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5.4: Concentric Spherical Capacitor

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 $$\frac{Q}{4\pi\epsilon}\left (\frac{1}{a}-\frac{1}{b}\right )$$, and so the capacitance is

$C=\frac{4\pi \epsilon}{\frac{1}{a}-\frac{1}{b}}.\label{5.4.1}$

If $$b \to \infty$$ we obtain for the capacitance of an isolated sphere of radius a:

$C=4\pi \epsilon a.\label{5.4.2}$

Exercise:  Calculate the capacitance of planet Earth, of radius 6.371 × 103 km, suspended in free space.  I make it 709 $$\mu\text{F}$$ -  which may be a bit smaller than you were expecting.