Two identical positive charges q are placed in a smooth horizontal circular groove at points separated by 90° at the centre. A third charge is placed in the same groove so that all three charges are in tangential equilibrium immediately after release. If the third charge is negative, where must it be placed and what must be its magnitude?
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Question

Two identical positive charges q are placed in a smooth horizontal circular groove at points separated by 90° at the centre. A third charge is placed in the same groove so that all three charges are in tangential equilibrium immediately after release. If the third charge is negative, where must it be placed and what must be its magnitude?

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G1 Physics · Accepted Answer

Let the original charges be at angular positions 0° and 90°. Symmetry requires a negative third charge to lie at 45°, the midpoint of the minor arc. Tangential equilibrium of either original charge gives |Q|/q = (2-√2)³ᐟ² / 2. Numerically, |Q| ≈ 0.224q. Hence the required charge is Q = -[(2-√2)³ᐟ²/2]q.