phy1020.DVI

Chapter 39

LC and LCR Circuits

We’ve previously looked at the RC circuit (a resistor and capacitor connected to a battery) and the RL circuit
(a resistor and inductor connected to a battery). The two circuits have complementary behavior: in the
RC circuit, the current starts out with a maximum value at the instant the switch is closed and decreases
exponentially toward zero. In the RL circuit, the current starts out small the instant the switch is closed,
increases with time, and eventually levels off to its maximum value.

39.1 LC Circuits

By connecting a charged capacitor and an inductor together, we create something called anLC circuit(Figure
39.1). In an LC circuit, the complementary behavior of the capacitor and the inductor give some interesting
results. As the capacitor discharges, a current is created in the circuit, which starts to build a magnetic field
in the inductor. As time goes on, the current will increase, but start to level off because the inhibiting effect
of the inductor: the inductor will create magnetic field an induced current in a direction that will oppose the
increase in the magnetic field in the inductor. By the time the capacitor has fully discharged, the magnetic
field in the inductor will have reached its maximum value.
At this point the current would stop, were it not for the presence of the magnetic field in the inductor.
Once the capacitor has fully discharged, it can no longer provide current to the inductor, and the magnetic
field in the inductor begins to collapse. But this change in the magnetic field induces a current in a direction
that opposes the collapse in the magnetic field—in other words, in a diction that will continue the current in
its original direction, so that the capacitor will begin to charge with the opposite polarity that it originally had.
By the time the magnetic field in the inductor has completely collapsed, the capacitor will be fully re-charged
(with opposite its original polarity), and the process begins again in reverse. Current will now begin to flow in
the opposite direction, creating a magnetic field in the inductor whose polarity is opposite what it was before.
The process will continue as before (but in the opposite direction) until the capacitor is fully charged with its
original polarity, and the cycle begins again, repeating over and over.
The result is an electrical form of simple harmonic motion, with energy moving back and forth between
the electric field stored in the capacitor and the magnetic field stored in the inductor. It can be shown that this
oscillation has angular frequency

!D

1

p

LC

; (39.1)

whereLis the inductance of the inductor andCis the capacitance of the capacitor. The period of oscillation
is thereforeTD2=!,or

TD2

p

LC: (39.2)