V the potential difference between them.

General method

1. Put +q on one conductor and −q on the other.
2. Use Gauss's law with a symmetric surface to find E between the conductors.
3. Compute V = V₊ − V₋ = ∫E·dl along a path from the positive to the negative conductor.
4. Find C = q/V; q cancels, leaving a purely geometric result.

Spherical capacitor

Concentric spheres: inner radius a carries +q, outer shell of inner radius b carries −q.

        .-----------------.
      /     outer (-q)      \
     |     .-----------.     |
     |    |  inner (+q) |    |
     |    |    radius a |    |
     |     '-----------'     |
      \       radius b       /
        '-----------------'

1. Gaussian sphere of radius r (a < r < b): E (4πr²) = q/ε0.
2. E = q / (4πε0 r²), radially outward.
3. V = ∫ₐᵇ E dr = (q/4πε0) ∫ₐᵇ dr/r² = (q/4πε0)(1/a − 1/b) = (q/4πε0)(b − a)/(ab)
4. C = q/V

C = 4πε0 ab / (b − a)

Special case: for b → ∞ (an isolated sphere), C = 4πε0 a. With a dielectric of constant k, C = 4πε0 k ab/(b − a).