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Step-by-Step Solution
Step 1: Write the expression for magnetic flux
The magnetic flux \phi through the coil is given by the product of the magnetic field B and the area S of the circular coil. Here, the coil has radius R = 1\,\text{m} , so its area is \pi R^2 = \pi \times 1^2 = \pi . The magnetic field is given by:
B = \frac{4}{\pi} \times 10^{-3} \left( 1 - \frac{t}{100} \right).
Thus, the flux \phi is:
\phi = B \times \pi R^2 = \left(\frac{4}{\pi} \times 10^{-3} \left(1 - \frac{t}{100}\right)\right) \times \pi \times 1^2.
Simplifying,
\phi = 4 \times 10^{-3} \left(1 - \frac{t}{100}\right).
Step 2: Find the induced EMF using Faraday's law
Faraday’s law states that the induced EMF \varepsilon is the negative rate of change of the flux:
\varepsilon = -\frac{d\phi}{dt}.
Differentiate \phi = 4 \times 10^{-3}\left(1 - \tfrac{t}{100}\right) with respect to t :
\varepsilon = -\frac{d}{dt}\Bigl[4 \times 10^{-3}\left(1 - \frac{t}{100}\right)\Bigr].
\varepsilon = 4 \times 10^{-3} \times \frac{1}{100} = 4 \times 10^{-5}\,\text{V}.
This EMF remains constant over the time interval from t = 0 to t = 100\text{ s} because the term 1 - \frac{t}{100} is linear in t , hence its derivative is a constant.
Step 3: Determine the total time interval
The magnetic field is completely switched off when B = 0 , which happens when the bracketed term becomes zero:
1 - \frac{t}{100} = 0 \quad \Longrightarrow \quad t = 100 \,\text{s}.
Hence, the field changes from its initial value to zero over 100\,\text{s} .
Step 4: Compute the heat (energy) dissipated in the coil
The resistance of the coil is given as R = 2\,\mu\Omega = 2 \times 10^{-6}\,\Omega . The power dissipated is P = \frac{\varepsilon^2}{R} , so the energy (heat) dissipated in time t is:
\text{Heat} = \frac{\varepsilon^2}{R} \times t.
Substituting \varepsilon = 4 \times 10^{-5}\,\text{V} , R = 2 \times 10^{-6}\,\Omega , and t = 100\,\text{s} :
\text{Heat} = \frac{\left(4 \times 10^{-5}\right)^2}{2 \times 10^{-6}} \times 100 \,\text{J}.
Calculate step by step:
\left(4 \times 10^{-5}\right)^2 = 16 \times 10^{-10} = 1.6 \times 10^{-9}.
Divide by the resistance:
\frac{1.6 \times 10^{-9}}{2 \times 10^{-6}} = 0.8 \times 10^{-3} = 8 \times 10^{-4}\,\text{W}.
Multiply by time (100 s):
8 \times 10^{-4} \,\text{W} \times 100\,\text{s} = 0.08\,\text{J}.
Thus, the total heat dissipated is 0.08\,\text{J} , which is:
0.08\,\text{J} = 80\,\text{mJ}.
Final Answer
The energy dissipated by the coil is 80 mJ.
