1. Physics 2113
Jonathan Dowling
Lecture 34: FRI 13 NOV Electrical Oscillations, LC Circuits, Alternating Current I
Nikolai Tesla

2. What are we going to learn? A road map
Electric charge  Electric force on other electric charges  Electric field, and electric potential
Moving electric charges : current
Electronic circuit components: batteries, resistors, capacitors
Electric currents  Magnetic field  Magnetic force on moving charges
Time-varying magnetic field  Electric Field
More circuit components: inductors.
Electromagnetic waves  light waves
Geometrical Optics (light rays).
Physical optics (light waves)

3. Energy Density in E and B Fields

4. Oscillators in Physics
Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system.
All oscillators can store energy in more than one way and exchange it back and forth between the different storage possibilities. For instance, in pendulums (and swings) one exchanges energy between kinetic and potential form.
We have studied that inductors and capacitors are devices that can store electromagnetic energy. In the inductor it is stored in a B field, in the capacitor in an E field.

5. PHYS2110: A Mechanical Oscillator
Newton’s law F=ma!

6. PHYS2113 An Electromagnetic LC Oscillator
Capacitor initially charged. Initially, current is zero, energy is all stored in the E-field of the capacitor.
Capacitor discharges completely, yet current keeps going. Energy is all in the B-field of the inductor all fluxed up.
A current gets going, energy gets split between the capacitor and the inductor.
The magnetic field on the coil starts to deflux, which will start to recharge the capacitor.
Finally, we reach the same state we started with (with
opposite polarity) and the cycle restarts.

7. Electric Oscillators: the Math
Energy Cons.
Or loop rule!
Both give Diffy-Q:
Solution to Diffy-Q:
LC Frequency
In Radians/Sec

9. Electric Oscillators: the Math
Voltage as Function of Time
Energy as Function of Time

10. LC Circuit: At t=0 1/3 Of Energy Utotal is on Capacitor C and Two Thirds On Inductor L. Find Everything! (Phase φ0=?)

11. Analogy Between Electrical And Mechanical Oscillations
Charqe q -> Position x
Current i=q’ -> Velocity v=x’
Dt-Current i’=q’’-> Acceleration a=v’=x’’

12. LC Circuit: Conservation of Energy
The energy is constant and equal to what we started with.

13. LC Circuit: Phase Relations
The current runs 90° out of phase with respect to the charge.

17. Example 1 : Tuning a Radio Receiver
FM radio stations: frequency
is in MHz.
The inductor and capacitor in my car radio have one program at L = 1 mH & C = 3.18 pF. Which is the FM station?
(b) WRKF 89.3
What is wavelength of radio wave?
How about for WJBO 1150 AM?

18. Example 2 ω = 2500 rad/s T = period of one complete cycle
In an LC circuit, L = 40 mH; C = 4 μF
At t = 0, the current is a maximum;
When will the capacitor be fully charged for the first time?
ω = 2500 rad/s
T = period of one complete cycle
T = 2π/ω = 2.5 ms
Capacitor will be charged after T=1/4 cycle i.e at
t = T/4 = 0.6 ms

19. Example 3
In the circuit shown, the switch is in position “a” for a long time. It is then thrown to position “b.”
Calculate the amplitude ωq0 of the resulting oscillating current. b a
E=10 V
1 mH
1 mF
Switch in position “a”: q=CV = (1 mF)(10 V) = 10 mC
Switch in position “b”: maximum charge on C = q0 = 10 mC
So, amplitude of oscillating current =
0.316 A

20. Example 4 In an LC circuit, the maximum current is 1.0 A.
If L = 1mH, C = 10 mF what is the maximum charge q0 on the capacitor during a cycle of oscillation?
Maximum current is i0=ωq0 Maximum charge: q0=i0/ω
Angular frequency w=1/√LC=(1mH 10 mF)–1/2 = (10-8)–1/2 = 104 rad/s
Maximum charge is q0=i0/ω = 1A/104 rad/s = 10–4 C