Question

A wheel with $8$ metallic spokes each $50\mathrm{cm}$ long is rotated with a speed of $120\mathrm{rev}/\min$ in a plane normal to the horizontal component of the earth's magnetic field. The earth's magnetic field at the place is $0.4\mathrm{g}$ and the angle of ${60}^{\circ }$. Calculate the EMF induced between the axle and the rim of the wheel. How will the value of EMF be affected if the number of spokes was increased?

Answer 1

Step 1: Given information

The number of metallic spokes is $8$.

Length = $50\mathrm{cm}$

Speed = $120\mathrm{rev}/\min$

Angle = ${60}^{\circ }$

Step 2: To find

We have to find the EMF Induced.

Step 3: Calculation

In a uniform magnetic field, the induced EMF in a rod of length $\mathrm{l}$ with angular speed is given by,

$$\begin{gathered} \mathrm{E}=\frac{1}{2}{\mathrm{Bl}}^2\mathrm{\omega } \\ \mathrm{\omega }=\frac{2\mathrm{\pi N}}{\mathrm{t}} \\ \mathrm{E}=\frac{1}{2}\times 0.2\times {10}^{-4}\times {(0.5)}^2\times \frac{2\times 3.14\times 120}{60} \\ \mathrm{E}=3.14\times {10}^{-5}\mathrm{V} \end{gathered}$$

The EMF induced between the axle and the rim of the wheel is $\mathbf{3}\mathbf{.}\mathbf{14}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{}\mathbf{V}$.

Because the EMFs across the spokes are parallel, the induced EMF is independent of the number of spokes. As a result, the increased number of spokes has no effect on it.