Capacitance of cylindrical capacitor

A capacitor's capacitance is C = q/V, where q is the magnitude of charge on either plate and V is the potential difference between the plates. It depends only on geometry and the medium.

General method

1. Place charge +q on one conductor and −q on the other.
2. Choose a Gaussian surface of suitable symmetry and use Gauss's law (∮E·dA = q/ε0) to find E between the conductors.
3. Find the potential difference V = V₊ − V₋ = ∫E·dl, integrated along a path from the positive to the negative conductor.
4. Substitute in C = q/V; q cancels, leaving C in terms of dimensions only.

Cylindrical capacitor

Two coaxial cylinders, inner radius a (charge +q), outer radius b (charge −q), length L ≫ (b − a), so edge effects are ignored.

      outer cylinder (-q), radius b
     ______________________________
    |   ______________________     |
    |  |  inner (+q), radius a |    |
    |  |<-- Gaussian surface r -->  |
    |  |______________________|    |
    |______________________________|
                 length L

1. Take a coaxial Gaussian cylinder of radius r (a < r < b) and length L. Flux passes only through the curved surface: E(2πrL) = q/ε0.
2. E = q/(2πε0 L r), directed radially outward.
3. V = ∫ₐᵇ E dr = (q/(2πε0 L)) ∫ₐᵇ dr/r = (q/(2πε0 L)) ln(b/a)
4. C = q/V

Answer: C = 2πε0 L / ln(b/a)

Capacitance per unit length is C/L = 2πε0/ln(b/a). With a dielectric of constant k, replace ε0 by kε0. It increases when the gap (b − a) is made small.