W9_parallel_resonance.eps

Week 5: Resistance 187

The argument of an exponential (or any transcendental function) has to be dimensionless, so the
units ofRCmust be atime, the so-calledexponential decay timefor the circuit:

τ=RC (389)

This is an important quantity to keep in mind when working withRCcircuits, as it provides an
instant estimate for how long it will take for the charge on the capacitor to decay.

Example 5.5.2: Charging Capacitor

C R

I(t)

t = 0

V 0

Figure 62: An initially uncharged capacitor being charged through a resistor by a battery with a
fixed voltageV 0.

In figure 62 we have added a battery and changed the initial condition toQ(0) = 0, an initially
uncharged capacitor. The solution to the problem proceedsalmost identicallyto the charging case.
From Kirchhoff’s loop rule:

V 0 −

Q

C−IR= 0 (390)

The current is now the rate at which the charge on the capacitorincreases:

I= +

dQ

dt

(391)

Substituting as before and rearranging, we get:

dQ

dt

+ Q

RC

=V^0

R

(392)

This is afirst order, linear,inhomogeneous, ordinary differential equation, in fact the equation for
exponential growth. It, too, can easily by solved by direct integration:

dQ

dt

=−

Q

RC

+

V 0

R

(393)

Now,pay attentionfor a second, as it took me years of solving this inefficiently before I finally
figured out how to do the algebraefficiently, and I’m going to share a little trick with you that will
help you get the right answer for this equation (which occurs over and over again in physics, both
last semester and this): Beforemultiplying out and trying to integratefactor the coefficient ofQ
out of the entire left hand side!:
dQ
dt

=−

1

RC

(Q−CV 0 ) (394)