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A large sheet has uniform surface charge density sigma. A rod of total length 2L carries linear charge density +lambda on one half and -lambda on the other half. It is hinged at its midpoint O and makes an angle theta with the normal to the sheet. Find the torque on the rod.

1. sigma*lambda*L²*cos(theta) / (2*e0)
2. sigma*lambda*L*cos²(theta) / e0
3. sigma*lambda*L²*sin(theta) / (2*e0)
4. sigma*lambda*L*sin²(theta) / e0

Correct answer: sigma*lambda*L²*sin(theta) / (2*e0)

Solution

The field of the sheet is uniform: E = sigma/(2*e0). The rod is a dipole: each half (length L, charge q = lambda*L) has its charge centroid at L/2 from O, so the separation of +q and -q centroids is L, giving dipole moment p = q*L = lambda*L*L = lambda*L². Torque = p*E*sin(theta) = lambda*L² * (sigma/(2*e0)) * sin(theta) = sigma*lambda*L²*sin(theta)/(2*e0).

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