A metallic rod of length 'l ' is tied to a string of length 2λ 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:

Engineering

Physics

Motional EMF

Question

$\frac{4 {Bωℓ}^{2}}{2}$

$\frac{3 {Bωℓ}^{2}}{2}$

$\frac{2 {Bωℓ}^{2}}{2}$

$\frac{5 {Bωℓ}^{2}}{2}$

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$d ε = (\vec{V} \times \vec{B}) d \vec{ℓ} = \int_{2 ℓ}^{3 ℓ} ωxBdx$

$= ω B {(\frac{x^{2}}{2})}_{2 ℓ}^{3 ℓ}$

$= \frac{ωB}{2} (9 ℓ^{2} - 4 ℓ^{2}) = \frac{5 {ωBℓ}^{2}}{2}$