Question

State whether the following statement is True or False?
In a circuit containing inductance $L$, resistance $R$ and Voltage $E$,the current in the circuit is given by $L\frac{di}{dt}Ri=E$.The current $i$ at time $t$ is given by $i=\frac{E}{R}(1-e^{-Rt/L})$.

• A. True
• B. False

Answer 1

The correct option is A True

The given equation

$\frac{di}{dt}+\frac{Ri}{L}=\frac{E}{L}$ is linear of the type $\frac{dy}{dx}Py=Q$.

$\therefore$ Its solution is

$i\cdot e^{\int \left(R/L\right)dt}=\int e^{\int \left(R/L\right)dt}\cdot \frac{E}{L}\cdot dt+c$

$\therefore i.e^{Rt/L}=\frac{E}{L}\int e^{Rt/L}dt+c$

$=\frac{E}{L}\cdot e^{Rt/L}\cdot \frac{L}{R}+c=\frac{E}{R}{e}^{Rt/L}+c$

when $t=0,i=0$ $\therefore =-\frac{E}{R}$

$\therefore i.e^{Rt/L}=\frac{E}{R}{r}^{Rt/L}-\frac{E}{R}=\frac{E}{R}(e^{Rt/L}-1)$

$\therefore i=\frac{E}{R}(1-e^{-Rt/L})$