phy1020.DVI

Figure 39.2 shows plots vs. time of the circuit current, voltages across the inductor and capacitor, charge
on the capacitor, and magnetic flux in the inductor. For these plots, attD 0 (the instant the switch is closed),
the capacitor in Fig. 39.1 is initially fully charged with the top plate positive. Positive current is taken to be
clockwise. All quantities vary sinusoidally with the same period, but may be shifted in phase with respect to
each other. If the initial charge on the capacitor isQ 0 , then the amplitudes for each quantity will be as shown
in the following table:

Quantity Symbol Amplitude

Current I!Q 0

Capacitor voltage VC Q 0 =C

Inductor voltage VL L!^2 Q 0

Capacitor charge QQ 0

Inductor mag. flux ˆB L!Q 0

Energy of an LC Circuit

In a simple harmonic oscillator formed by a mass on a spring, energy is continuously sloshing back and forth
between kinetic and potential energy, with the sum of the two (the total energy) being constant. Similarly,
in an LC circuit, energy is continuously sloshing back and forth between electric energy in the capacitor and
magnetic energy in the inductor. The electric energy in the capacitor is given by Eq. (25.12):

UeD

1

2

Q^2

C

: (39.3)

From Figure 39.2 and the above table, we have the charge on the capacitor (lower plate) as a function of time
is given by

Q.t/DQ 0 cos!t: (39.4)

and so the electric energyUeas a function of time is

Ue.t/D

Q 02

2C

cos^2 !t: (39.5)

Similarly, the magnetic energy in the inductor is given by Eq. (37.15):

UmD

1

2

LI^2 ; (39.6)

where the current at timetis

I.t/D!Q 0 sin!t: (39.7)

Substituting this expression forI.t/into the formula forUmgives an expression for the magnetic energy of
the inductor as a function of time:

Um.t/D

L!^2 Q^20

2

sin^2 !t (39.8)

D

Q^20

2C

sin^2 !t; (39.9)

where we have used the fact that!^2 D1=LC. The total energyUis then

UDUeCUm (39.10)

D

Q^20

2C

cos^2 !tC

Q^20

2C

sin^2 !tD

Q^20

2C

; (39.11)

which is a constant, as expected.