A uniform magnetic field B exists in a direction perpendicular to the plane of a square loop made of a metal wire. The wire has a diameter of 4 mm and a total length of 30 cm. The magnetic field changes with time at a steady rate \(\frac{dB}{dt} = 0.032\ \mathrm{T\,s^{-1}}\). The induced current in the loop is close to (Resistivity of the metal wire is \(1.23 \times 10^{-8}\ \Omega\,\mathrm{m}\))
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- 0.53 A
- 0.43 A
- 0.34 A
- 0.61 A
The Correct Option is D
Solution and Explanation
Induced emf: \(\varepsilon = \frac{dB}{dt}\cdot A = 0.032 \times 5.625 \times 10^{-3} = 1.8 \times 10^{-4}\) V.
Wire cross-section: \(A_c = \pi\left(2 \times 10^{-3}\right)^2 = 1.257 \times 10^{-5}\ \mathrm{m^2}\).
Resistance: \(R = \frac{\rho L}{A_c} = \frac{1.23 \times 10^{-8} \times 0.30}{1.257 \times 10^{-5}} = 2.94 \times 10^{-4}\ \Omega\).
Current: \(I = \frac{\varepsilon}{R} = \frac{1.8 \times 10^{-4}}{2.94 \times 10^{-4}} \approx 0.61\) A.
Answer: (4) 0.61 A
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