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A rod OP of length L rotates in the OXY plane about OZ with a constant angular velocity $ω_{0}$ . A uniform magnetic field $B_{0} \hat{k}$ exists in the region. The emf induced across points PQ (PQ =a) is

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$\frac{ω_{0} B_{0} a (2 L - a)}{2}$

$\frac{ω_{0} B_{0} a^{2}}{2}$

$\frac{ω_{0} B_{0} (L^{2} + a^{2})}{2}$

$\frac{ω_{0} B_{0} (L^{2} - a^{2})}{2}$

answer is C.

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Detailed Solution

$dE = {vB}_{0} dr = ω_{0} B_{0} rdr E_{FQ} = ω_{0} B_{0} \int_{L - a}^{L} rdr = \frac{ω_{0} B_{0} a (2 L - a)}{2}$

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