Week 5: Resistance 185
Note well thatI 2 comes outnegative– this simply means that we guessed its directionincorrectly
in our original decoration of the figure. The second battery is actually being charged as the first
one discharges. This is (as you can see from the numbers) about asnasty a problem of this sort as
you are likely to see. Usually problems like this on a quiz or exam will have voltages and resistances
that are chosen to give rational answers that one can work out without needing a calculator.
5.5:RCCircuits
So far everything we have done with charges and currents has beenstatic. True, we have studied
flowing currents but those currents have beenconstantin time, as have all potential differences. We
now illustrate the use of Kirchhoff’s Loop Rule to obtain anequation of motionfor the charging
or discharging of a capacitor through a resistance. We begin with a discharging capacitor, as the
slightly easier problem.
Example 5.5.1: Discharging Capacitor
Q
C RI(t)t = 00Figure 61: The capacitorCis initially charged toQ 0. Att= 0 the switch is closed and it discharges
through the resistor, building up a currentI(t).
The capacitor in figure 61 is initially charged toQ 0. Att= 0, the switch is closed and charge
begins to flow off of the capacitor and is driven through the resistor, so that at timetthere is a
chargeQ(t) left on the capacitor and a currentI(t) in the circuit. Our goal is to basically understand
everythingabout this problem. We want to knowI(t),Q(t),VC(t),VR(t), the powerPC(t) delivered
by the capacitor, the powerPR(t) consumed by the resistor, and a full understanding of energy as
a function of time in the circuit.
To find all of this, we begin by writingKirchhoff’s Loop Rulefor the loop above (going clockwise
around the circuit in the direction of the current), at some timetafter the switch is closed:
Q
C−IR= 0 (372)
The current and charge are not independent. The current is, in fact, the rate at which the charge
on the capacitor decreases:
I=−dQ
dt(373)
If we substitute this relation into Kirchhoff’s loop rule, divide byR, and rearrange, we get the
following equation of motion forQ:
dQ
dt
+
Q
RC
= 0 (374)
This is afirst order, linear, homogeneous, ordinary differential equation, in fact the equation for
exponential decay. It can easily by solved by direct integration. The solution proceedsas follows.
Rearrange the equation as follows:
dQ
dt