Find the current at any instant if the relation between L, R, E is $L \frac{di}{dt} + Ri = E$

Solution:

$\mathrm{L} \frac{d i}{d t}+R i=E$

$\therefore \quad \frac{d i}{d t}+\frac{R i}{L}=\frac{E}{L}$

Solution is given by

$\boldsymbol{i} . e^{\int\left(\frac{R}{L}\right) d t}=\int e^{\int\left(\frac{R}{L}\right) d t} \cdot \frac{E}{L} d t+c$

$\therefore \quad \boldsymbol{i} . \boldsymbol{e}^{(R t / L)}=\frac{E e^{(R t / L)}}{R}+c$

At $t=0, i=0 \quad \therefore c=\frac{E}{R}$

$\therefore \boldsymbol{i} \cdot e^{(R t / L)}=\frac{E e^{(R t / L)}}{L}+\frac{-E}{R}$

$\therefore \boldsymbol{i}=\frac{E}{\boldsymbol{R}}\left(\mathbf{1}-\boldsymbol{e}^{-(\boldsymbol{R} t / L)}\right)$

For given condition $R=100, L=0.5, E=20$

$\therefore i=0.2\left(1-e^{-200 t}\right)$