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A metallic rod of length $l$ is tied to a string of length $2 l$ and made to rotate with angular speed $ω$ on a horizontal table with one end of the string fixed at centre. If there is a vertical magnetic field B in the region, the induced emf across the ends of the rod is

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a

$\frac{2 Bω l^{3}}{2}$

b

$\frac{3 Bω l^{3}}{2}$

c

$\frac{4 Bω l^{2}}{2}$

d

$\frac{5 Bω l^{2}}{2}$

answer is D.

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Detailed Solution

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$∵$ Induced emf is rate of change of magnetic flux

Question Image

$\begin{array}{l} e = \int Bvdv (∵ v = ωx) \Rightarrow e = \int_{2 t}^{3 t} (ωx) \cdot B \cdot dx \\ Using. \int x^{n} dx = \frac{x^{n + 1}}{n + 1} \Rightarrow e = Bω {[\frac{x^{2}}{2}]}_{2 l}^{3 l} \\ e = Bω \frac{[(3 l)^{2} - (2 l)^{2}]}{2} = \frac{5 B l^{2} ω}{2} \end{array}$

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