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An isolated coductor, initially free fro...

An isolated coductor, initially free from charge, is charge by repeated conacts with a plate, which afrer each contact has a charge Q due to some mechanism.If q is the charge on the conductor after the first operation, prove that the maximum charge that can be give to the conductor in this way is `Qq//Q-q`.

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At each contact, the potential both becomes same. At the first contact, charges on the conductor and the plate are `Q-q` and `q`, respectively. So
`V_(1)=(Q-q)/C_(1)=q/C_(2)` or `C_(1)/C_(2)=(Q-q)/q` (i)
In the second contact, the charge transferred to the plate is `q_(2)`. Then
`V_(2)=(Q-q_(2))/C_(1)=(q+q_(2))/C_(2)` or `(Q-q)/q=(Q-q_(2))/(q+q_(2))`
or `(q+q_(2))(Q-q)=q(Q-q_(2))`
or `qQ-q^(2)+q_(2)Q-q_(2)q=qQ-qq_(2)`
or `q_(2)Q=q^(2)`
`:. q_(2)=q^(2)/Q`
The total charge transferred in a large number of contacts is
`q_(max)=q+q^(2)/Q+q^(3)/Q_(2)+q^(4)/Q_(3)+...+oo`
`=q/(1-q/Q)=(qQ)/(Q-q)`.
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