Two metal spheres of radius R and 2R are placed at very large distance from each other and they are connected by a coil of inductance L as shown in figure. One of the spheres (radius R) is given charge Q. At what time after closing switch S does charge on this sphere becomes zero?

\infty

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0.0

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π \sqrt{\frac{8 π ϵ_{0} L R}{3}}

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π \sqrt{2 π ϵ_{0} L R}

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Solution

The correct option is C $π \sqrt{\frac{8 π ϵ_{0} L R}{3}}$

$V_{1} - V_{2} = L \frac{d i}{d t} \frac{k (Q - q)}{R} - \frac{k q}{2 R} = L \frac{d i}{d t} \frac{k Q}{R} - \frac{3 k q}{2 R} = L \frac{d i}{d t} \Rightarrow \frac{- 3 k}{2 R L} i = \frac{d^{2} i}{d t^{2}} ∴ w = \sqrt{\frac{3 k}{2 R L}} t = \frac{T}{2} = \frac{1}{2} \times 2 π \sqrt{\frac{2 R L \times 4 π ϵ_{0}}{3}} = π \sqrt{\frac{8 R L π ϵ_{0}}{3}}$

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