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.
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problems.proof.prevStepproblems.proof.nextStep
I∞=lim⁡t→∞I(t)=ER,I(t90)I∞=0.90I_\infty=\lim_{t\to\infty}I(t)=\frac{\mathcal E}{R},\quad \frac{I(t_{90})}{I_\infty}=0.90
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As t→∞t\to\infty, the exponential vanishes and I∞=E/RI_\infty=\mathcal E/R. Set the current fraction at the desired instant to 0.90 to find t90t_{90}.