Chapter 21Chapter 21Alternating Current Alternating Current

Circuits Circuits

and Electromagnetic and Electromagnetic WavesWavesConceptual questions: 2,3,4,5, 8,11,15

Quick quizzes: 1,2,4,5 Problems: 2a,11,23,48

AC CircuitAC Circuit

Resistor in an AC CircuitResistor in an AC Circuit

Resistors in an AC CircuitResistors in an AC Circuit

The direction of the current has no The direction of the current has no effect on the behavior of the effect on the behavior of the resistorresistor

The rate at which electrical energy The rate at which electrical energy is dissipated in the circuit is given is dissipated in the circuit is given byby P = iP = i22 R R

rms Current and Voltagerms Current and Voltage The The rms currentrms current is the direct is the direct

current that would dissipate the current that would dissipate the same amount of energy in a same amount of energy in a resistor as is actually dissipated by resistor as is actually dissipated by the AC currentthe AC current

Alternating voltages can also be Alternating voltages can also be discussed in terms of rms valuesdiscussed in terms of rms values

maxmax

rms I707.02

II

maxmax

rms V707.02

VV

Ohm’s Law in an AC Ohm’s Law in an AC CircuitCircuit

Ohm’s Law for a resistor, R, in an Ohm’s Law for a resistor, R, in an AC circuitAC circuit ΔVΔVrmsrms = I = Irmsrms R R

What is the resistance of a lightbulb that uses an average power of 75 W when connected to a 60 Hz power source with a peak voltage of 170V?

Problem 21.2.a

Which of the following statements might be true for a resistor connected to an AC generator? (a) Paverage = 0 and iaverage = 0 (b) Pav = 0 and iav > 0 (c) Pav > 0 and iav = 0 (d) Pav > 0 and iav > 0.

QUICK QUIZ 21.1

Capacitors in an AC CircuitCapacitors in an AC Circuit

The current starts The current starts out at a large value out at a large value and charges the and charges the plates of the plates of the capacitor, initially capacitor, initially there is no resistancethere is no resistance

As the charge on the As the charge on the plates increases, the plates increases, the voltage across the voltage across the plates increases and plates increases and the current flowing in the current flowing in the circuit decreasesthe circuit decreases

Capacitors in an AC CircuitCapacitors in an AC Circuit

The voltage lags The voltage lags the current by 90the current by 90oo

Capacitive Reactance and Capacitive Reactance and Ohm’s LawOhm’s Law

Ohm’s Law for a capacitor in an AC Ohm’s Law for a capacitor in an AC circuitcircuit ΔVΔVrmsrms = I = Irmsrms X XCC

Cƒ2

1XC

Inductors in an AC Inductors in an AC CircuitCircuit

The current in the The current in the circuit is impeded by circuit is impeded by the emf of the the emf of the inductorinductor

The voltage across The voltage across the inductor always the inductor always leads the current by leads the current by 90°90°

Inductive Reactance, XInductive Reactance, XLL, , and Ohm’s Lawand Ohm’s Law

Ohm’s Law for the inductorOhm’s Law for the inductor ΔVΔVrmsrms = I = Irmsrms X XLL

XXLL = 2 = 2ƒLƒL

The RLC Series CircuitThe RLC Series Circuit

The current in the The current in the circuit is the same circuit is the same at any time and at any time and varies sinusoidally varies sinusoidally with timewith time

Current and Voltage Current and Voltage Relationships in an RLC Relationships in an RLC CircuitCircuit

The instantaneous The instantaneous voltage across the voltage across the resistor is in phase resistor is in phase with the currentwith the current

The instantaneous The instantaneous voltage across the voltage across the inductor leads the inductor leads the current by 90°current by 90°

The instantaneous The instantaneous voltage across the voltage across the capacitor lags the capacitor lags the current by 90°current by 90°

Phasor DiagramsPhasor Diagrams Represent the voltage Represent the voltage

across each element across each element as a rotating vector, as a rotating vector, called a called a phasorphasor

Its projection on the y-Its projection on the y-axis represents the axis represents the varying instantaneous varying instantaneous voltage in the circuitvoltage in the circuit

The diagram is called The diagram is called a a phasor diagramphasor diagram

Phasor Diagram for RLC Phasor Diagram for RLC Series CircuitSeries Circuit

The voltage across the The voltage across the resistor is on the +x resistor is on the +x axis since it is in phase axis since it is in phase with the currentwith the current

The voltage across the The voltage across the inductor is on the +y inductor is on the +y since it leads the since it leads the current by 90°current by 90°

The voltage across the The voltage across the capacitor is on the –y capacitor is on the –y axis since it lags axis since it lags behind the current by behind the current by 90°90°

Phasor Phasor DiagramDiagram

The phasors are The phasors are added as vectors to added as vectors to account for the account for the phase differences in phase differences in the voltagesthe voltages

ΔVΔVLL and ΔV and ΔVCC are on are on the same line and the same line and so the net y so the net y component is ΔVcomponent is ΔVL L - - ΔVΔVCC

ΔVΔVmaxmax From the Phasor From the Phasor DiagramDiagram

The voltages are not in phase, so they The voltages are not in phase, so they cannot simply be added to get the cannot simply be added to get the voltage across the combination of the voltage across the combination of the elements or the voltage sourceelements or the voltage source

is the is the phase anglephase angle between the between the current and the maximum voltagecurrent and the maximum voltage

2 2max ( )

tan

R L C

L C

R

V V V V

V V

V

For the circuit of the figure below, is the voltage of the source equal to (a) the sum of the maximum voltages across the elements, (b) the sum of the instantaneous voltages across the elements, or (c) the sum of the rms voltages across the elements?

QUICK QUIZ 21.2

Impedance and Ohm’s Impedance and Ohm’s LawLaw

Ohm’s Law can be applied to the Ohm’s Law can be applied to the impedanceimpedance ΔVΔVmaxmax = I = Imaxmax Z Z

R

XXtan

)XX(RZ

CL

2CL

2

Problem 21.11. What value of capacitor must be inserted in a 60 Hz circuit in series with a generator of 170 V maximum voltage to produce an rms current output of 0.75 A?

Problem 21.31. A 60.0 Q resistor, a 3.00 mF capacitor, and a 0.400 H inductor are connected in series to a 90.0 V 60 Hz source. Find the voltage across the LC combination. Repeat for the RC combination.

Problems 11 and 23Problems 11 and 23

QuestionsQuestions

Power in an AC CircuitPower in an AC Circuit No power losses are associated with No power losses are associated with

capacitors and pure inductors in an AC capacitors and pure inductors in an AC circuitcircuit

The average power delivered by the The average power delivered by the generator is converted to internal energy in generator is converted to internal energy in the resistorthe resistor PPavav = I = IrmsrmsΔVΔVRR = = IIrmsrmsΔVΔVrmsrms cos cos cos cos is called the is called the power factorpower factor of the circuit of the circuit

We may maximize P by adjusting the power We may maximize P by adjusting the power factorfactor V

IV

R

rms

rmsV

V

L

C

Resonance in an AC CircuitResonance in an AC Circuit

ResonanceResonance occurs at occurs at the frequency, ƒthe frequency, ƒoo, , where the current where the current has its maximum has its maximum valuevalue To achieve maximum To achieve maximum

current, Z must be current, Z must be minimumminimum

This occurs when XThis occurs when XLL = = XXCC

LC2

1ƒo

2 2( )L CZ R X X

TransformerTransformer

1 1V Nt

2 2V N

t

The use of iron core results in a the same magnetic flux in both Primary and secondary windings

22 1

1

NV V

N

The power input into the primary equals The power input into the primary equals the power output at the secondarythe power output at the secondary

II11ΔVΔV11 = I = I22ΔVΔV22

TransformerTransformer

2. What is the impedance of an RLC circuit at the resonance frequency?

3. When a dc voltage is applied to a transformer, the primary coil sometimes overheats and burns. Why?

4. Why are the primary and secondary coils of a transformer wrapped on an iron core that passes through both coils?

Conceptual questionsConceptual questions

Hertz’s Basic LC CircuitHertz’s Basic LC Circuit When the switch is When the switch is

closed, oscillations closed, oscillations occur in the current occur in the current and in the charge on and in the charge on the capacitorthe capacitor

When the capacitor is When the capacitor is fully charged, the total fully charged, the total energy of the circuit is energy of the circuit is stored in the electric stored in the electric field of the capacitorfield of the capacitor At this time, the current At this time, the current

is zero and no energy is is zero and no energy is stored in the inductorstored in the inductor

EM Waves by an AntennaEM Waves by an Antenna

EM Waves by an AntennaEM Waves by an Antenna

Because the Because the oscillating charges oscillating charges in the rod produce in the rod produce a current, there is a current, there is also a magnetic also a magnetic field generatedfield generated

As the current As the current changes, the changes, the magnetic field magnetic field spreads out from spreads out from the antennathe antenna

Electromagnetic Waves Electromagnetic Waves are Transverse Wavesare Transverse Waves

TheThe E E and and BB fields fields are perpendicular to are perpendicular to each othereach other

Both fields are Both fields are perpendicular to the perpendicular to the direction of motiondirection of motion Therefore, em Therefore, em

waves are waves are transverse wavestransverse waves

Properties of EM WavesProperties of EM Waves

Speed of ALL electromagnetic waves is Speed of ALL electromagnetic waves is

The ratio of the electric field to the magnetic The ratio of the electric field to the magnetic field is equal to the speed of lightfield is equal to the speed of light

Electromagnetic waves carry energy as they Electromagnetic waves carry energy as they travel through spacetravel through space

Ec

B

1

o o

c

2 2max max max max

2 2 2o o o

Average power per un

E B E c

it ar

c

a

B

e

In an apparatus such as that in the figure below, suppose the black disk is replaced by one with half the radius. Which of the following are different after the disk is replaced? (a) radiation pressure on the disk; (b) radiation force on the disk; (c) radiation momentum delivered to the disk in a given time interval.

QUICK QUIZ 21.4

The Spectrum of EM The Spectrum of EM WavesWaves

c = ƒc = ƒλλ Wavelengths for Wavelengths for

visible light range visible light range from 400 nm to 700 from 400 nm to 700 nmnm

There is no sharp There is no sharp division between one division between one kind of em wave and kind of em wave and the nextthe next

15. Does a wire connected to a battery emit an EM wave?

8. When light (or any EM wave) travels across a given region, what is that moves?

11. Suppose a creature from another planet had eyes that were sensitive to infrared radiation. Describe what he would see if he looked around the room you are in now. What would be bright and what would be dim?

QuestionsQuestions

Question 21.5Question 21.5Receiving radio antennas can be in the form of conducting lines or loops. What should the orientation of each of these antennas be, relative to a broadcasting antenna that is perpendicular to the Earth?

Problem 21.48. Assume that the solar radiation incident on the Earth is 1340 W/m2. Calculate the power radiated by the Sun. The average Sun-Earth separation is 1.49 1011 m.

MCADMCAD

A time varying magnetic field

1. Produces an electric field

2. Provides power to rotate an ac generator

3. Can induce currents to flow in a conductive loop

4. Both 1 and 3

A transformer is used to change

1. Voltage

2. Power

3. Current

4. Voltage and current

Which of the following most Which of the following most accurately describes light?accurately describes light?

a.a. An electric and magnetic wave An electric and magnetic wave parallelparallel to to each other and each other and perpendicularperpendicular to the direction to the direction of propagationof propagation

b.b. An electric and magnetic wave An electric and magnetic wave parallelparallel to to each other and each other and parallelparallel to the direction of to the direction of propagationpropagation

c.c. An electric and magnetic wave An electric and magnetic wave perpendicularperpendicular to each other and to each other and perpendicularperpendicular to the to the direction of propagationdirection of propagation

d.d. An electric and magnetic wave An electric and magnetic wave perpendicularperpendicular to each other and to each other and parallelparallel to the direction of to the direction of propagationpropagation