Question

If a wheel with 24 metallic spokes each \( 40 \) cm long is rotated with a speed of \( 180 \) rev/min in a plane normal to the horizontal component of Earth's magnetic field, the emf induced between the axle and the rim of the wheel is \( E \). If the number of spokes is made \( 12 \) and the wheel is rotated with a speed of \( 90 \) rev/min in the same field, the induced emf is:

• A. \( E \)
• B. \( 2E \)
• C. \( 4E \)
• D. \( 0.5E \)

Hint:

For rotating metallic wheels, induced emf is proportional to angular velocity, meaning halving the rotation speed halves the induced emf.

Answer 1

The Correct Option is D

Step 1: Induced EMF in a Rotating Wheel The induced emf between the axle and rim is given by: \[ E = B L \omega \] where: - \( B \) = Magnetic field strength, - \( L \) = Length of the spoke, - \( \omega \) = Angular velocity. Step 2: Comparing Initial and Final Cases Since number of spokes does not affect the induced emf, we compare the effect of rotation speed. Initial conditions: \[ E = B L \omega_{\text{initial}} \] where \( \omega_{\text{initial}} = 180 \) rev/min. Final conditions: \[ E' = B L \omega_{\text{final}} \] Since \( \omega_{\text{final}} = 90 \) rev/min, \[ E' = \frac{90}{180} E = 0.5E \] Conclusion Thus, the correct answer is: \[ 0.5E \]