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Cylindrical Capacitor

Capacitance of two coaxial cylinders of length L, inner radius a, outer radius b. Derived from the field of an infinite line charge via Gauss's law.
Class 12
Derivation

Setup

Two coaxial cylinders of length LL, inner radius aa, outer radius bb. Charge +Q+Q on inner, −Q-Q on outer. Linear charge density: λ=Q/L\lambda = Q/L.

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

From Gauss's law applied to a coaxial cylinder of radius rr and length LL (formula ef15-field-infinite-line):

E=λ2πε0r=Q2πε0LrE = \frac{\lambda}{2\pi\varepsilon_0 r} = \frac{Q}{2\pi\varepsilon_0 L r}

Potential difference

V=∫abE dr=Q2πε0L∫abdrr=Q2πε0Lln⁡baV = \int_a^b E\, dr = \frac{Q}{2\pi\varepsilon_0 L}\int_a^b \frac{dr}{r} = \frac{Q}{2\pi\varepsilon_0 L}\ln\frac{b}{a}

Capacitance

C=QV=2πε0Lln⁡(b/a)C = \frac{Q}{V} = \frac{2\pi\varepsilon_0 L}{\ln(b/a)} C=2πε0Lln⁡(b/a)\boxed{C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)}}

Capacitance per unit length

CL=2πε0ln⁡(b/a)[F m−1]\frac{C}{L} = \frac{2\pi\varepsilon_0}{\ln(b/a)} \quad [\text{F m}^{-1}]

This is the standard specification for coaxial cables.

Note
Unlike the parallel plate and spherical cases, capacitance here depends logarithmically on the ratio $b/a$. Coaxial cables are a direct application: the dielectric between the conductors increases $C/L$ by a factor $K$.