Problem 2

An ideal DC source of emf

\mathcal E

is connected in series with a resistor

R>0

and ideal inductor

L>0

. Before the switch closes the inductor current is zero; at

t=0

the switch closes, with positive current chosen along the source. Neglect internal resistance and take all elements as constant. (a) Use Kirchhoff's law to find

I(t)

for

t\ge0

. (b) Determine the steady current

I_\infty

. (c) Find when the current reaches

0.90I_\infty

and check the initial and long-time limits.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

\boxed{I(t)=\frac{\mathcal E}{R}(1-e^{-Rt/L}),\quad I(0)=0,\quad I(\infty)=I_\infty=\frac{\mathcal E}{R},\quad t_{90}=\frac{L}{R}\ln10}

problems.proof.analysis

The boxed results give the transient current, both limits $I(0)=0$ and $I(\infty)=I_\infty$ , and the 90% time. The time has scale $L/R$ and grows with the circuit time constant.