Question

A circular conducting coil of radius 1 m is being heated by the change of magnetic field $\vec{B}$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is 2 $\mu\Omega$. The magnetic field is slowly switched off such that its magnitude changes in time as \[ B = \frac{4}{\pi} \times 10^{-3} \text{T} \left( 1 - \frac{t}{100} \right) \] The energy dissipated by the coil before the magnetic field is switched off completely is E = ________ mJ.

Hint:

When the magnetic field varies linearly with time, the induced EMF is constant, which makes energy calculations much simpler (\(P \times t\)).

Answer 1

Correct Answer: 80

Step 1: Understanding the Concept:
A changing magnetic field induces an electromotive force (EMF) in a closed loop. This EMF causes current to flow, leading to energy dissipation via resistance (Joule heating).
Step 2: Key Formula or Approach:
1. Flux: \(\phi = BA\).
2. Induced EMF: \(e = \left| \frac{d\phi}{dt} \right|\).
3. Energy: \(E = \int \frac{e^2}{R} dt\).
Step 3: Detailed Explanation:
Area \(A = \pi r^2 = \pi \text{ m}^2\).
Flux \(\phi = \pi \times \frac{4}{\pi} \times 10^{-3} \left( 1 - \frac{t}{100} \right) = 4 \times 10^{-3} \left( 1 - \frac{t}{100} \right) \text{ Wb}\).
Induced EMF \(e\):
\[ e = \left| \frac{d}{dt} \left[ 4 \times 10^{-3} \left( 1 - \frac{t}{100} \right) \right] \right| = \frac{4 \times 10^{-3}}{100} = 4 \times 10^{-5} \text{ V} \]
Since \(e\) is constant, the energy dissipated over time \(t_{total} = 100 \text{ s}\) is:
\[ E = \frac{e^2}{R} \Delta t = \frac{(4 \times 10^{-5})^2}{2 \times 10^{-6}} \times 100 \]
\[ E = \frac{16 \times 10^{-10}}{2 \times 10^{-6}} \times 100 = 8 \times 10^{-4} \times 100 = 0.08 \text{ J} \]
\[ E = 80 \text{ mJ} \]
Step 4: Final Answer:
The energy dissipated is 80 mJ.