Two conducting spheres of radius $r$ and $2 r$ at very large separation. Each sphere is having charge $Q$ . These spheres are connected with a conducting wire of resistance $R$ . Then, the current flowing in the wire at $t$ is

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$I = \frac{Q}{8 {πε}_{0} rR} e^{- \frac{3 t}{4 {πε}_{0} rR}}$

$I = \frac{Q}{4 {πε}_{0} rR} e^{- \frac{3 t}{8 {πε}_{0} rR}}$

$I = \frac{Q}{8 {πε}_{0} rR} e^{- \frac{3 t}{8 {πε}_{0} rR}}$

$I = \frac{Q}{4 {πε}_{0} rR} e^{- \frac{3 t}{4 {πε}_{0} rR}}$

answer is B.

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Detailed Solution

$\begin{aligned} \frac{K (2 Q - q)}{r} = \frac{kq}{2 r} = \frac{dq}{dt} R \\ \int \frac{K}{rR} dt = \int_{Q}^{q} \frac{dq}{2 Q - \frac{3 q}{2}} \end{aligned}$
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$4 Q - 3 q = Q e^{- \frac{3 k t}{2 r R}}$
Differentiating w.r.t time
$- \frac{3 dq}{dt} = - \frac{3 KQ}{2 rR} e^{- \frac{3 kt}{2 RR}}$
$I = \frac{Q}{8 {πε}_{0} rR} e^{- \frac{3 t}{8 π_{0} rR}}$