A coil of inductance $2 H$ having negligible resistance is connected to a source of supply whose voltage is given by $V = 3 t$ volt (where t is in second). If the voltage is applied when $t = 0$ , then the energy stored in the coil after $4 s$ is ______J.

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answer is 144.

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Detailed Solution

$L \frac{d i}{d t} = ε = 3 t \Rightarrow L \int di = 3 \int tdt \Rightarrow Li = \frac{3 t^{2}}{2} \Rightarrow i = \frac{3 t^{2}}{2 L}$

$Energy, E = \frac{1}{2} {Li}^{2} = \frac{1}{2} L {(\frac{3 t^{2}}{2 L})}^{2} = \frac{1}{2} \times \frac{9 t^{4}}{4 L} = \frac{9}{8} \times \frac{{(4)}^{4}}{4 \times 2} = 144 J$

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