A metallic rod of length ' $\ell$ ' is tied to a string of length $2 \ell$ and made to rotate with angular speed $\omega$ 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:

(C) $\frac{5 \mathrm{~B} \omega \ell^{2}}{2}$

(A) $\frac{3 \mathrm{~B} \omega \ell^{2}}{2}$

(B) $\frac{4 \mathrm{~B} \omega \ell^{2}}{2}$

(D) $\frac{2 \mathrm{~B} \omega \ell^{2}}{2}$

JEE Main · 2013 · 07 Apr 2013 · Offline · Electromagnetic Induction

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. If there is a vertical magnetic field ' B ' in the region, the e.m.f. induced across the ends of the rod is:

(A) $\frac{3 B ω ℓ ^{2}}{2}$
(B) $\frac{4 B ω ℓ ^{2}}{2}$
(C) $\frac{5 B ω ℓ ^{2}}{2}$ ✓ Correct
(D) $\frac{2 B ω ℓ ^{2}}{2}$

Answer: (C)

Solution

The rod is not pinned at the axis; it hangs at the far end of a string of length

2 ℓ

, so as it sweeps round at angular speed

ω

it spans radial distances from

2 ℓ

3 ℓ

. A small element at radius

x

moves with speed

ω x

and contributes

B ω x d x

to the emf, so integrating across the rod,

e = \int_{2 ℓ}^{3 ℓ} B ω x d x = \frac{B ω}{2} [(3 ℓ)^{2} - (2 ℓ)^{2}] = \frac{B ω}{2} (9 ℓ^{2} - 4 ℓ^{2}) = \frac{5 B ω ℓ ^{2}}{2}

JEE Main · 2013 · 07 Apr 2013 · Offline · Single correct option

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JEE Main · 2013 · 07 Apr 2013 · Offline · Electromagnetic Induction

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