Problem 2

A capacitor of capacitance

C

, initially uncharged, is connected in series to a resistor

R

and an ideal DC source of constant emf

\mathcal E

. The switch closes at

t=0

; leads are ideal and source internal resistance and parasitic effects are neglected. Take positive current in the charging direction, and let

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)

I(t)

, and the time constant. (c) Give the values just after switching and at long times, and check energy conservation.

problems.proof.stepOf

problems.proof.prevStep problems.proof.nextStep

W_{source}=\mathcal E Q,\quad U_C=\frac{Q^2}{2C},\quad W_R=\mathcal E Q-\frac{Q^2}{2C}

problems.proof.analysis

The source supplies work $\mathcal E Q$ , the capacitor stores $Q^2/(2C)$ , and the remainder is resistor heat. The source, leads, and components are ideal as stated.