The magnetic flux linked with a coil at any instant t is given by $\phi= (4t^3 - 3t^2 + 2t+ 5) Wb$. The emf induced in the coil at $t = 3 s$ is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → -92 V
Given magnetic flux: $\phi(t) = 4t^3 - 3t^2 + 2t + 5 \, \text{Wb}$
Induced emf: $ \mathcal{E} = - \frac{d\phi}{dt} $
Derivative of flux:
$\frac{d\phi}{dt} = \frac{d}{dt} (4t^3 - 3t^2 + 2t + 5) = 12 t^2 - 6 t + 2$
At $t = 3 \, \text{s}$:
$\frac{d\phi}{dt} = 12(3)^2 - 6(3) + 2 = 12(9) - 18 + 2 = 108 - 18 + 2 = 92 \, \text{Wb/s}$
Induced emf: $\mathcal{E} = - \frac{d\phi}{dt} = -92 \, \text{V}$
Answer: $\mathcal{E} = -92 \, \text{V}$