Consider a thin metallic sheet perpendicular to the plane of the paper moving with speed \(v\) in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities \(\sigma_1\) and \(\sigma_2\) are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects) :
Show Hint
- \(\sigma_1 = \epsilon_0 vB,\ \sigma_2 = -\epsilon_0 vB\)
- \(\sigma_1 = \frac{\epsilon_0 vB}{2},\ \sigma_2 = \frac{-\epsilon_0 vB}{2}\)
- \(\sigma_1 = \sigma_2 = \epsilon_0 vB\)
- \(\sigma_1 = \frac{-\epsilon_0 vB}{2},\ \sigma_2 = \frac{\epsilon_0 vB}{2}\)
The Correct Option is A
Solution and Explanation
Charges accumulate until the electric force balances the magnetic force: \(qE = qvB\), so \(E = vB\).
The two faces form a parallel-plate arrangement with surface charge densities \(+\sigma\) and \(-\sigma\): \(E = \sigma/\epsilon_0\). Hence \(\sigma = \epsilon_0 vB\).
\(\sigma_1 = \epsilon_0 vB,\ \sigma_2 = -\epsilon_0 vB\).
Answer: (1) \(\sigma_1 = \epsilon_0 vB,\ \sigma_2 = -\epsilon_0 vB\)
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