Physics 121: Electricity & Magnetism – Lecture 13E-M Oscillations and AC

CurrentDale E. GaryWenda Cao

NJIT Physics Department

December 5, 2007

Electromagnetic Oscillations

C

qU E 2

2

2

2

1LiU B C

qU E 2

2

2

2

1LiU B

December 5, 2007

Oscillating Quantities We will write oscillating quantities with a lower-case symbol, and

the corresponding amplitude of the oscillation with upper case.

Examples:

Oscillating Quantity Amplitude

Voltage v V

Current i I

Charge q Q

)cos( tQq

)(cos22

222

tC

Q

C

q

dt

tdI

dt

di )cos(

December 5, 2007

Derivation of Oscillation Frequency We have shown qualitatively that LC circuits act like an

oscillator. We can discover the frequency of oscillation by looking at the

equations governing the total energy.

Since the total energy is constant, the time derivative should be zero:

But and , so making these substitutions: This is a second-order, homogeneous differential equation,

whose solution is i.e. the charge varies according to a cosine wave with amplitude

Q and frequency . Check by taking two time derivatives of charge:

Plug into original equation:

22

2

1

2Li

C

qUUU BE

0dt

diLi

dt

dq

C

q

dt

dU

dt

dqi

2

2

dt

qd

dt

di 0

2

2

C

q

dt

qdL

)cos( tQq

)sin( tQdt

dq)cos(2

2

2

tQdt

qd

0)cos()cos(22

2

tC

QtLQ

C

q

dt

qdL 0

12 C

LLC

1

dt

dqi

2

2

dt

qd

dt

di

December 5, 2007

1. The expressions below give the charge on a capacitor in an LC circuit. Choose the one that will have the greatest maximum current?

A. q = 2 cos 4t

B. q = 2 cos(4t+/2)

C. q = 2 sin t

D. q = 4 cos 4t

E. q = 2 sin 5t

Which Current is Greatest?

December 5, 2007

2. The three circuits below have identical inductors and capacitors. Rank the circuits according to the time taken to fully discharge the capacitor during an oscillation, greatest first.

A. I, II, III.B. II, I, III.C. III, I, II.D. III, II, I.E. II, III, I.

Time to Discharge Capacitor

I. II. III.

December 5, 2007

Charge, Current & Energy Oscillations

The solution to the equation is , which gives the charge oscillation.

From this, we can determine the corresponding oscillation of current:

And energy

But recall that , so .

That is why our graph for the energy oscillation had the same amplitude for both UE and UB. Note that

)cos( tQq02

2

C

q

dt

qdL

)sin( tQdt

dqi

)(cos22

222

tC

Q

C

qU E

)(sin2

1

2

1 2222 tLQLiU B

LC

1 )(sin

22

2

tC

QU B

C

Qtt

C

QUU BE 2

)](sin)([cos2

222

2

Constant

December 5, 2007

Damped Oscillations Recall that all circuits have at least a

little bit of resistance. In this general case, we really have

an RLC circuit, where the oscillations get smaller with time. They are said to be “damped oscillations.”

Damped Oscillations

Then the power equation becomesLRte 2/

Ridt

diLi

dt

dq

C

q

dt

dU 2

Power lost due to resistive heating As before, substituting and gives the differential equation for q dt

dqi

2

2

dt

qd

dt

di

02

2

C

q

dt

dqR

dt

qdL

22 )2/( LR )cos(2/ tQeq LRt

Solution:

December 5, 2007

3. How does the resonant frequency for an ideal LC circuit (no resistance) compare with ’ for a non-ideal one where resistance cannot be ignored?

A. The resonant frequency for the non-ideal circuit is higher than for the ideal one (’ > ).

B. The resonant frequency for the non-ideal circuit is lower than for the ideal one (’ < ).

C. The resistance in the circuit does not affect the resonant frequency—they are the same (’ = ).

Resonant Frequency

December 5, 2007

Alternating Current The electric power out of a home or office power socket is in the form of

alternating current (AC), as opposed to the direct current (DC) of a battery.

Alternating current is used because it is easier to transport, and easier to “transform” from one voltage to another using a transformer.

In the U.S., the frequency of oscillation of AC is 60 Hz. In most other countries it is 50 Hz.

The figure at right shows one way to make an alternating current by rotating a coil of wire in a magnetic field. The slip rings and brushes allow the coil to rotate without twisting the connecting wires. Such a device is called a generator.

It takes power to rotate the coil, but that power can come from moving water (a water turbine), or air (windmill), or a gasoline motor (as in your car), or steam (as in a nuclear power plant).

tdm sin )sin( tIi d

December 5, 2007

RLC Circuits with AC Power When an RLC circuit is driven with an AC

power source, the “driving” frequency is the frequency of the power source, while the circuit can have a different “resonant” frequency .

Let’s look at three different circuits driven by an AC EMF. The device connected to the EMF is called the “load.”

What we are interested in is how the voltage oscillations across the load relate to the current oscillations.

We will find that the “phase” relationships change, depending on the type of load (resistive, capacitive, or inductive).

d

2)2/(/1 LRLC

December 5, 2007

A Resistive Load Phasor Diagram: shows the

instantaneous phase of either voltage or current.

For a resistor, the current follows the voltage, so the voltage and current are in phase ().

If

Then

tR

VtIi d

RdRR sinsin

tVv dRR sin

December 5, 2007

4. The plot below shows the current and voltage oscillations in a purely resistive circuit. Below that are four curves. Which color curve best represents the power dissipated in the resistor?

A. The green curve (straight line).B. The blue curve.C. The black curve.D. The red curve.E. None are correct.

Power in a Resistive Circuit

PR

t

December 5, 2007

For a capacitive load, the voltage across the capacitor is proportional to the charge

But the current is the time derivative of the charge

In analogy to the resistance, which is the proportionality constant between current and voltage, we define the “capacitive reactance” as

So that .

The phase relationship is that º, and current leads voltage.

A Capacitive Load

tX

Vi d

C

CC cos

CX

dC

1

tC

Q

C

qv dC sin

tCVdt

dqi dCdC cos

December 5, 2007

An Inductive Load For an inductive load, the voltage across the

inductor is proportional to the time derivative of the current

But the current is the time derivative of the charge

Again in analogy to the resistance, which is the proportionality constant between current and voltage, we define the “inductive reactance” as

So that .

The phase relationship is that º, and current lags voltage.

tX

Vi d

L

LL cos

LX dL

dt

diLv L

L

tL

Vdtt

L

Vi d

d

Ld

LL

cos sin

December 5, 2007

5. We just learned that capacitive reactance is and inductive reactance is . What are the units of reactance?

A. Seconds per coulomb.B. Henry-seconds.C. Ohms.D. Volts per Amp.E. The two reactances have different units.

Units of Reactance

LX dL C

Xd

C 1

December 5, 2007

Summary Table

Circuit Element

Symbol Resistance or Reactance

Phase of Current

Phase Constant

Amplitude Relation

Resistor R R In phase with vR

0º (0 rad) VR=IRR

Capacitor C XC=1/dC Leads vR by 90º

90º (/2) VC=ICXC

Inductor L XL=dL Lags vR by 90º

90º (/2) VL=ILXL

December 5, 2007

Summary Energy in inductor:

LC circuits: total electric + magnetic energy is conserved LC circuit:

LRC circuit:

Resistive, capacitive, inductive

2

2

1LiU B Energy in magnetic field

22

2

1

2Li

C

qUUU BE

)cos( tQqLC

1

Charge equation Current equationOscillation frequency

)sin( tQi

Charge equation Oscillation frequency

22 )2/( LR )cos(2/ tQeq LRt

tR

VtIi d

RdRR sinsin t

X

Vi d

C

CC cos t

X

Vi d

L

LL cos

CX

dC

1

LX dL RX R Reactances:

December 5, 2007

6. How did you like using the clickers in this class?

A. Great!B. It had its moments.C. I could take it or leave it.D. I would rather leave it.E. It was the worst!

Thoughts on Clickers

December 5, 2007

7. Which answer describes the most important way that the clicker aided you in learning the material?

A. It helped me to think about the material presented just before each question.

B. It broke up the lecture and kept me awake.C. It tested my understanding.D. It provided immediate feedback.E. It showed me what others were thinking.

Thoughts on Clickers

December 5, 2007

8. Which answer describes the second most important way that the clicker aided you in learning the material?

A. It helped me to think about the material presented just before each question.

B. It broke up the lecture and kept me awake.C. It tested my understanding.D. It provided immediate feedback.E. It showed me what others were thinking.

Thoughts on Clickers

December 5, 2007

9. How would you react to clickers being used in other classes at NJIT?

A. I think it would be excellent.B. I think it is a good idea.C. I wouldn’t mind.D. I would rather not.E. I definitely hope not.

Thoughts on Clickers

December 5, 2007

10. What problems did you have with your clicker?

A. I had no problems with my clicker.B. It was too big or bulky, a pain to carry around.C. I had trouble remembering to bring it to class.D. My clicker had mechanical problems.E. I lost or misplaced it (for all or part of the

semester).

Thoughts on Clickers

December 5, 2007

11. If you had the choice between using a clicker versus having a lecture quiz where you had to fill in a scantron, which would you prefer?

A. I would prefer the clicker.B. I would prefer the scantron quiz.

Thoughts on Clickers

December 5, 2007

12. Please click any button on your clicker as you turn your clicker in. This will register your name as having turned in your clicker.

Have a Nice Day