A time dependent magnetic flux passing through the perpendicular plane of a coil is given by \( \Phi_{B} = 2t^{3} - 3t^{2} - 12t - 2 \). Arrange the magnitudes of induced emf, at the following instants of time, in ascending order:
(A) at \( t = 1\ \mathrm{s} \)
(B) at \( t = 2\ \mathrm{s} \)
(C) at \( t = 3\ \mathrm{s} \)
(D) at \( t = 5\ \mathrm{s} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (B), (A), (C), (D)
By Faraday’s law, the magnitude of the induced emf equals the rate of change of magnetic flux.
\( \displaystyle \varepsilon = -\frac{d\Phi_{B}}{dt} = -\frac{d}{dt}(2t^{3} - 3t^{2} - 12t - 2) \)
\( |\varepsilon| = |6t^{2} - 6t - 12| \)
Values at the given instants:
\( t = 1\ \mathrm{s}:\ |6 - 6 - 12| = 12\ \mathrm{V} \)
\( t = 2\ \mathrm{s}:\ |24 - 12 - 12| = 0 \)
\( t = 3\ \mathrm{s}:\ |54 - 18 - 12| = 24\ \mathrm{V} \)
\( t = 5\ \mathrm{s}:\ |150 - 30 - 12| = 108\ \mathrm{V} \)
Ascending order: \( 0 \lt 12 \lt 24 \lt 108 \), that is, (B), (A), (C), (D).