A conducting circular loop is placed in a uniform magnetic field of 0.5 T, with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at the rate of 4 mm/s. When the radius of the loop is 4 cm, the induced emf in the loop is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 160\pi\,\mu\mathrm{V} \)
The field is constant, but the area of the loop changes, so the flux \( \Phi = B\pi r^{2} \) changes. By Faraday's law the magnitude of the emf is the rate of change of flux.
\( \displaystyle |\varepsilon| = \frac{d\Phi}{dt} = B\,\frac{d(\pi r^{2})}{dt} = B \cdot 2\pi r\,\frac{dr}{dt} \)
\( B = 0.5\,\mathrm{T} \)
\( r = 4\,\mathrm{cm} = 0.04\,\mathrm{m} \)
\( dr/dt = 4\,\mathrm{mm\,s^{-1}} = 4\times10^{-3}\,\mathrm{m\,s^{-1}} \)
\( \displaystyle |\varepsilon| = 0.5 \times 2\pi \times 0.04 \times 4\times10^{-3} = 1.6\pi\times10^{-4}\,\mathrm{V} \)
\( \displaystyle |\varepsilon| = 160\pi\times10^{-6}\,\mathrm{V} = 160\pi\,\mu\mathrm{V} \)