Find current I at time t if t = 0 I = U and L, R, E are constants.

Mumbai University > First Year Engineering > sem 2 > Applied Maths 2

Marks : 6

Year : 2013

$$L\frac {di}{dt}+Ri=E$$ $$\therefore\frac{di}{dt}+\frac{Ri}{L}=\frac{E}{L}$$ Which is same as that of $$\frac{di}{dt}+pi=Q$$

Here $p=\frac{R}{L} \ and \ Q=\frac{E}{L}$

The given D.E is linear equation

$I.f$ $=e^{\int pdt} \\ =e^{\int\frac{R}{L}dt} \\ =e^{\frac{Rt}{L}}$

The solution is

$i{\times} I.f =\int I.f\times Qdt$

$i{\times}e^{Rt/L}$ $=\int\frac{E}{L}{\times} e^{Rt/L}dt \\ =\frac{E}{L}\int e^{Rt/L}dt \\ =\frac{E}{L}\frac {e^{Rt/L}}{R/L} +C \\ =\frac{E}{R}e^{Rt/L}+C \\ \therefore i=\dfrac ER+ce^{-Rt/L}...........(1)$

At $t=0 \ \ \ \ i=0$

$\therefore 0=\dfrac ER +C \\ \therefore C=\dfrac {-E}R$

Eq (1) becomes

$i=\dfrac ER-\dfrac ER\Big(e^{-RT/L}\Big) \\ i=\dfrac ER\Big(e^{-RT/L}\Big)$