Physic Labs

Problem 2

An ideal DC source of emf E\mathcal E is connected in series with a resistor R>0R>0 and ideal inductor L>0L>0. Before the switch closes the inductor current is zero; at t=0t=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)I(t) for t≥0t\ge0. (b) Determine the steady current I∞I_\infty. (c) Find when the current reaches 0.90I∞0.90I_\infty and check the initial and long-time limits.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
I(0)=0⇒C=−ER,I(t)=ER(1−e−Rt/L)I(0)=0\Rightarrow C=-\frac{\mathcal E}{R},\quad I(t)=\frac{\mathcal E}{R}(1-e^{-Rt/L})
problems.proof.analysis

Setting t=0t=0 gives C=−E/RC=-\mathcal E/R and hence the time-dependent current. It begins at zero, as inductor current cannot jump instantaneously.