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A conducting sphere A of radius $r_{A}$ carries a charge Q. Another conducting sphere B of radius $r_{B}$ is uncharged initially. Both the spheres are separated by a very large distance. Sphere A is connected with sphere B through a long conducting wire of zero resistance through a switch S as shown in the figure. At t=0 switch S is closed.
As soon as the current flow through the wire vanishes, switch S is opened, sphere A is again charged to a total charge Q and then switch S is again closed and the process is continued for infinite times. If q is the charge transferred to sphere B when the switch S is closed for the first time, then total charge on the sphere B after large number of such processes is

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$\frac{q Q}{Q - q}$

$Q \frac{r_{B}}{r_{A}}$

None of these

answer is B.

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Detailed Solution

$\frac{Q_{A}}{Q_{B}} = \frac{r_{A}}{r_{B \}}$

$I f Q_{T} i s t o t a l c h a r g e p r e s e n t o n t h e s p h e r e s (A + B) a t a n y t i m e$

$\Rightarrow Q_{B} = Q_{T} (\frac{r_{B}}{r_{A} + r_{B}}) = k Q_{T} (k = \frac{r_{B}}{r_{A} + r_{B}} = c o n s \tan t .)$

$1 s t c o n n e c t i o n \to q = Q k \Rightarrow k = q / Q$

$(i + 1) t h c o n n e c t i o n \to q_{i + 1} = (q_{i + Q}) k = (q_{i + Q}) q / Q$

$F o r i \to \infty, q_{i} = q_{i + 1} \Rightarrow q_{\infty} = (q_{\infty} + Q) (q / Q) \Rightarrow q_{\infty} = Q_{q} / (Q - q)$

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