Question

A metal loop of area \(10 \, \text{cm}^2\) is placed in a region such that its area vector points along \( \hat{k} \). The region contains a uniform magnetic field of magnitude \(1.73 \, \text{T}\) that points in the direction \( \hat{i} + \hat{j} + \hat{k} \). When the magnetic field is switched off, the field decreases to zero at a steady rate in 10 seconds. The magnitude of emf induced in the loop is:

• A. \( 0.10 \, \text{mV} \)
• B. \( 0.17 \, \text{mV} \)
• C. \( 1 \, \text{mV} \)
• D. \( 1.7 \, \text{mV} \)

Hint:

Only the magnetic field component along the area vector contributes to the flux. Use \( \text{EMF} = \frac{\Delta \Phi}{\Delta t} \).

Answer 1

The Correct Option is A

Only the component of \( \vec{B} \) along the area vector \( \hat{k} \) contributes to flux. From direction: \[ \vec{B} = 1.73 (\hat{i} + \hat{j} + \hat{k}) \Rightarrow B_k = \frac{1.73}{\sqrt{3}} \] Area: \[ A = 10\, \text{cm}^2 = 10 \times 10^{-4} \, \text{m}^2 = 10^{-3} \, \text{m}^2 \] EMF: \[ \mathcal{E} = \left| \frac{d\Phi}{dt} \right| = \left| \frac{A \cdot B_k}{t} \right| = \frac{10^{-3} \cdot \frac{1.73}{\sqrt{3}}}{10} \approx \frac{1.73 \times 10^{-3}}{17.32} \approx 0.1 \times 10^{-3} = 0.10 \, \text{mV} \]