The magnetic flux in a closed circuit of resistance 20 $\mathrm{\Omega }$ varies with time t as $\mathrm{\varphi }=4{\mathrm{t}}^3+2{\mathrm{t}}^2-15\mathrm{t}+3$. Calculate the induced emf at t = 1s

Given : Resistance = $20\mathrm{\Omega };\mathrm{\varphi }=4{\mathrm{t}}^3+2{\mathrm{t}}^2-15\mathrm{t}+3$
Required equation $\to \mathrm{\epsilon }=-\mathrm{d\varphi }/\mathrm{dt}$
$\mathrm{\epsilon }=-\frac{\mathrm{d}}{\mathrm{dt}}\left(4{\mathrm{t}}^3+2{\mathrm{t}}^2-15\mathrm{t}+3\right)\mathrm{dt}=-\left(12{\mathrm{t}}^2+4\mathrm{t}-15\right)$
$\begin{array}{l}=-12-4+15(\text{ sincet }=1\mathrm{s})\\ =-1\mathrm{V}\end{array}$
Therefore, the magnitude of induced emf is – 1 V