Spherical Capacitor

The capacitance for spherical or cylindrical conductors can be obtained by evaluating the voltage difference between the conductors for a given charge on each. By applying Gauss' law to an charged conducting sphere, the electric field outside it is found to be

The voltage between the spheres can be found by integrating the electric field along a radial line:

From the definition of capacitance, the capacitance is

Does an isolated charged sphere have capacitance?
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Capacitor Concepts
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Isolated Sphere Capacitor?

An isolated charged conducting sphere has capacitance. Applications for such a capacitor may not be immediately evident, but it does illustrate that a charged sphere has stored some energy as a result of being charged. Taking the concentric sphere capacitance expression:

and taking the limits gives

Further confirmation of this comes from examining the potential of a charged conducting sphere:

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Capacitor Concepts
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