A conducting circular loop is placed in a uniform magnetic field B = 0.02 T with its plane perpendicular to the field. Somehow, the radius of the loop starts shrinking at a constant rate of 1.0 mm s−1. At an instant when the radius of the loop is 2 cm, the induced emf in it is.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 2.5 μV
By Faraday’s law, the induced emf equals the rate of change of magnetic flux. The field is constant, but the flux changes because the area of the loop is shrinking.
\( \displaystyle \phi = B\pi r^{2} \)
\( \displaystyle |\varepsilon| = \frac{d\phi}{dt} = B \times 2\pi r\,\frac{dr}{dt} \)
\( B = 0.02\,\mathrm{T} \)
\( r = 2\,\mathrm{cm} = 2\times10^{-2}\,\mathrm{m} \)
\( dr/dt = 1.0\,\mathrm{mm\,s^{-1}} = 1.0\times10^{-3}\,\mathrm{m\,s^{-1}} \)
\( \displaystyle |\varepsilon| = 0.02\,\mathrm{T} \times 2 \times 3.14 \times 2\times10^{-2}\,\mathrm{m} \times 1.0\times10^{-3}\,\mathrm{m\,s^{-1}} \)
\( \displaystyle |\varepsilon| \approx 2.5\times10^{-6}\,\mathrm{V} = 2.5\,\mu\mathrm{V} \)