Physics 2102

Jonathan Dowling

Lecture 29: WED 25 MAR 09Lecture 29: WED 25 MAR 09Ch. 31.1Ch. 31.1––4: Electrical Oscillations, LC4: Electrical Oscillations, LC

Circuits, Alternating CurrentCircuits, Alternating Current

EXAM 03: 6PM THU 02 APR 2009

The exam will cover:Ch.28 (second half) throughCh.32.1-3 (displacement current,and Maxwell's equations).

The exam will be based on:HW08 – HW11

Final Day to Drop Course: FRI 27 MAR

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 practicalapplications, for instance, to keep time, orto focus energy in a system.

All oscillators can store energy inmore than one way and exchangeit back and forth between thedifferent storage possibilities. Forinstance, in pendulums (and swings)one exchanges energy betweenkinetic and potential form.

Oscillators in PhysicsOscillators in Physics

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

Utot =Ukin +Upot = const Utot=1

2mv

2+1

2k x

2

dUtot

dt= 0 =

1

2m 2v

dv

dt

!"#

$%&+1

2k 2x

dx

dt

!"#

$%&

v = !x (t)

a = !v (t) = !!x (t)

! mdv

dt+ k x = 0

)cos()( :Solution00

!" += txtx

phase :

frequency :

amplitude :

0

0

!

"

x

m

k=!

PHYS2101: A Mechanical OscillatorPHYS2101: A 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.

PHYS2101 An Electromagnetic LC OscillatorPHYS2101 An Electromagnetic LC Oscillator

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

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

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

Energy!Conservation:!Utot=U

B+U

E

Utot =1

2L i

2+1

2

q

C

2

UB =1

2L i

2!!!!!UE =

1

2

q

C

2

Utot=U

B+U

E Utot =1

2L i

2+1

2

q

C

2

dUtot

dt= 0 =

1

2L 2i

di

dt

!"#

$%&+1

2C2q

dq

dt

!"#

$%&

VL +VC = 0 = Ldi

dt

!"#

$%&+1

Cq( )

i = !q (t)

!i (t) = !!q (t)C

q

dt

qdL +=

2

2

0

! "1

LC

q = q0cos(! t +"

0)

Electric Oscillators: the MathElectric Oscillators: the Math

Or loop rule!

i = !q (t) = "q0# sin(# t +$

0)

!i (t) = !!q (t) = "# 2q0cos(# t +$

0)

Energy Cons.

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

LC FrequencyIn Radians/Sec

UB =1

2L i[ ]

2=1

2L q

0! cos(! t +"

0)[ ]2

VL = L !i (t) = " 2q0sin(" t +#

0)$% &'2

q = q0cos(! t +"

0)

Electric Oscillators: the MathElectric Oscillators: the Math

i = !q (t) = "q0# sin(# t +$

0)

!i (t) = !!q (t) = "# 2q0cos(# t +$

0)

UE =1

2

q[ ]C

2

=1

2Cq0cos(! t +"

0)[ ]2

Energy as Function of Time Voltage as Function of Time

VC =1

Cq(t)[ ] =

1

Cq0cos(! t +"

0)[ ]

02

2

=+ xkdt

xdm

Analogy Between Electrical And Mechanical Oscillations

q! x 1 /C! k

i! v L! m

LC

1=!

)cos()( 00 !" += txtx

m

k=!

C

q

dt

qdL +=

2

2

0

q = q0cos(! t +"

0)

i = !q (t) = "q0# sin(# t +$

0)

!i (t) = !!q (t) = "# 2q0cos(# t +$

0)

v = !x (t) = "x0# sin(# t +$

0)

a = !!x (t) = "# 2x0cos(# t +$

0)

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

-1.5

-1

-0.5

0

0.5

1

1.5

Time

Charge

Current

)cos( 00 !" += tqq

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

dqi

UB =1

2Li

2=1

2L! 2

q0

2sin

2(! t +"

0)

0

0.2

0.4

0.6

0.8

1

1.2

Time

Energy in capacitor

Energy in coil

UE =1

2

q

C

2

=1

2Cq0

2cos

2(! t +"

0)

LCxx

1 and ,1sincos

that,grememberin And

22==+ !

Utot =UB +UE =1

2Cq0

2

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

LC Circuit: Conservation of EnergyLC Circuit: Conservation of Energy

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 favorite FMstation?

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

FM radio stations: frequency is in MHz.

! =1

LC

=1

1"10#6" 3.18 "10

#12rad/s

= 5.61"108rad/s

f =!

2"

= 8.93#107Hz

= 89.3!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 thefirst time?

! =1

LC=

1

16x10"8rad/s

• ω = 2500 rad/s• T = period of onecomplete cycle•T = 2π/ω = 2.5 ms• Capacitor will becharged after T=1/4cycle i.e at• t = T/4 = 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 along time. It is then thrownto position “b.”

• Calculate the amplitude ωq0of the resulting oscillatingcurrent.

• 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 =

!q0=

1

(1mH)(1µF)(10µC) = 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 q0 on

the capacitor during a cycle of oscillation?

)cos( 00 !" += tqq

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

dqi

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