Question

A wheel with $20$ metallic spokes each $1\, m$ long is rotated with a speed of $120\, rpm$ in a plane perpendicular to a magnetic field of $0.4\, G$. The induced emf between the axle and rim of the wheel will be, $(1 G = 10^{-4}\, T)$

• A. $2.51 \times 10^{-4}\,V$
• B. $2.51 \times 10^{-5}\, V$
• C. $4.0\times 10^{-5}\, V$
• D. $2.51 \,V$

Answer 1

The Correct Option is A

To find the induced electromotive force (emf) between the axle and rim of the wheel, we can use the concept of electromagnetic induction. The formula for the induced emf (\(E\)) in a rotating wheel with spokes perpendicular to the magnetic field is given by:

\(E = \frac{1}{2} B \omega r^2\)

where:

• \(B\) is the magnetic field strength in tesla.
• \(\omega\) is the angular velocity in radians per second.
• \(r\) is the length of each spoke in meters.

Given:

• \(B = 0.4\, G = 0.4 \times 10^{-4}\, T\)
• Number of revolutions per minute (rpm) = 120. We need to convert this to radians per second:

\(\omega = 120\, \text{rpm} \times \frac{2\pi \, \text{rad}}{1\, \text{rev}} \times \frac{1\, \text{min}}{60\, \text{sec}} = 4\pi\, \text{rad/sec}\)

• \(r = 1\, m\)

Substituting these values into the formula:

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

\(E = 0.5 \times 0.4 \times 10^{-4} \times 4\pi\)

\(E = 0.8\pi \times 10^{-4}\, V\)

Calculating the numerical value:

\(\pi \approx 3.14159\)

\(E \approx 0.8 \times 3.14159 \times 10^{-4} \, V = 2.51327 \times 10^{-4}\, V\)

So, the induced emf is approximately:

\(E \approx 2.51 \times 10^{-4}\, V\)

Thus, the correct option is \(2.51 \times 10^{-4}\, V\).