In fig, CODF is a semicircular loop of a conducting wire of resistance R and radius r. It is placed in a uniform magnetic field B, which is directed into the page (perpendicular to the plane of the loop).
The loop is rotated with a constant angular speed $ω$ about an axis passing through the centre $O$ , and perpendicular to the page. Then the induced current in the wire loop is

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zero

${Br}^{2} ω / R$

$B π r^{2} ω / R$

${Br}^{2} ω / 2 R$

answer is C.

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Detailed Solution

$\begin{array}{l} The area swept by radius OC in one half circle is π r^{2} / 2 . \\ The flux change in time T / 2 is thus (π r^{2} B / 2) . \\ The induced emf is then e = π r^{2} B / T = B ω r^{2} / 2 [∵ T = \frac{2 π}{ω}] \\ The induced current is then I = e / R = B ω r^{2} / 2 R \end{array}$

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