# Question 1.P.5: The given figures show the inductor and the waveform of the ......

The given figures show the inductor and the waveform of the current passing through an inductor of resistance 1 ohm and inductance 2 H. The energy absorbed by the inductor up to 4 s is

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Given that R = 1 Ω; 0 ≤ t ≤ 2 seconds

\begin{aligned} E_{ R 1} & =\int_0^2(3 t)^2 d t \\ & =\int_0^2 9 t^2 \cdot d t=9\left|\frac{t^3}{3}\right|_0^2=\frac{9 \times 8}{3}=24 J \end{aligned}

\begin{aligned} E_{ R 2} & =\int_2^4 6^2 \cdot 1 d t=36 \times 2=72 J \\ E_{ L 1} & =\int_0^2 L \cdot i\left\lgroup \frac{d i}{d t}\right\rgroup \cdot d t \\ & =\int_0^2 2 \cdot 3 t \cdot 3 \cdot d t=18 \int_0^2 t \cdot d t=18\left|\frac{t^2}{2}\right|_0^2=18 \times 2=36 \\ E_{ L 2} & =\int_2^4 L \cdot i\left(\frac{d i}{d t}\right) d t \\ & =\int_2^4 2 \cdot 6 \cdot 0 \cdot d t=0 J \end{aligned}

Therefore,

$E=E_{ R 1}+E_{ R 2}+E_{ L 1}+E_{ L 2}=24+72+36+0=132 J$

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