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Chapter 31
Maxwell’s Equations and
Electromagnetic Waves
Copyright © 2009 Pearson Education, Inc.
Units of Chapter 31
• 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
Copyright © 2009 Pearson Education, Inc.
Units of Chapter 31
• Measuring the Speed of Light
• Energy in EM Waves; the Poynting Vector
• Radiation Pressure
• Radio and Television; Wireless
Communication
Copyright © 2009 Pearson Education, Inc.
E&M Equations to date
Gauss' Law:
 E  dA 
Qenc
0
dB
Faraday's Law:  E  d  
dt
Ampere's Law:
 Bd
 0 I enc
Two for the electric field;
only one for the magnetic
field – not very symmetric!
Copyright © 2009 Pearson Education, Inc.
ConcepTest 31.1a
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?
EM Waves I
1) the plastic loop
2) the copper loop
3) voltage is same in both
Plastic
Copper
ConcepTest 31.1a
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?
Faraday’s law says nothing about
the material:
d
%  N
B
dt
The change in flux is the same (and
N is the same), so the induced emf
is the same.
EM Waves I
1) the plastic loop
2) the copper loop
3) voltage is same in both
Plastic
Copper
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):
Qmag  0 
Copyright © 2009 Pearson Education, Inc.
mag
B

dA



Q
0 enc  0

E&M Equations to date - updated
 E  dA 
Qenc
0
 B  dA    Q
0
mag
enc
dB
 E  d   dt
 Bd
Copyright © 2009 Pearson Education, Inc.
 0 I enc
0
E&M Equations to date - updated
 E  dA 
Qenc
0
mag
B

dA



Q
0 enc

No effect since RHS
identically zero
dB
 E  d   dt

 Bd

 0 I enc
dQ
dQ mag
mag
Now, I 
suggests I 
0
dt
dt
Copyright © 2009 Pearson Education, Inc.
These two not pretty,
i.e., not symmetric
E&M Equations to date – more
updated
 E  dA 
Qenc
0
mag
B

dA



Q
0 enc

mag
d  B I enc
 E  d   dt   0
 Bd

???  0 I enc
Wouldn’t it be nice if we could replace ??? with something?
Copyright © 2009 Pearson Education, Inc.
31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
Ampère’s law
relates the
magnetic field
around a current
to the current
through a
surface.
 Bd
 0 I encl
Copyright © 2009 Pearson Education, Inc.
31-1 Changing Electric Fields Produce
Magnetic Fields; Ampère’s Law and
Displacement Current
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:
Copyright © 2009 Pearson Education, Inc.
31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
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:
Copyright © 2009 Pearson Education, Inc.
31-1 Changing Electric Fields
Produce Magnetic Fields; Ampère’s
Law and Displacement Current
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.
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
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):
Copyright © 2009 Pearson Education, Inc.
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:
Copyright © 2009 Pearson Education, Inc.
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:
Qenc
 E  dA 
0
mag
B

dA



Q
0 enc

d  B I emag
nc
E

d




dt
0
dE
 B  d  0 0 dt  0 I enc
Copyright © 2009 Pearson Education, Inc.
31-4 Production of Electromagnetic
Waves
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.
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31-4 Production of Electromagnetic
Waves
Oscillating charges
will produce
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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31-4 Production of Electromagnetic
Waves
Far from the source, the waves
are plane waves:
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31-4 Production of Electromagnetic
Waves
The electric and magnetic waves are
perpendicular to each other, and to the
direction of propagation.
Copyright © 2009 Pearson Education, Inc.
31-5 Electromagnetic Waves, and
Their Speed, Derived from Maxwell’s
Equations
In the absence of currents and charges,
Maxwell’s equations become:
Copyright © 2009 Pearson Education, Inc.
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
Copyright © 2009 Pearson Education, Inc.
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:
.
Copyright © 2009 Pearson Education, Inc.
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
.
Copyright © 2009 Pearson Education, Inc.
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,
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
The frequency of an electromagnetic wave
is related to its wavelength and to the
speed of light:
Copyright © 2009 Pearson Education, Inc.
31-6 Light as an Electromagnetic Wave
and the Electromagnetic Spectrum
Electromagnetic waves can have any
wavelength; we have given different names to
different parts of the wavelength spectrum.
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
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?
Copyright © 2009 Pearson Education, Inc.
31-8 Energy in EM Waves; the
Poynting Vector
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:
Copyright © 2009 Pearson Education, Inc.
31-8 Energy in EM Waves; the
Poynting Vector
This energy is
transported by
the wave.
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31-8 Energy in EM Waves; the
Poynting Vector
The energy transported through a unit area
per unit time is called the intensity:
Its vector form is the Poynting vector:
Copyright © 2009 Pearson Education, Inc.
31-8 Energy in EM Waves; the
Poynting Vector
Typically we are interested in the average
S
value of S:
.
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
31-9 Radiation Pressure
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:
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
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?
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 31
• Maxwell’s equations are the basic equations
of electromagnetism:
 E  dA 
Qencl
o
mag
B

dA



Q
o encl

 E d

1
o
I
mag
dB

dt
dE
 B  d  o I  o o dt
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 31
• 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.
Copyright © 2009 Pearson Education, Inc.
Summary of Chapter 31
• 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:
Copyright © 2009 Pearson Education, Inc.