Question

The plane of a circular coil of resistance 7.5 \( \Omega \) is placed perpendicular to a uniform magnetic field. The flux \( \phi \) (in weber) through the coil varies with time t (in second) as \( \phi = 2t^2 + 3t - 2 \). The induced power in the coil at time t = 3s is

• A. 7.5 W
• B. 15 W
• C. 30 W
• D. 20 W

Hint:

Remember that power depends on the square of the EMF. Always calculate the EMF first by differentiating the flux equation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Induced power in a coil due to changing magnetic flux is given by \( P = \frac{e^2}{R} \), where \( e \) is the induced EMF and \( R \) is the resistance. The induced EMF is the rate of change of magnetic flux, \( e = -\frac{d\phi}{dt} \).
Step 2: Key Formula or Approach:
1. Induced EMF: \( |e| = \left| \frac{d\phi}{dt} \right| \) 2. Induced Power: \( P = \frac{e^2}{R} \)
Step 3: Detailed Explanation:
Given flux equation: \( \phi = 2t^2 + 3t - 2 \). Differentiate \( \phi \) with respect to \( t \) to find the induced EMF \( e \): \[ e = \frac{d}{dt}(2t^2 + 3t - 2) = 4t + 3 \] At \( t = 3 \) s: \[ e = 4(3) + 3 = 12 + 3 = 15 \, \text{V} \] Given resistance \( R = 7.5 \, \Omega \). Calculate Power \( P \): \[ P = \frac{e^2}{R} = \frac{(15)^2}{7.5} = \frac{225}{7.5} = 30 \, \text{W} \]
Step 4: Final Answer:
The induced power is 30 W.