Cylindrical and spherical capacitor

The parallel-plate capacitor is only the simplest case. Changing the geometry of the plates — coaxial cylinders or concentric spheres — the physics stays the same (Gauss’s law, integrating the field to obtain $\Delta V$, then $C = Q/\Delta V$), but the form of the capacitance changes. It is worth seeing the two most important cases, since they recur in practice: coaxial cables are everywhere, and the spherical capacitor is the bridge back to the flat-plate geometry.

Cylindrical capacitor

Two coaxial conducting cylinders of radii $r_1 < r_2$ and length $L$ carry charges $\pm Q$. By Gauss’s theorem, between the cylinders the field is radial: $E = \frac{\lambda}{2\pi\varepsilon_0\,\rho} = \frac{Q}{2\pi\varepsilon_0 L\,\rho} \qquad (r_1 < \rho < r_2)$ The potential difference is found by integrating along the radius: $\Delta V = \int_{r_1}^{r_2} E\,\dd\rho = \frac{Q}{2\pi\varepsilon_0 L}\ln\!\frac{r_2}{r_1}$ from which the capacitance per unit length: $\frac{C}{L} = \frac{2\pi\varepsilon_0}{\ln(r_2/r_1)}$ This is the geometry of coaxial cables (TV, data networks) and some particle detectors.

Spherical capacitor

Two concentric shells of radii $r$ and $R$ (with $r < R$) carry charges $\pm Q$. By Gauss’s law $|E| = kQ/\rho^2$ between the spheres. Integrating gives $|\Delta V| = kQ(1/r - 1/R)$, from which: $\ev{C_{\text{sfer}} = 4\pi\varepsilon_0\,\frac{rR}{R-r}}$ In the limit where the gap is thin, $R - r \ll r$, this formula reduces exactly to that of the parallel-plate capacitor: the two nearly facing plates no longer “see” the curvature.

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Topics: Electric field and potential Concepts: Capacitance and capacitor Objects: Parallel-plate capacitor

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