Question

A wheel with $10$ spokes each of length '$L$' m is rotated with a uniform angular velocity '$\omega$' in a plane normal to the magnetic field '$B$'. The emf induced between the axle and the rim of the wheel.

• A. $\frac{1}{2}N\omega B{L}^2$
• B. $\frac{1}{2}\omega B{L}^2$
• C. $\omega b{L}^2$
• D. $N\omega B{L}^2$

Answer 1

Answer: (B)

$\frac{1}{2}\omega B{L}^2$

Since the angular velocity is constant, the linear velocity
v increases linearly from $v=0$ to $v=L\omega$, from axle to rim of the wheel. $L$ is spokes length.
As such the mean velocity will be
$$v_a=\frac{L\omega }{2}$$
Emf induced by Faraday's law of Electromagnetic induction:
$$E=B{v}_{a}l=\frac{\omega B{L}^2}{2}$$
Emf across the spokes are in parallel so number of spokes are not relevant to the question. Option B is correct.