Question

A metal wheel with $8$ metallic spokes, each $40\text{cm}$ long, is rotated at a speed of $120\text{rev/min}$ in a plane perpendicular to earth's magnetic field of $0.3\times {10}^{-4}\text{T}.$ Find the magnitude of the induced emf between the axis and the rim of the wheel (approximately).

• A. $3\times {10}^{-5}\text{V}$
• B. $24\times {10}^{-5}\text{V}$
• C. $0.37\times {10}^{-5}\text{V}$
• D. $6\times {10}^{-5}\text{V}$

Answer 1

The correct option is A $3\times {10}^{-5}\text{V}$

The emf developed between the ends of a spoke is given by

$E=\frac{1}{2}B\omega l^2$

Here $\omega =120\frac{\text{rev}}{\text{min}}=120\times \frac{2\pi }{60}\text{rad/s}$

$\Rightarrow \omega =4\pi \text{rad/s}$

$\therefore E=\frac{1}{2}\times 0.3\times {10}^{-4}\times 4\pi \times (0.4{)}^2$

$\Rightarrow E=0.096\pi \times {10}^{-4}$

$\Rightarrow E\approx 3\times {10}^{-5}\text{V}$

The same emf is induced between the ends of each spoke because all the spokes in the wheel are joined in parallel.

Hence,
$\text{emf between rim and axle = emf across each spoke}$

$=3\times {10}^{-5}\text{V}$

Hence, option $\left(\text{A}\right)$ is correct.