5.4: Concentric Spherical Capacitor
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Unlike the coaxial cylindrical capacitor, I don’t know of any very obvious practical application, nor quite how you would construct one and connect the two spheres to a battery, but let’s go ahead all the same. Figure V.4 will do just as well for this one.
The two spheres are of inner and outer radii a and b, with a potential difference V between them, with charges +Q and −Q on the inner and outer spheres respectively. The potential difference between the two spheres is then Q4πϵ(1a−1b), and so the capacitance is
C=4πϵ1a−1b.
If b→∞ we obtain for the capacitance of an isolated sphere of radius a:
C=4πϵa.
Exercise: Calculate the capacitance of planet Earth, of radius 6.371 × 103 km, suspended in free space. I make it 709 μF - which may be a bit smaller than you were expecting.