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Question

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

A
2Bωl32
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B
3Bωl32
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C
4Bωl22
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D
5Bωl22
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Solution

The correct option is D 5Bωl22
Let us try to solve it by dimensional analysis first rather than going for conventional approach.
We know that the emf induced in a rod rotating with angular velocity ω in a region of magnetic field is given by
Emf = Bωl2
Now, if we see options (a) and (b) are not dimensionally correct so they must be omitted.

Option (c) can be written as 2Bωl2
2Bωl2=B(2ωl)l = B(ω×2l)l
But this would be the emf across metallic string and not across the metallic rod. Emf across the rod needs to be more than 2Bωl2 because length increases further after string.
So, option (d) is the correct answer.

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