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Three concentric thin metal shells A, B and C have radii a, 2a and 3a respectively. Shell B is earthed and shell C is given a charge Q. Shell A is initially uncharged and isolated. Now shell C is connected to shell A by a thin wire. What is the final charge on shell B?
- -4Q/13
- -8Q/11
- -5Q/3
- -3Q/7
Correct answer: -8Q/11
Solution
When C (radius 3a) is wired to A (radius a) they become one conductor at a common potential. Let the charge on A be q_A and on C be q_C with q_A + q_C = Q (charge supplied originally to C is conserved on the A-C system; A started neutral). B is earthed so its potential is zero and it acquires an induced charge q_B. Writing the potentials of A, B, C from the superposition of all three shells and imposing V_B = 0 together with V_A = V_C and q_A + q_C = Q, solving the linear system gives the induced charge on the earthed middle shell as q_B = -8Q/11.
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