Electric charges QA and QB have been put on two conducting concentric spherical shells – see the figure. The charges of each of the surfaces S1, S2, S3, S4 respectively are
0, QA, – QA, QA + QB
In the figure the inner thick shell carries QA and the outer thick shell carries QB. S1 and S2 are the inner and outer surfaces of the inner shell; S3 and S4 are the inner and outer surfaces of the outer shell.
In electrostatic equilibrium the field inside the conducting material is zero, so by Gauss's law the net charge enclosed by any Gaussian surface drawn inside the metal must be zero.
S1: the cavity inside the inner shell contains no charge, so a Gaussian surface in the inner shell's metal encloses only the charge on S1; hence S1 carries 0.
S2: the whole charge QA of the inner shell therefore sits on its outer surface S2.
S3: a Gaussian surface inside the outer shell's metal encloses QA + q3 = 0, so q3 = –QA (induced charge).
S4: the outer shell has total charge QB = q3 + q4, so q4 = QB + QA.
Thus the four surfaces carry 0, QA, –QA and QA + QB; a charge on an inner cavity wall or an equal splitting between surfaces would violate the zero-field condition inside the metal.
Hence, the answer is 0, QA, – QA, QA + QB.
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