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Wize University Physics Textbook (Master) > Electric Potential and Potential Energy

Capacitance (spherical and cylindrical)

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Example: Capacitance of a Spherical Capacitor


Consider a solid sphere and a spherical shell which are concentric. The inner sphere has charge Q and the outer spherical shell has charge -Q. The inner sphere has radius a, and the outer spherical shell has an inner radius of b. What is the capacitance of the spheres?

To find capacitance, we need the voltage. To get the voltage between the two spheres, we integrate over the electric field, using the inner and outer radii as the limits of integration. In this case, the electric field is the same as that for a point charge (you can prove that with Gauss's law). That is, E=Q4πϵoR2E=\frac{Q}{4\pi\epsilon_oR^2}.
V=∫abEdr=∫abQ4πϵor2dr=Q4πϵo∫ab1r2dr=Q4πϵo(1a−1b)\begin{aligned} V&=\int_a^bEdr\\ &=\int_a^b\frac{Q}{4\pi\epsilon_or^2}dr\\ &= \frac{Q}{4\pi\epsilon_o}\int_a^b\frac{1}{r^2}dr\\ &=\frac{Q}{4\pi\epsilon_o}\left(\frac{1}{a}-\frac{1}{b}\right) \end{aligned}
Capacitance:
C=QV=QQ4πϵo(1a−1b)=4πϵo1a−1bC=\frac{Q}{V}=\frac{Q}{\frac{Q}{4\pi\epsilon_o}\left(\frac{1}{a}-\frac{1}{b}\right)}=\frac{4\pi\epsilon_o}{\frac{1}{a}-\frac{1}{b}}
Like always, capacitance is purely a geometric property - the only variables that matter here are the radii of the two spheres.

Practice: Capacitance of a Cylindrical Capacitor


What is the capacitance per unit length of two very long concentric cylinders with surface charges +Q and -Q? The radii of the two cylinders are R1 and R2.