A solid conducting sphere carrying a charge Q is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of -4Q, the new potential di…

Question

A solid conducting sphere carrying a charge Q is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge of -4Q, the new potential difference between the same two surfaces is

Accepted Answer

(2) V. In the gap between the sphere (radius a) and the shell (inner radius b) only the enclosed charge Q matters: E=(kQ)/(x²). So V=∫ₐ^b(kQ)/(x²)dx=kQ(1/a-1/b), and the region between the shell's inner and outer surfaces is conducting, so it adds nothing. Adding -4Q to the shell changes the charges on the shell's surfaces, but the charge enclosed by any Gaussian surface inside the gap is still just Q. The gap field, and hence the difference, is unchanged: the new potential difference is V.

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