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Chapter 34
Electromagnetic Waves
Maxwell’s Theory
• Electricity and magnetism were originally thought to be
unrelated
• Maxwell’s theory showed a close relationship between
all electric and magnetic phenomena and proved that
electric and magnetic fields play symmetric roles in
nature
• Maxwell hypothesized that a changing electric field
would produce a magnetic field
• He calculated the speed of light – 3x108 m/s
– and concluded that light and other
electromagnetic waves consist of
James Clerk Maxwell
fluctuating electric and magnetic fields
1831-1879
Maxwell’s Theory
• Stationary charges produce only electric fields
• Charges in uniform motion (constant velocity) produce
electric and magnetic fields
• Charges that are accelerated produce electric and
magnetic fields and electromagnetic waves
• A changing magnetic field produces an electric field
• A changing electric field produces a magnetic
field
• These fields are in phase and, at any point,
they both reach their maximum value at the
James Clerk Maxwell
same time
1831-1879
Modifications to Ampère’s Law
• Ampère’s Law is used to analyze magnetic fields
created by currents
• But this form is valid only if any electric fields present
are constant in time
• Maxwell modified the equation to include time-varying
electric fields and added another term, called the
displacement current, Id
• This showed that magnetic fields are produced both by
conduction currents and by time-varying electric fields
 
d E
 B  ds  0 I  0 0 dt
d E
Id  0
dt
Maxwell’s Equations
• In his unified theory of electromagnetism, Maxwell
showed that the fundamental laws are expressed in
these four equations:
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Maxwell’s Equations
• Gauss’ Law relates an electric field to the charge
distribution that creates it
• The total electric flux through any closed surface equals
the net charge inside that surface divided by o
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Maxwell’s Equations
• Gauss’ Law in magnetism: the net magnetic flux
through a closed surface is zero
• The number of magnetic field lines that enter a closed
volume must equal the number that leave that volume
• If this wasn’t true, there would be magnetic monopoles
found in nature
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Maxwell’s Equations
• Faraday’s Law of Induction describes the creation of an
electric field by a time-varying magnetic field
• The emf (the line integral of the electric field around any
closed path) equals the rate of change of the magnetic
flux through any surface bounded by that path
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Maxwell’s Equations
• Ampère-Maxwell Law describes the creation of a
magnetic field by a changing electric field and by
electric current
• The line integral of the magnetic field around any closed
path is the sum of o times the net current through that
path and oo times the rate of change of electric flux
through any surface bounded by that path
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Maxwell’s Equations
• Once the electric and magnetic fields are known at
some point in space, the force acting on a particle of
charge q can 
be found



F  qE  qv  B
• Maxwell’s equations with the Lorentz Force Law
completely describe all classical electromagnetic
interactions
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Maxwell’s Equations
• In empty space, q = 0 and I = 0
• The equations can be solved with wave-like solutions
(electromagnetic waves), which are traveling at the
speed of light
• This result led Maxwell to predict that light waves were
a form of electromagnetic radiation
 
 B  dA  0
 
d E
 B  ds  0 I  0 0 dt
  q
 E  dA 
0
 
d B
E

d
s



dt
Electromagnetic Waves
• From Maxwell’s equations applied to empty space, the
following relationships can be found:
 E
 E
 0 0 2
2
x
t
2
2
 B
 B
 0 0 2
2
x
t
2
2
• The simplest solutions to these partial differential
equations are sinusoidal waves – electromagnetic
waves:
E  Emax cos(kx  t ); B  Bmax cos(kx  t )
• The speed of the electromagnetic wave is:

E Emax
v 
 c  2.99792  10 m/s  
k
B Bmax
 0 0
1
8
Plane Electromagnetic Waves
• The vectors for the electric and magnetic
fields in an em wave have a specific spacetime behavior consistent with Maxwell’s
equations
• Assume an em wave that travels in the x
direction
• We also assume that at any point in space,
the magnitudes E and B of the fields depend
upon x and t only
• The electric field is assumed to be in the y
direction and the magnetic field in the z
direction
Plane Electromagnetic Waves
• The components of the electric and
magnetic fields of plane electromagnetic
waves are perpendicular to each other and
perpendicular to the direction of
propagation
• Thus, electromagnetic waves are
transverse waves
• Waves in which the electric and magnetic
fields are restricted to being parallel to a
pair of perpendicular axes are said to be
linearly polarized waves
Poynting Vector
• Electromagnetic waves carry energy
John Henry Poynting
can 1852 – 1914
• As they propagate through space, they
transfer that energy to objects in their path
• The rate of flow of energy in an em wave is
described by a vector, S, called the
Poynting vector defined as: S  1 E  B
μo
• Its direction is the direction of propagation
and its magnitude varies in time
• The SI units: J/(s.m2) = W/m2
• Those are units of power per unit area
Poynting Vector
• Energy carried by em waves is shared equally by the
electric and magnetic fields
• The wave intensity, I, is the time average of S (the
Poynting vector) over one or more cycles
• When the average is taken, the time average of cos2(kx ωt) = ½ is involved
I  Savg
2
2
Emax Bmax Emax
c Bmax



2μo
2μo c
2μo
Energy Density
• The energy density, u, is the energy per unit volume
2
1
B
2
u

u

ε
E

• It can be shown that B
E
o
2
2μo
• The instantaneous energy density associated with the
magnetic field of an em wave equals the instantaneous
energy density associated with the electric field and in a
given volume this energy is shared equally by E and B
• The total instantaneous energy density is the sum of the
energy densities associated with each field:
u =uE + uB = εoE2 = B2 / μo
Energy Density
• When this is averaged over one or more cycles, the total
average becomes
uavg = εo(E2)avg = ½ εoE2max = B2max / 2μo
• The intensity of an em wave equals the average energy
density multiplied by the speed of light
I = Savg = cuavg
• Electromagnetic waves transport linear momentum as
well as energy
• As this momentum is absorbed by some surface,
pressure is exerted on the surface
Chapter 34
Problem 6
An electron moves through a uniform electric field E = (2.50 i^ + 5.00
j^) V/m and a uniform magnetic field B = (0.400 k^) T. Determine the
acceleration of the electron when it has a velocity v = 10.0 i^ m/s.
Hertz’s Experiment
• In 1887 Hertz was the first to experimentally
generate and detect electromagnetic waves
• An induction coil was connected to two
large spheres forming a capacitor
• Oscillations were initiated by short voltage
pulses by the coil
• As the air in the gap is ionized, it becomes a
better conductor
• At a very high frequencies the discharge
between the electrodes exhibited an
oscillatory behavior
Heinrich Rudolf
Hertz
1857 – 1894
Hertz’s Experiment
• The inductor and capacitor formed the
transmitter, equivalent to an LC circuit from
a circuit viewpoint
• Several meters away from the transmitter
was the receiver (a single loop of wire
connected to two spheres) with its own
inductance and capacitance
• When the resonance frequencies of the
transmitter and receiver matched, energy
transfer occurred between them
Heinrich Rudolf
Hertz
1857 – 1894
Hertz’s Results
• Hertz hypothesized the energy transfer was in the form
of waves (now known to be electromagnetic waves)
• Hertz confirmed Maxwell’s theory by showing the waves
existed and had all the properties of light waves (with
different frequencies and wavelengths)
• Hertz measured the speed of the waves from the
transmitter (used the waves to form an interference
pattern and calculated the wavelength)
• The measured speed was very close to 3 x 108 m/s, the
known speed of light, which provided evidence in
support of Maxwell’s theory
Electromagnetic Waves Produced by
an Antenna
• Neither stationary charges nor steady currents can
produce electromagnetic waves
• The fundamental mechanism responsible for this
radiation: when a charged particle undergoes an
acceleration, it must radiate energy in the form of
electromagnetic waves
• Electromagnetic waves are radiated by any circuit
carrying alternating current
• An alternating voltage applied to the wires of an
antenna forces the electric charge in the antenna to
oscillate
Electromagnetic Waves Produced by
an Antenna
• Half-wave antenna: two rods are connected to an ac
source, charges oscillate between the rods (a)
• As oscillations continue, the rods become less charged,
the field near the charges decreases and the field
produced at t = 0 moves away from the rod (b)
• The charges and field reverse (c) and the oscillations
continue (d)
Electromagnetic Waves Produced 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
• The magnetic field lines form concentric
circles around the antenna and are
perpendicular to the electric field lines at
all points
• The antenna can be approximated by an
oscillating electric dipole
The Spectrum of EM Waves
• Types of electromagnetic
waves are distinguished
by their frequencies
(wavelengths): c = ƒ λ
• There is no sharp
division between one
kind of em wave and the
next – note the overlap
between types of waves
The Spectrum of EM Waves
• Radio waves are used in
radio and television
communication systems
• Microwaves (1 mm to 30
cm) are well suited for
radar systems +
microwave ovens are an
application
• Infrared waves are
produced by hot objects
and molecules and are
readily absorbed by most
materials
The Spectrum of EM Waves
• Visible light (a small
range of the spectrum
from 400 nm to 700 nm) –
part of the spectrum
detected by the human
eye
• Ultraviolet light (400 nm
to 0.6 nm): Sun is an
important source of uv
light, however most uv
light from the sun is
absorbed in the
stratosphere by ozone
The Spectrum of EM Waves
• X-rays – most common
source is acceleration of
high-energy electrons
striking a metal target,
also used as a diagnostic
tool in medicine
• Gamma rays: emitted by
radioactive nuclei, are
highly penetrating and
cause serious damage
when absorbed by living
tissue
Chapter 34
Problem 11
In SI units, the electric field in an electromagnetic wave is described
by Ey = 100 sin (1.00 × 107 x – ωt). Find (a) the amplitude of the
corresponding magnetic field oscillations, (b) the wavelength λ, and
(c) the frequency f.
Answers to Even Numbered Problems
Chapter 34:
Problem 10
733 nT
Properties of em Waves, 3
• Electromagnetic waves obey the superposition
principle