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Transcript of Maxwell’s Equations and - people.Virginia.EDUpeople.virginia.edu/~ben/2415131/Lecture_20.pdf ·...
Copyright © 2009 Pearson Education, Inc.
Chapter 31
Maxwell’s Equations and
Electromagnetic Waves
Copyright © 2009 Pearson Education, Inc.
• Changing Electric Fields Produce Magnetic
Fields; Ampère’s Law and Displacement
Current
• Gauss’s Law for Magnetism
• Maxwell’s Equations
• Production of Electromagnetic Waves
• Electromagnetic Waves, and Their Speed,
Derived from Maxwell’s Equations
• Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
Units of Chapter 31
Copyright © 2009 Pearson Education, Inc.
• Measuring the Speed of Light
• Energy in EM Waves; the Poynting Vector
• Radiation Pressure
• Radio and Television; Wireless
Communication
Units of Chapter 31
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E&M Equations to date
0
0
enc
B
QE dA
dE d
dt
B d I
Copyright © 2009 Pearson Education, Inc.
31-2 Gauss’s Law for Magnetism
Gauss’s law relates the electric field on a
closed surface to the net charge enclosed
by that surface. The analogous law for
magnetic fields is different, as there are no
single magnetic point charges
(monopoles):
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E&M Equations to date - updated
0
0
0
mag
e
enc
B
nc
QE dA
B dA
dE d
dt
B d I
Q
No effect since RHS
identically zero
These two not pretty,
i.e., not symmetric
Now, I suggest 0s mag
magd dQI
t
Q
dt d
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E&M Equations to date – more
updated
0
0
0
0
???
mag
enc
mag
enc
B
QE dA
B dA
dE d
t
d I
Q
I
d
B
Wouldn’t it be nice if we could replace ??? with something?
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Ampère’s law
relates the
magnetic field
around a current
to the current
through a
surface.
31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
0???B d I
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In order for Ampère’s
law to hold, it can’t
matter which surface
we choose. But look
at a discharging
capacitor; there is a
current through
surface 1 but none
through surface 2:
31-1 Changing Electric Fields Produce
Magnetic Fields; Ampère’s Law and
Displacement Current
0???B d I
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Therefore, Ampère’s law is modified to include
the creation of a magnetic field by a changing
electric field – the field between the plates of the
capacitor in this example:
31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
Copyright © 2009 Pearson Education, Inc.
31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
The second term in Ampere’s law has the
dimensions of a current (after factoring out
the μ0), and is sometimes called the
displacement current:
where
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Example 31-1: Charging capacitor.
A 30-pF air-gap capacitor has circular plates of area
A = 100 cm2. It is charged by a 70-V battery through a
2.0-Ω resistor. At the instant the battery is connected,
the electric field between the plates is changing most
rapidly. At this instant, calculate
(a) the current into the plates, and
(b) the rate of change of electric field between the
plates.
(c) Determine the magnetic field induced between the
plates. Assume E is uniform between the plates at
any instant and is zero at all points beyond the edges
of the plates.
31-1 Ampère’s Law and
Displacement Current
E
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31-3 Maxwell’s Equations
We now have a complete set of equations
that describe electric and magnetic fields,
called Maxwell’s equations. In the absence of
dielectric or magnetic materials, they are:
0
0
0
0 0 0
mag
enc
enc
B
mag
E
QE dA
B dA
dE d
dt
d
Q
Id
I
B dt
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Since a changing electric field produces
a magnetic field, and a changing
magnetic field produces an electric field,
once sinusoidal fields are created they
can propagate on their own.
These propagating fields are called
electromagnetic waves.
31-4 Production of Electromagnetic
Waves
ConcepTest 31.1a EM Waves I
Plastic
Copper
A loop with an AC current produces
a changing magnetic field. Two
loops have the same area, but one
is made of plastic and the other
copper. In which of the loops is
the induced voltage greater?
1) the plastic loop
2) the copper loop
3) voltage is same in both
Faraday’s law says nothing about
the material:
The change in flux is the same (and
N is the same), so the induced emf
is the same.
ConcepTest 31.1a EM Waves I
Plastic
Copper
A loop with an AC current produces
a changing magnetic field. Two
loops have the same area, but one
is made of plastic and the other
copper. In which of the loops is
the induced voltage greater?
1) the plastic loop
2) the copper loop
3) voltage is same in both
BdN
dt
%
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Oscillating charges
will produce
electromagnetic
waves:
31-4 Production of Electromagnetic
Waves
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31-4 Production of Electromagnetic
Waves
Close to the antenna,
the fields are
complicated, and are
called the near field:
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Far from the source, the waves
are plane waves:
31-4 Production of Electromagnetic
Waves
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The electric and magnetic waves are
perpendicular to each other, and to the
direction of propagation.
31-4 Production of Electromagnetic
Waves
ConcepTest 31.2 Oscillations
The electric field in an EM
wave traveling northeast
oscillates up and down. In
what plane does the
magnetic field oscillate?
1) in the north-south plane
2) in the up-down plane
3) in the NE-SW plane
4) in the NW-SE plane
5) in the east-west plane
The magnetic field oscillates perpendicular to BOTH the
electric field and the direction of the wave. Therefore the
magnetic field must oscillate in the NW-SE plane.
ConcepTest 31.2 Oscillations
The electric field in an EM
wave traveling northeast
oscillates up and down. In
what plane does the
magnetic field oscillate?
1) in the north-south plane
2) in the up-down plane
3) in the NE-SW plane
4) in the NW-SE plane
5) in the east-west plane
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
In the absence of currents and charges,
Maxwell’s equations become:
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
This figure shows an electromagnetic wave of
wavelength λ and frequency f. The electric and
magnetic fields are given by
where
.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Applying Faraday’s law to the rectangle of
height Δy and width dx in the previous figure
gives a relationship between E and B:
.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Similarly, we apply
Maxwell’s fourth
equation to the
rectangle of length Δz
and width dx, which
gives
.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Using these two equations and the
equations for B and E as a function of time
gives
Here, v is the velocity of the wave.
Substituting,
.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
The magnitude of this speed is
3.0 x 108 m/s – precisely equal
to the measured speed of light.
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31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
Example 31-2: Determining E and B in EM
waves.
Assume a 60-Hz EM wave is a sinusoidal
wave propagating in the z direction with E
pointing in the x direction, and E0 = 2.0 V/m.
Write vector expressions for E and B as
functions of position and time.
E
E
E
B
B
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The frequency of an electromagnetic wave
is related to its wavelength and to the
speed of light:
31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
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Electromagnetic waves can have any
wavelength; we have given different names to
different parts of the wavelength spectrum.
31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Example 31-3: Wavelengths of EM waves.
Calculate the wavelength
(a) of a 60-Hz EM wave,
(b) of a 93.3-MHz FM radio wave, and
(c) of a beam of visible red light from a
laser at frequency 4.74 x 1014 Hz.
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Example 31-4: Cell phone antenna.
The antenna of a cell phone is often ¼
wavelength long. A particular cell phone has
an 8.5-cm-long straight rod for its antenna.
Estimate the operating frequency of this
phone.
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31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Example 31-5: Phone call time lag.
You make a telephone call from New York
to a friend in London. Estimate how long it
will take the electrical signal generated by
your voice to reach London, assuming the
signal is (a) carried on a telephone cable
under the Atlantic Ocean, and (b) sent via
satellite 36,000 km above the ocean.
Would this cause a noticeable delay in
either case?
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The speed of light
was known to be
very large,
although careful
studies of the
orbits of Jupiter’s
moons showed
that it is finite.
One important
measurement, by
Michelson, used a
rotating mirror:
31-7 Measuring the Speed of Light
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Over the years, measurements have become
more and more precise; now the speed of light
is defined to be
c = 2.99792458
108 m/s.
This is then used to define the meter.
31-7 Measuring the Speed of Light
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Energy is stored in both electric and magnetic
fields, giving the total energy density of an
electromagnetic wave:
Each field contributes half the total energy
density:
31-8 Energy in EM Waves; the
Poynting Vector
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This energy is
transported by
the wave.
31-8 Energy in EM Waves; the
Poynting Vector
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The energy transported through a unit area
per unit time is called the intensity:
31-8 Energy in EM Waves; the
Poynting Vector
Its vector form is the Poynting vector:
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31-8 Energy in EM Waves; the
Poynting Vector
Typically we are interested in the average
value of S: S
.
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31-8 Energy in EM Waves; the
Poynting Vector
Example 31-6: E and B from the Sun.
Radiation from the Sun reaches the Earth
(above the atmosphere) at a rate of about
1350 J/s·m2 (= 1350 W/m2). Assume that this is
a single EM wave, and calculate the maximum
values of E and B.
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In addition to carrying energy, electromagnetic
waves also carry momentum. This means that a
force will be exerted by the wave.
The radiation pressure is related to the average
intensity. It is a minimum if the wave is fully
absorbed:
and a maximum if it is fully reflected:
31-9 Radiation Pressure
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31-9 Radiation Pressure
Example 31-7: Solar pressure.
Radiation from the Sun that reaches
the Earth’s surface (after passing
through the atmosphere) transports
energy at a rate of about 1000 W/m2.
Estimate the pressure and force
exerted by the Sun on your
outstretched hand.
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31-9 Radiation Pressure
Example 31-8: A solar sail.
Proposals have been made to use the
radiation pressure from the Sun to help
propel spacecraft around the solar
system. (a) About how much force
would be applied on a 1 km x 1 km
highly reflective sail, and (b) by how
much would this increase the speed of
a 5000-kg spacecraft in one year? (c) If
the spacecraft started from rest, about
how far would it travel in a year?
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This figure illustrates the process by which a
radio station transmits information. The audio
signal is combined with a carrier wave.
31-10 Radio and Television; Wireless
Communication
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The mixing of signal and carrier can be done
two ways. First, by using the signal to modify
the amplitude of the carrier (AM):
31-10 Radio and Television; Wireless
Communication
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Second, by using the signal to modify the
frequency of the carrier (FM):
31-10 Radio and Television; Wireless
Communication
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At the receiving end, the wave is received,
demodulated, amplified, and sent to a
loudspeaker.
31-10 Radio and Television; Wireless
Communication
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The receiving
antenna is
bathed in
waves of many
frequencies; a
tuner is used to
select the
desired one.
31-10 Radio and Television; Wireless
Communication
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31-10 Radio and Television; Wireless
Communication
A straight antenna will have a current induced
in it by the varying electric fields of a radio
wave; a circular antenna will have a current
induced by the changing magnetic flux.
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31-10 Radio and Television; Wireless
Communication
Example 31-9: Tuning a station.
Calculate the transmitting wavelength
of an FM radio station that transmits
at 100 MHz.
ConcepTest 31.3 TV Antennas
Before the days of cable,
televisions often had two
antennae on them, one straight
and one circular. Which antenna
picked up the magnetic
oscillations?
1) the circular one
2) the straight one
3) both equally; they were
straight and circular for
different reasons
The varying B field in the loop
means the flux is changing and
therefore an emf is induced.
ConcepTest 31.3 TV Antennas
Before the days of cable,
televisions often had two
antennae on them, one straight
and one circular. Which antenna
picked up the magnetic
oscillations?
1) the circular one
2) the straight one
3) both equally; they were
straight and circular for
different reasons
ConcepTest 31.4 Radio Antennas
If a radio transmitter has a vertical
antenna, should a receiver’s
antenna be vertical or horizontal
to obtain the best reception?
1) vertical
2) horizontal
3) doesn’t matter
If a wave is sent out from a vertical
antenna, the electric field oscillates
up and down. Thus, the receiver’s
antenna should also be vertical so
that the arriving electric field can set
the charges in motion.
ConcepTest 31.4 Radio Antennas
E field of wave
E field of wave
If a radio transmitter has a vertical
antenna, should a receiver’s
antenna be vertical or horizontal
to obtain the best reception?
1) vertical
2) horizontal
3) doesn’t matter
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• Maxwell’s equations are the basic equations
of electromagnetism:
Summary of Chapter 31
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• Electromagnetic waves are produced by
accelerating charges; the propagation speed
is given by
• The fields are perpendicular to each other
and to the direction of propagation.
Summary of Chapter 31
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• The wavelength and frequency of EM waves
are related:
• The electromagnetic spectrum includes
all wavelengths, from radio waves through
visible light to gamma rays.
• The Poynting vector describes the
energy carried by EM waves:
Summary of Chapter 31