Q.

Two small metal spheres having equal charge and mass are suspended from some point on the ceiling of a damp room with silk threads of equal length. Let centre to centre distance between sphere be x, x << l, l is length of silk thread. Due to ionization of medium, charge leaks off from each sphere and they keep on coming closer to each other at a constant rate.Let their approach velocity y varies as v∝x−1/2If mass of each sphere is m then the rate at which charge varies with respect to time is dqdt∝N22πε0mgl. The value of N is _________.

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answer is 3.

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Detailed Solution

At equilibrium, q24πε0x2cos⁡θ=mgsin⁡θ For small θ,cos⁡θ→1 and sin⁡θ≈θ=x/2l⇒q24πε0x2=mgx2l⇒q=x3/2⋅2πε0mgl⇒ dqq=32x3/2⋅2πε0mgl⋅dxdt Put v=dxdt∝x1/2⇒dqdt∝322πε0mgl Hence, N=3
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