Lecture 20Lecture 20

Physics 2102

Jonathan Dowling

Ch. 31: Electrical Oscillations, LCCh. 31: Electrical Oscillations, LCCircuits, Alternating CurrentCircuits, Alternating Current

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)

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

All oscillators operate along the sameprinciple: they are systems that can storeenergy in more than one way andexchange it back and forth between thedifferent storage possibilities. For instance,in pendulums (and swings) one exchanges energybetween kinetic and potential form.

Oscillators in PhysicsOscillators in Physics

In this course we have studied that coils and capacitors are devicesthat can store electromagnetic energyelectromagnetic energy. In one case it is stored in a magnetic field, in the other in an electric field.

potkintotEEE += 22

2

1

2

1xkvmE

tot+=

!"

#$%

&+!"

#$%

&==

dt

dxxk

dt

dvvm

dt

dEtot

22

12

2

10

dt

dxv =

0=+ xkdt

dvm

)cos()( :Solution00!" += txtx

phase :

frequency :

amplitude :

0

0

!

"

x

m

k=!

A mechanical oscillatorA mechanical oscillator

02

2

=+ xkdt

xdm

Newton’s lawF=ma!

The magnetic field on the coil starts to collapse,which will start to recharge the capacitor.

Finally, we reach the same state we started with (withopposite polarity) and the cycle restarts.

An electromagnetic oscillatorAn electromagnetic oscillator

Capacitor discharges completely, yet current keepsgoing. Energy is all in the inductor.

Capacitor initially charged. Initially, current iszero, energy is all stored in the capacitor.

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

elecmagtotEEE +=

2

2

2

1

2

1

C

qiLE

tot+=

!"

#$%

&+!"

#$%

&==

dt

dqq

Cdt

diiL

dt

dEtot

22

12

2

10

( )qCdt

diL

10 +!

"

#$%

&=

dt

dqi =

C

q

dt

qdL +=

2

2

0 02

2

=+ xkdt

xdm

Compare with:

Analogy between electrical and mechanical oscillations:

MLvi

kCxq

!!

!!

/1

LC

1=!

)cos( 00 !" += tqq

Electric Oscillators: the mathElectric Oscillators: the math

(the loop rule!)

)cos()( 00 !" += txtx

m

k=!

-1.5

-1

-0.5

0

0.5

1

1.5

Time

Charge

Current

)cos( 00 !" += tqq

)sin( 00 !"" +#== tqdt

dqi

)(sin2

1

2

10

22

0

22 !"" +== tqLLiEmag

0

0.2

0.4

0.6

0.8

1

1.2

Time

Energy in capacitor

Energy in coil

)(cos2

1

2

10

22

0

2

!" +== tqCC

qEele

LCxx

1 and ,1sincos

that,grememberin And

22==+ !

2

0

2

1q

CEEEelemagtot=+=

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

Electric Oscillators: the mathElectric Oscillators: the math

Example 1 : tuning a radio receiverExample 1 : tuning a radio receiver

The inductor and capacitorin my car radio are usuallyset at L = 1 mH & C = 3.18pF. Which is my favoriteFM station?

(a) QLSU 91.1(b) WRKF 89.3(c) Eagle 98.1 WDGL

FM radio stations: frequency is in MHz.

srad

srad

LC

/1061.5

/

1018.3101

1

1

8

126

!=

!!!=

=

""

#

MHz

Hz

f

3.89

1093.8

2

7

=

!=

="

#

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 firsttime?

srad

xLC/

1016

11

8!=="

• ω = 2500 rad/s• T = period of one complete cycle = 2π/ω = 2.5 ms• Capacitor will be charged after1/4 cycle i.e at t = 0.6 ms

-1.5

-1

-0.5

0

0.5

1

1.5

Time

Charge

Current

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 of theresulting oscillating current.

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

== )10()1)(1(

10 C

FmHq µ

µ! 0.316 A

)sin( 00 !"" +#= tqi

b a

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 on the

capacitor during a cycle of oscillation?

)cos( 00 !" += tqq

)sin( 00 !"" +#== tqdt

dqi

Maximum current is i0=ωq0 → 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

Damped LC OscillatorDamped LC OscillatorIdeal LC circuit without resistance:

oscillations go on for ever; ω =(LC)-1/2

Real circuit has resistance, dissipatesenergy: oscillations die out, or are“damped”

Math is complicated! Important points:– Frequency of oscillator shifts away fromω = (LC)-1/2

– Peak CHARGE decays with timeconstant = 2L/R

– For small damping, peak ENERGYdecays with time constant = L/R

0 4 8 12 16 200.0

0.2

0.4

0.6

0.8

1.0

UE

time (s)

L

Rt

eC

QU

!

=2

2

max

C

RL

SummarySummary• Capacitor and inductor combination

produces an electrical oscillator, naturalfrequency of oscillator is ω=1/√LC

• Total energy in circuit is conserved:switches between capacitor (electric field)and inductor (magnetic field).

• If a resistor is included in the circuit, thetotal energy decays (is dissipated by R).