A square coil of \( 50 \) turns, each of side \( 10\ \mathrm{cm} \) is held with its plane perpendicular to a magnetic field that varies with time as \( B(t) = 2t + 0.5t^{2} \) (in SI unit). The magnitude of induced emf in the coil at \( t = 1.0\ \mathrm{s} \) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 1.5\ \mathrm{V} \)
By Faraday’s law, the induced emf equals the rate of change of flux linkage. The coil area is fixed and the plane is perpendicular to \( \vec{B} \), so only \( B \) changes.
\( \displaystyle |\varepsilon| = N\frac{d\Phi}{dt} = NA\frac{dB}{dt} \)
\( A = (10\,\mathrm{cm})^{2} = (0.10\,\mathrm{m})^{2} = 1.0\times10^{-2}\,\mathrm{m^{2}} \)
\( \displaystyle \frac{dB}{dt} = \frac{d}{dt}(2t + 0.5t^{2}) = 2 + t \)
\( \text{At } t = 1.0\,\mathrm{s}\text{: } dB/dt = 3\,\mathrm{T\,s^{-1}} \)
\( |\varepsilon| = 50\times(1.0\times10^{-2}\,\mathrm{m^{2}})\times3\,\mathrm{T\,s^{-1}} = 1.5\,\mathrm{V} \)