First slide

Magnetic flux and induced emf

Question

A line charge I per unit length is pasted uniformly onto the rim of a wheel of mass m and radius R. The wheel has light non-conducting spokes and is free to rotate about a
vertical axis as shown in figure. A uniform magnetic field B exists as shown in figure. What is the angular velocity of the wheel when the field is suddenly switched off?

Moderate

A

$\frac{2 π λ a^{2} B}{m R}$
B

$\frac{π λ a^{2} B}{m R}$
C

$\frac{3 π λ a^{2} B}{m R}$
D

$\frac{π λ a^{2} B}{2 m R}$

Solution

$\begin{array}{l} τ = \frac{Δ L}{Δ t} = \frac{I ω}{Δ t} = \frac{m R^{2} ω}{Δ t} \\ but τ = λ (2 π R) E R = λ (π a^{2} \frac{Δ B}{Δ t}) R = \frac{λ π a^{2} B R}{Δ t} \\ \Rightarrow \frac{m R^{2} ω}{Δ t} = \frac{λ π a^{2} B R}{Δ t} \Rightarrow ω = \frac{λ π a^{2} B}{m R} \end{array}$

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