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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problems.proof.prevStepproblems.proof.nextStep
Q(t)=CE(1−e−t/(RC)),I(t)=ERe−t/(RC),τ=RC;Q(0)=0, I(0+)=ER, Q(∞)=CE, I(∞)=0;Wsrc=CE2, UC=12CE2, WR=12CE2\boxed{Q(t)=C\mathcal E(1-e^{-t/(RC)}),\quad I(t)=\frac{\mathcal E}{R}e^{-t/(RC)},\quad \tau=RC;\quad Q(0)=0,\ I(0^+)=\frac{\mathcal E}{R},\ Q(\infty)=C\mathcal E,\ I(\infty)=0;\quad W_{src}=C\mathcal E^2,\ U_C=\frac12C\mathcal E^2,\ W_R=\frac12C\mathcal E^2}
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The boxed expressions give the time-dependent quantities and time constant. They satisfy the initially uncharged condition.