A conducting circular loop is placed in a uniform magnetic field of 10 T with its plane normal to the field. Now, the radius of the loop starts shrinking at the rate 5 mm/s. When the radius is 5 cm then the induced emf is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 15.7 mV
The plane of the loop is normal to the field, so the magnetic flux is \( \Phi = B\,\pi r^{2} \). As the radius shrinks, the area and hence the flux change, and Faraday's law gives the size of the induced emf.
\( \displaystyle |\varepsilon| = \frac{d\Phi}{dt} = B\,\frac{d(\pi r^{2})}{dt} = B \cdot 2\pi r\,\frac{dr}{dt} \)
\( B = 10\,\mathrm{T} \)
\( r = 5\,\mathrm{cm} = 0.05\,\mathrm{m} \)
\( dr/dt = 5\,\mathrm{mm\,s^{-1}} = 5\times10^{-3}\,\mathrm{m\,s^{-1}} \)
\( \displaystyle |\varepsilon| = 10 \times 2\pi \times 0.05 \times 5\times10^{-3} = 5\pi\times10^{-3}\,\mathrm{V} \)
\( \displaystyle |\varepsilon| = 1.57\times10^{-2}\,\mathrm{V} = 15.7\,\mathrm{mV} \)