Question

A circuit area 0.01 $m²$ is kept inside a magnetic field which is normal to its plane. The magnetic field changes from 2 T to 1 T in 1 ms. If the resistance of the circuit is 2 $Ω$. The amount of heat evolved is

• A. 0.05 J
• B. 50 J
• C. 0.50 J
• D. 500 J

Hint:

A circuit area 0.01 $m

Answer 1

The Correct Option is A

To solve this problem, we need to calculate the amount of heat evolved in the circuit due to the change in magnetic field. This involves several steps and the application of Faraday's Law of Electromagnetic Induction along with Joule's Law for heating.

1. First, determine the change in magnetic flux through the circuit. The magnetic field changes from 2 T to 1 T, so the change in magnetic field, \(\Delta B = B_{\text{final}} - B_{\text{initial}} = 1 \, \text{T} - 2 \, \text{T} = -1 \, \text{T}\)
2. The area of the circuit is 0.01 \(\text{m}^2\). The change in magnetic flux, \(\Delta \Phi\), is given by: \(\Delta \Phi = A \times \Delta B = 0.01 \, \text{m}^2 \times (-1 \, \text{T}) = -0.01 \, \text{Wb}\)
3. By Faraday's Law of Electromagnetic Induction, the induced electromotive force (EMF), \(\mathcal{E}\), is given by: \(\mathcal{E} = -\frac{\Delta \Phi}{\Delta t}\)
4. Substitute the values (note the time duration, \(\Delta t = 1 \, \text{ms} = 1 \times 10^{-3} \, \text{s}\)): \(\mathcal{E} = -\frac{-0.01 \, \text{Wb}}{1 \times 10^{-3} \, \text{s}} = 10 \, \text{V}\)
5. Now, use Ohm's Law to find the current, \(I\), generated in the circuit: \(I = \frac{\mathcal{E}}{R} = \frac{10 \, \text{V}}{2 \, \Omega} = 5 \, \text{A}\)
6. The energy (or heat) evolved in the circuit can be calculated using Joule’s Law: \(H = I^2 \times R \times \Delta t\)
7. Substitute the values: \(H = (5 \, \text{A})^2 \times 2 \, \Omega \times 1 \times 10^{-3} \, \text{s} = 0.05 \, \text{J}\)

Thus, the amount of heat evolved in the circuit is 0.05 J.