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A solid conducting sphere of radius a carries a net positive charge of 2Q. A conducting spherical shell (inner radius b, outer radius c) is concentric with the solid sphere and carries a net charge of -Q. What are the surface charge densities on the inner and outer surfaces of the shell?
- Inner: -2Q/(4*pi*b²), Outer: Q/(4*pi*c²)
- Inner: -Q/(4*pi*b²), Outer: Q/(4*pi*c²)
- Inner: 0, Outer: Q/(4*pi*c²)
- Inner: 2Q/(4*pi*b²), Outer: -Q/(4*pi*c²)
Correct answer: Inner: -2Q/(4*pi*b²), Outer: Q/(4*pi*c²)
Solution
The solid sphere has charge +2Q. Inside the conducting material of the shell, E = 0. Applying Gauss's law with a surface inside the conductor: charge enclosed = 0. So the inner surface of the shell has charge -2Q (to cancel +2Q of the solid sphere). The shell has total charge -Q. Therefore charge on outer surface = -Q - (-2Q) = +Q. Surface charge density: inner = -2Q/(4*pi*b²), outer = +Q/(4*pi*c²).
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