A fixed smooth non-conducting track is a quarter circle of radius R in a vertical plane. A small metal ball A is fixed at the bottom. An identical ball B can move on the track. Initially A and B are in contact and a total charge Q is given to A, which is shared equally, so each ball gets Q/2. Ball B finally comes to rest at a point where radius OB makes angle θ (<90°) with the downward vertical. Each ball has mass m. The charge Q is:
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Question

A fixed smooth non-conducting track is a quarter circle of radius R in a vertical plane. A small metal ball A is fixed at the bottom. An identical ball B can move on the track. Initially A and B are in contact and a total charge Q is given to A, which is shared equally, so each ball gets Q/2. Ball B finally comes to rest at a point where radius OB makes angle θ (<90°) with the downward vertical. Each ball has mass m. The charge Q is:

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G1 Physics · Accepted Answer

The separation between A and B is the chord AB = 2R sin(θ/2). Each ball carries charge Q/2, so F e F_e F e ​ = k(Q/2)² / [2R sin(θ/2)]² = kQ² / [16R² sin²(θ/2)]. Tangential equilibrium of B gives F e F_e F e ​ cos(θ/2) = mg sinθ = 2mg sin(θ/2)cos(θ/2). Thus, F e F_e F e ​ = 2mg sin(θ/2). Therefore, kQ²/[16R² sin²(θ/2)] = 2mg sin(θ/2), Q² = 32mgR² sin³(θ/2)/k. Using k = 1/(4πε₀), Q = 8R√(2πε₀mg) · sin³ᐟ²(θ/2).