Question

A wheel with $10$ spokes each of length $L$ is rotated with a uniform angular velocity $\omega $ in a plane normal to the magnetic field $B$. The emf induced between the axis and the rim of the wheel.
(A) $\dfrac{1}{2}N\omega B{L^2}$
(B) $\dfrac{1}{2}\omega B{L^2}$
(C) $\omega B{L^2}$
(D) $N\omega B{L^2}$

Answer 1

Hint: Since we need to find the emf induced between the axis and the rim, we need to apply the faraday’s law of electromagnetic induction. In order to apply Faraday's law, at first we need to find the velocity with which the wheel moves.

Complete Step-By-Step Solution:
In the question, it is given that the wheel moves with constant angular velocity, this means that the angular velocity does not change with time. Therefore, the linear velocity of the wheel increases.
Let us consider that initially, the linear velocity was $0$
In the question, it is given that the angular velocity is $\omega $, thus, we know that the linear velocity as and angular velocity is related as:
${V_L} = r.{V_A}$
Where,
${V_L}$ is the linear velocity and ${V_A}$ is the angular velocity and $r$is the distance
Thus, putting the values we get:
$v = L\omega $
Where, $L$ is the spokes length
Thus, the average velocity is calculated as $ = \dfrac{{0 + L\omega }}{2}$
Now, from Faraday’s law of electromagnetic induction, we need that the emf induced can be calculated by multiplying the length, magnetic field and velocity.
So, we get:
$E = BL{V_L}$
Now, putting the values, we get:
$E = \dfrac{{\omega B{L^2}}}{2}$
Thus, the expression of emf induced in given as shown above. Since, the direction of induced emf is parallel so the number of spokes are not required while calculating.

Option (B) is correct.

Note:
We know, Faraday's law of electromagnetic induction, states that when the flux linkages in a coil is changed an emf is induced. The emf induced continues to exist as long as the change in magnetic flux linkages continues. The magnitude of induced emf is proportional to the change in magnetic flux.