Daily Physicist

Leon Cooper image courtesy of Kenneth C. Zirkel under the Creative Commons Attribution-ShareAlike 3.0 Unported License.

Today is the birthday of

• Leon Cooper (1930-2024).
• Steven Chu (1948-).
• Daniel C. Tsui (1939-).

Today’s Problem

Find the capacitance of a spherical capacitor with inner sphere radius \(a\) and outer sphere radius \(b\).

Answer

From Gauss’s law we can calculate the electric field of a spherically symmetric total charge \(Q\). Using a spherical surface surrounding \(Q\) we have

\[\oint \vec{E}\cdot\vec{dA} = E\oint dA = E4\pi r^2 = \frac{Q}{\epsilon_0}\]

So we have

\[E=\frac{Q}{4\pi\epsilon_0 r^2}\]

To get the voltage across the capacitor we integrate this from \(r=a\) to \(r=b\). We have

\[V=\frac{Q}{4\pi\epsilon_0}\int_a^b\frac{dr}{r^2}=\frac{Q}{4\pi\epsilon_0}\left(\frac{1}{a}-\frac{1}{b}\right)\]

The capacitance is then

\[C=\frac{Q}{V}=\frac{4\pi\epsilon_0}{\frac{1}{a}-\frac{1}{b}}\]