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Chapter 21
Alternating Current Circuits
and Electromagnetic Waves
Conceptual questions: 2,3,4,5, 8,11,15
Quick quizzes: 1,2,4,5
Problems: 2a,11,23,48
AC Circuit
Resistor in an AC Circuit
Resistors in an AC Circuit
The direction of the current has no
effect on the behavior of the resistor
 The rate at which electrical energy is
dissipated in the circuit is given by


P = i2 R
rms Current and Voltage

The rms current is the direct current
that would dissipate the same amount
of energy in a resistor as is actually
dissipated by the AC current
Irms

Imax

 0.707 Imax
2
Alternating voltages can also be
discussed in terms of rms values
Vrms
Vmax

 0.707 Vmax
2
Ohm’s Law in an AC Circuit

Ohm’s Law for a resistor, R, in an AC
circuit

ΔVrms = Irms R
Problem 21.2.a
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?
QUICK QUIZ 21.1
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.
Capacitors in an AC Circuit
The current starts out
at a large value and
charges the plates of
the capacitor, initially
there is no resistance
 As the charge on the
plates increases, the
voltage across the
plates increases and
the current flowing in
the circuit decreases

Capacitors in an AC Circuit

The voltage lags the
current by 90o
Capacitive Reactance and
Ohm’s Law

Ohm’s Law for a capacitor in an AC
circuit

ΔVrms = Irms XC
1
XC 
2 ƒC
Inductors in an AC Circuit
The current in the
circuit is impeded by
the emf of the inductor
 The voltage across the
inductor always leads
the current by 90°

Inductive Reactance, XL, and
Ohm’s Law

Ohm’s Law for the inductor

ΔVrms = Irms XL
XL
= 2ƒL
The RLC Series Circuit

The current in the
circuit is the same at
any time and varies
sinusoidally with
time
Current and Voltage
Relationships in an RLC Circuit
The instantaneous
voltage across the
resistor is in phase with
the current
 The instantaneous
voltage across the
inductor leads the
current by 90°
 The instantaneous
voltage across the
capacitor lags the
current by 90°

Phasor Diagrams



Represent the voltage
across each element as
a rotating vector, called
a phasor
Its projection on the yaxis represents the
varying instantaneous
voltage in the circuit
The diagram is called a
phasor diagram
Phasor Diagram for RLC
Series Circuit
The voltage across the
resistor is on the +x
axis since it is in phase
with the current
 The voltage across the
inductor is on the +y
since it leads the
current by 90°
 The voltage across the
capacitor is on the –y
axis since it lags behind
the current by 90°

Phasor Diagram
The phasors are
added as vectors to
account for the
phase differences in
the voltages
 ΔVL and ΔVC are on
the same line and so
the net y component
is ΔVL - ΔVC

ΔVmax From the Phasor Diagram

The voltages are not in phase, so they cannot
simply be added to get the voltage across the
combination of the elements or the voltage
source
2
Vmax  VR  (VL  VC ) 2
VL  VC
tan  
VR

 is the phase angle between the current and
the maximum voltage
QUICK QUIZ 21.2
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?
Impedance and Ohm’s Law

Ohm’s Law can be applied to the
impedance

ΔVmax = Imax Z
Z  R 2  ( XL  X C ) 2
XL  X C
tan  
R
Problems 11 and 23
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.
Questions
Power in an AC Circuit
No power losses are associated with
capacitors and pure inductors in an AC circuit
 The average power delivered by the
generator is converted to internal energy in
the resistor




Pav = IrmsΔVR = IrmsΔVrms cos 
cos  is called the power factor of the circuit
We may maximize P by adjusting the power
factor
V
VL
I
rms
rms
V
R
V
C
Resonance in an AC Circuit
Z  R 2 ( X L  X C ) 2

Resonance occurs at
the frequency, ƒo,
where the current has
its maximum value


To achieve maximum
current, Z must be
minimum
This occurs when XL = XC
1
ƒo 
2 LC
Transformer
The use of iron core results in a the same magnetic flux in both
Primary and secondary windings

V2   N 2
t

V1   N1
t
N2
V2 
V1
N1
The power input into the primary equals the
power output at the secondary
I1ΔV1 = I2ΔV2
Transformer
Conceptual questions
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?
Hertz’s Basic LC Circuit


When the switch is
closed, oscillations
occur in the current and
in the charge on the
capacitor
When the capacitor is
fully charged, the total
energy of the circuit is
stored in the electric
field of the capacitor

At this time, the current
is zero and no energy is
stored in the inductor
EM Waves by an Antenna
EM Waves by an Antenna
Because the
oscillating charges in
the rod produce a
current, there is also
a magnetic field
generated
 As the current
changes, the
magnetic field
spreads out from
the antenna

Electromagnetic Waves are
Transverse Waves
The E and B fields are
perpendicular to each
other
 Both fields are
perpendicular to the
direction of motion


Therefore, em waves
are transverse waves
Properties of EM Waves
c
1

Speed of ALL electromagnetic waves is

The ratio of the electric field to the magnetic field is
equal to the speed of light
 o o
E
c
B

Electromagnetic waves carry energy as they travel
through space
Average power per unit area 
2
max
2
max
Emax Bmax E
c B


2o
2o c
2o
QUICK QUIZ 21.4
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.
The Spectrum of EM Waves

c = ƒλ
Wavelengths for visible
light range from 400 nm
to 700 nm
 There is no sharp
division between one
kind of em wave and the
next

Questions
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?
Question 21.5
Receiving 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.
MCAD
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
accurately describes light?
a.
b.
c.
d.
An electric and magnetic wave parallel to each
other and perpendicular to the direction of
propagation
An electric and magnetic wave parallel to each
other and parallel to the direction of propagation
An electric and magnetic wave perpendicular to
each other and perpendicular to the direction of
propagation
An electric and magnetic wave perpendicular to
each other and parallel to the direction of
propagation