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Two thin concentric conducting spherical shells have radii r and 3r. The outer shell carries charge q while the inner shell is initially uncharged (neutral). The inner shell is then connected to earth through a key K. How much charge flows from the inner shell to earth once K is closed?

1. -q/3
2. q/3
3. 3q
4. -3q

Correct answer: -q/3

Solution

When the inner shell (radius r) is earthed, its potential must be zero. The potential at the inner shell is due to its own charge q' (at radius r) plus the outer shell charge q (at radius 3r): V = k*q'/r + k*q/(3r) = 0. Solving gives q' = -q/3. Since the inner shell started neutral (0), the charge that flowed from inner shell to earth is the negative of what remained: charge to earth = 0 - q' wait, the charge appearing on the inner shell is -q/3, which came from earth. The net charge flowing from inner shell to earth is q' - 0 evaluated as the charge that left earth; conventionally the charge that flows from the inner shell to earth equals -(final charge) measured... the induced final charge on inner shell is -q/3, so the charge flowing TO the inner shell from earth is -q/3, equivalently charge -q/3 flows from inner shell to earth.

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