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Question

A sphercial capacitor is made of two conducting spherical shells of radii a and b. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure (31-E27). Calculate the capacitance.

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Solution


We have two capacitors: one made by the shells a and c and the other made by the shells b and c.

The capacitance of the capacitor Cac is given by
Cac=4π∈0acK(c-a)

The capacitance of the capacitor Ccb is given by
Ccb=4π∈0bcKK(b-c)

The two capacitors are in series; thus, the equivalent capacitance is given by
1C=1Cac+1Ccb⇒1C=(c-a)4π∈0acK+(b-c)4π∈0cb⇒1C=b(c-a)+Ka(b-c)K4π∈0abc⇒C=K4π∈0abcb(c-a)+Ka(b-c)

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