Physic Labs

Problem 2

A capacitor of capacitance CC, initially uncharged, is connected in series to a resistor RR and an ideal DC source of constant emf E\mathcal E. The switch closes at t=0t=0; leads are ideal and source internal resistance and parasitic effects are neglected. Take positive current in the charging direction, and let Q(t)Q(t) be the charge on the plate connected to the positive terminal. (a) Write Kirchhoff’s loop equation and relate current to the rate of change of charge. (b) Determine Q(t)Q(t), I(t)I(t), and the time constant. (c) Give the values just after switching and at long times, and check energy conservation.
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E−RI−QC=0,I=dQdt\mathcal E-RI-\frac QC=0,\quad I=\frac{dQ}{dt}
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Kirchhoff’s loop law makes the resistor and capacitor voltage drops sum to the emf; for charging, I=dQ/dtI=dQ/dt. The source, leads, and components are ideal as stated.

problems.proof.pitfall. Do not treat the current as constant or the capacitor as a short circuit throughout charging; its voltage rises, so the current decays.