Lecture 35: FRI 17 APRLecture 35: FRI 17 APRElectrical Oscillations, LC Circuits, Alternating Current IElectrical Oscillations, LC Circuits, Alternating Current I

Physics 2113

Jonathan Dowling

Nikolai Tesla

What are we going to learn?What are we going to learn?A road mapA 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)

Energy Density in E and B Fields

Oscillators are very useful in practical applications, for instance, to keep time, or to focus energy in a system.

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.

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.

Oscillators in PhysicsOscillators in Physics

We have studied that inductors and capacitors are devices that can store electromagnetic energyelectromagnetic energy. In the inductor it is stored in a B field, in the capacitor in an E field.

PHYS2110: A Mechanical OscillatorPHYS2110: A Mechanical Oscillator

Newton’s law F=ma!

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 (withopposite polarity) and the cycle restarts.

PHYS2113 An Electromagnetic LC OscillatorPHYS2113 An Electromagnetic LC Oscillator

Capacitor discharges completely, yet current keeps going. Energy is all in the B-field of the inductor all fluxed up.

Capacitor initially charged. Initially, current is zero, energy is all stored in the E-field of the capacitor.

A current gets going, energy gets split between the capacitor and the inductor.

Electric Oscillators: the MathElectric Oscillators: the Math

Or loop rule!Or loop rule!

Energy Cons.Energy Cons.

Both give Diffy-Q: Solution to Diffy-Q:

LC FrequencyIn Radians/Sec

Electric Oscillators: the MathElectric Oscillators: the Math

Energy as Function of TimeEnergy as Function of Time Voltage as Function of TimeVoltage as Function of Time

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

Analogy Between Electrical And Mechanical Oscillations

Charqe q -> Position xCurrent i=q’ -> Velocity v=x’Dt-Current i’=q’’-> Acceleration a=v’=x’’

The energy is constant and equal to what we started with.

LC Circuit: Conservation of EnergyLC Circuit: Conservation of Energy

LC Circuit: Phase RelationsLC Circuit: Phase Relations

The current runs 90° out of phase with respect to the charge.

Example 1 : Tuning a Radio ReceiverExample 1 : Tuning a Radio Receiver

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?

FM radio stations: frequency is in MHz.

ExampleExample 22• 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 = πω = 2.5 ms• Capacitor will be charged after T=1/4 cycle i.e at • t = T/4 = 0.6 ms

Example 3Example 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.

• Switch in position “a”: q=CV = (1 μF)(10 V) = 10 μC• Switch in position “b”: maximum charge on C = q0 = 10 μC• So, amplitude of oscillating current =

0.316 A

ba

E=10 V1 mH 1 μF

Example 4Example 4In an LC circuit, the maximum current is 1.0 A.

If L = 1mH, C = 10 μF 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 ω=1/√LC=(1mH 10 μF)–1/2 = (10-8)–1/2 = 104 rad/s

Maximum charge is q0=i0/ω = 1A/104 rad/s = 10–4 C