Physics 2102

Jonathan Dowling

Lecture 29: WED 25 MAR 09Lecture 29: WED 25 MAR 09Ch. 31.1–4: Electrical Ch. 31.1–4: Electrical

Oscillations, LC Circuits, Oscillations, LC Circuits, Alternating CurrentAlternating Current

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EXAM 03: 6PM THU 02 APR 2009

The exam will cover: Ch.28 (second half) through Ch.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 What are we going to learn?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 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.

Oscillators in PhysicsOscillators in Physics

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

Utot =Ukin +U pot =const Utot =12m v2 +

12kx2

dUtotdt

=0 =12m 2v

dvdt

⎛⎝⎜

⎞⎠⎟ +

12k 2x

dxdt

⎛⎝⎜

⎞⎠⎟

v = ′x (t)a= ′v(t)= ′′x (t)

→ mdvdt

+ k x = 0

)cos()( :Solution 00 φω += txtx

phase : frequency : amplitude :

0

0

φωx

mk=ω

PHYS2101: A Mechanical PHYS2101: A Mechanical OscillatorOscillator

02

2

=+ xkdtxdm

Newton’s law F=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 PHYS2101 An Electromagnetic LC OscillatorOscillator

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 the capacitor and the inductor.

Energy Conservation: Utot =UB +UE

Utot =12Li2 +

12qC

2

UB =12Li2     UE =

12qC

2

Utot =UB +UE Utot =12Li2 +

12qC

2

dUtot

dt=0 =

12L 2i

didt

⎛⎝⎜

⎞⎠⎟ +

12C

2qdqdt

⎛⎝⎜

⎞⎠⎟

VL +VC =0 =Ldidt

⎛⎝⎜

⎞⎠⎟ +

1C

q( )

i = ′q (t)′i (t)= ′′q (t)C

qdtqdL += 2

2

0

ω ≡1LC

q =q0 cos(ω t+ϕ 0 )

Electric Oscillators: the Electric Oscillators: the MathMath

Or loop rule!

i = ′q (t)=−q0ωsin(ω t+ϕ 0 )

′i (t) = ′′q (t) = −ω 2q0 cos(ω t +ϕ 0 )

Energy Cons.

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

LC FrequencyIn Radians/Sec

UB =12L i[ ]

2 =12L q0ω cos(ω t+ϕ 0 )[ ]

2VL =L′i (t)= ω 2q0 sin(ω t+ϕ 0 )⎡⎣ ⎤⎦

2

q =q0 cos(ω t+ϕ 0 )

Electric Oscillators: the Electric Oscillators: the MathMath

i = ′q (t)=−q0ωsin(ω t+ϕ 0 )

′i (t) = ′′q (t) = −ω 2q0 cos(ω t +ϕ 0 )

UE =12

q[ ]C

2

=12C

q0 cos(ω t+ϕ 0 )[ ]2

Energy as Function of TimeVoltage as Function of Time

VC =1C

q(t)[ ] =1C

q0 cos(ω t+ϕ 0 )[ ]

02

2

=+ xkdtxdm

Analogy Between Electrical And Mechanical Oscillations

q→ x 1 / C→ ki→ v L→ m

LC1=ω

)cos()( 00 φω += txtx

mk=ω

Cq

dtqdL += 2

2

0

q =q0 cos(ω t+ϕ 0 )

i = ′q (t)=−q0ωsin(ω t+ϕ 0 )

′i (t) = ′′q (t) = −ω 2q0 cos(ω t +ϕ 0 )

v = ′x (t)=−x0ωsin(ω t+ϕ 0 )

a = ′′x (t)=−ω 2x0 cos(ω t+ϕ 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

ChargeCurrent

)cos( 00 φω += tqq

)sin( 00 φωω +−== tqdtdqi

UB =12Li2 =

12Lω 2q0

2 sin2(ω t+ϕ 0 )

0

0.2

0.4

0.6

0.8

1

1.2

Time

Energy in capacitorEnergy in coil

UE =12qC

2

=12C

q02 cos2(ω t+ϕ 0 )

LCxx 1 and ,1sincos

that,grememberin And

22 ==+ ω

Utot =UB +UE =12C

q02

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

LC Circuit: Conservation LC Circuit: Conservation of Energyof Energy

Example 1 : Tuning a Radio Example 1 : Tuning a Radio ReceiverReceiver

The inductor and capacitor in my car radio are usually set at L = 1 mH & C = 3.18 pF. Which is my favorite FM station?

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

FM radio stations: frequency is in MHz.

ω =1LC

=1

1 × 10−6 × 3.18 × 10−12rad/s

= 5.61 ×108 rad/s

f =ω2p

=8.93×107Hz=89.3 MHz

ExampleExample 22• In an LC circuit, L = 40 mH; C = 4 mF

• At t = 0, the current is a maximum;

• When will the capacitor be fully charged for the first time?

ω =1LC

=1

16x10−8rad/s

• ω = 2500 rad/s• T = period of one complete cycle •T = 2p/ω = 2.5 ms• Capacitor will be charged after T=1/4 cycle i.e at • t = T/4 = 0.6 ms

-1.5

-1

-0.5

0

0.5

1

1.5

Time

ChargeCurrent

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 mF)(10 V) = 10 mC• Switch in position “b”: maximum charge on C = q0 = 10 mC• So, amplitude of oscillating current =ωq0 =

1(1mH)(1μF)

(10μC) = 0.316 A

)sin( 00 φωω +−= tqi

b a

E=10 V1 mH 1 mF

Example 4Example 4In 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?

)cos( 00 φω += tqq

)sin( 00 φωω +−== tqdtdqi

Maximum current is i0=ωq0 Maximum charge: q0=i0/ω

Angular frequency ω=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