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A simple pendulum of length L with a bob of mass m hangs in front of a large nonconducting sheet carrying surface charge density sigma. The bob initially hangs vertically. A charge +q is suddenly placed on the bob. Find the maximum angle by which the string swings away from the vertical.

1. sin⁻¹( q*sigma / (2*m*g))
2. tan⁻¹( q*sigma / (2*m*g))
3. 2 * tan⁻¹( q*sigma / (2*m*g*epsilon0))
4. q*sigma / (2*m*g)

Correct answer: 2 * tan⁻¹( q*sigma / (2*m*g*epsilon0))

Solution

The field of an infinite charged sheet is E = sigma/(2*epsilon0), giving a constant horizontal force F = qE = q*sigma/(2*epsilon0) on the bob. The equilibrium deflection is theta0 = tan⁻¹(F/(mg)) = tan⁻¹( q*sigma/(2*m*g*epsilon0)). When released from rest at the vertical, the bob (like a pendulum under a constant transverse force) overshoots and reaches a maximum angle equal to twice the equilibrium angle: theta_max = 2*theta0 = 2*tan⁻¹( q*sigma/(2*m*g*epsilon0)).

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