Physic Labs

Problem 2

At t=0t=0, an initially uncharged capacitor of capacitance CC is connected in series with a resistor RR and an ideal battery of emf EE. Take current in the charging direction as positive and neglect internal resistance. Find the charge q(t)q(t) on the positive plate, current i(t)i(t), and energy stored in the capacitor for t≥0t\geq0. Assume ideal components with constant values R>0R>0, C>0C>0, and E>0E>0, a switch closed at t=0t=0, and no initial capacitor charge. (a) Derive the differential equation from Kirchhoff’s loop rule. (b) Find q(t)q(t) and then the current, identifying the circuit time constant. (c) Find the capacitor energy and check the limits at t=0t=0 and as tt becomes very large.
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i(t)=dqdt=ERe−t/(RC),UC(t)=q22C=12CE2(1−e−t/(RC))2i(t)=\frac{dq}{dt}=\frac{\mathcal E}{R}e^{-t/(RC)},\qquad U_C(t)=\frac{q^2}{2C}=\frac12C\mathcal E^2\left(1-e^{-t/(RC)}\right)^2
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Differentiating q(t)q(t) gives the charging current. The capacitor's stored electric-field energy is q2/(2C)q^2/(2C), yielding the expression shown.