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$\text { The switch in the circuit shown in Fig. has been closed for a long time before it is }$

$\text { opened at } t=0 . \text { Find the following: }$

$\begin{array}{l}{\text { a) } i_{\mathrm{L}}(t) \text { for } t \geq 0} \\ {\text { b) } i_{\mathrm{o}}(t) \text { for } t \geq 0} \\ {\text { c) } v_{\mathrm{o}}(t) \text { for } t \geq 0}\end{array}$

$\text { For } t<0$
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DC

$\frac{d i}{d t}=0$                                    $V=0 \Rightarrow S \cdot C$

(1) $I_{0}=20 \mathrm{A}$

$\text { For }t>0$
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(2)

(3) $\tau=\frac{L}{R}=\frac{2}{10}=0.2 sec$

(4) $i(t)=I_{0} e^{\frac{-t}{\tau}}=20 e^{-\frac{t}{0.2}}=20 e^{-5 t} \quad (A)$

$i_{L}(t)=20 e^{-5 t}$

$i_{1} (t)=\frac{10}{40+10} i_{L}(t)$

$i_{1} (t)=\frac{10}{50} * 20 e^{-5 t}$

$i_{1}(t)=4 e^{-5 t}$

$i_{0}(t)=-i_{1}(t)=-4 e^{-5 t}$

$V_{0}(t)=i_{0}(t) R$

$V_{0}(t)=-4 e^{-5 t} * 40$

$V_{0}(t)=-160 e^{-5 t} \quad (V)$