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Physics 1051
Lecture 24
Electromagnetic Waves
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Recommended Problems
Fishbane, Chapter 34, Problems 7,10,14,16,21,24,30
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The story so far
So far we have talked about oscillations and waves,
electric forces and fields, and magnetic forces and
fields.
In the next couple of lectures we use all of this
material to develop a description ofelectromagnetic
waves.
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What do we know about
electricity and magnetism? Charged particles produce electric fields
Moving charges produce magnetic fields
Electric fields cause a force on a charged particle
Magnetic fields cause a force on moving charges
particles
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What do we know about
electricity and magnetism? The electric flux through a closed surface is
proportional to the charge inside the surface
(Gausss Law).
This says that electric fields are due to electric
charges. Electric field lines begin and end atcharges. We can use it to show that the electric
field of a point charge is proportional to 1/r2.
0
Q E dA
I !
&&
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What do we know about
electricity and magnetism? The magnetic flux through a closed surface is zero.
This says that there are no magnetic monopoles,
and that magnetic field lines have no beginning or
end.
0B
dA !
&&
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What do we know about
electricity and magnetism? Integrating the magnetic field around a closed path gives a
result that is proportional to the current enclosed by the
path. (Ampres Law)
This says that magnetic fields are caused by currents.
There is another part to this that we have not discussed:Magnetic fields can also be caused by changing electric
fields. Including this mathematically would give us another
term on the rhs that included the time derivative of a surface
integral of the electric field. I will spare you the details.
0 encB ds IQ !& &
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What do we know about
electricity and magnetism? Changed in magnetic flux cause an induced emf.
(Faradays Law).
This says that changing magnetic fields produce an
electric potential, or in other words changing
magnetic fields produce an electric field.
surface
d
dt
d E ds B dA
dt
I *!
! && &&
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What do we know about
electricity and magnetism? These laws show that there is a strong connection
between electricity and magnetism. In factelectricity and magnetism are two aspects of thesame phenomenon, which we can callelectromagnetism.
The electric and magnetic fields are coupled theydepend on each other.
In a vacuum, if we have no electric charges and noelectric currents, then the equations weve writtendown above are perfectlysymmetric with respect to
Eand B. This means thatEand Bbehave the same.
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Maxwells Equations
The equations weve written down here are a form of
Maxwells Equations.
Maxwell saw that all of these electromagnetic
phenomena could be described in a single
mathematical framework.
The solution to Maxwells equations in a vacuum
turns out to be traveling waves
both the electric andmagnetic field propagate through space in the form of
waves.
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Electromagnetic Waves
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0
0
( , ) sin
( , ) sin
E r t E kx t
B r t B kx t
[
[
!
!
& &&
& &&
k!2T
P
[ !2T
T
! 2Tf
The plane wave solutions to Maxwells Equations have the
following form
where E0 and B0 are theamplitudes of the electric and
magnetic components of the wave.
kis the wave number of the electromagnetic waves. (Really it
should be a vector pointing in the direction of propagation ofthe wave.)
k! wave number
P = wave length
f ! frequency
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Maxwells equations tell us several things:
The electric and magnetic fields propagate as traveling
waves
These waves travel at the speed of light. The speed of the
waves is given by
The electric and magnetic fields are always perpendicular to
each other.
The amplitudes of the fields are also related:
0 0
1v c
Q I! !
0 0 0E B !
& &
E cB!
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Electromagnetic Waves
Maxwells equations, published in 1864, had far reaching
consequences. They showed in detail how electricity and
magnetism were connected. Maxwells equations predicted the
existence of electromagnetic waves, which traveled with the
speed of light. Maxwell conjectured that light was a form ofelectromagnetic radiation.
In 1887 Heinrich Hertz
produced radio waves and
from the measured frequency
and wavelength confirmed
Maxwells conjecture.
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Electromagnetic Spectrum
Radio Waves
Microwaves
Infrared
Visible light
Ultraviolet
X-rays Gamma rays
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Example 1
Fishbane, Chapter34, Problem 5
If the electric field for a plane electromagnetic wave
is given by
what are and the direction of propagation of the
wave?
0
0
cos( )
x
y
E
E E kz t [
!
!
B&
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Example 2
Fishbane, Chapter34, Problem 61.
Use dimensional analysis to show that
has the dimensions of speed.
0 0
1
Q I
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Electromagnetic Energy
Electromagnetic waves carry energy and can transport
energy from one place to another. The energy carried
by an electromagnetic waves is shared equallybetween the magnetic field part of the wave and the
electric field part of the wave. The average value of
the total energy density in the wave is
220 0
0
0
1
2 2
Eu B
I
Q! !
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Intensity and Energy Transport
c
c
x
R
Since electromagnetic fields
represent a form of energy,
and electromagnetic waves
propagate through space, then
electromagnetic waves
transport energy.
Consider a segment of acircular beam of circular light
of radius R. How much
energy is transported across a
surface in time (t?
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Electromagnetic Energy
The power per unit area delivered by an
electromagentic wave to a surface perpendicular to the
direction of propagation is
This is called the intensity of the em wave.
220 0 0 0
0
0 02 2 2
c E EBc I cu B
I
Q Q! ! ! !
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Example
The intensity of sunlight on the
earths surface is approximately
1 kW/m2.
Calculate the energy density of
sunlight at the earths surface.
Estimate the amplitude of the
electric and magnetic fields of
the sunlight striking the earth.
u !I
c
!10
3
3v
10
8 J m-3
! 3.3v106 J m-3
I! uc
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u }I0E0
2
2
To calculate the amplitude of
the electric field we use the
relation
E}2u
I0
}2 v 3.3 v106
8.85v1012V m
-1
} 864 V m -1
Thus we obtain
E0
B0
! c
To calculate the amplitude of
the magnetic field we use the
relation
B0 !E0
c
! 8643v108
T
! 2.87 v106 T
Thus we obtain
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Example 3
A typical laser pointer has a power of around 0.5 mW
and a beam width of around 1mm.
Calculate the intensity of the beam.
Calculate the amplitude of the electric and magnetic
fields of the beam.
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Production of EM Waves
Electromagnetic waves
are produced by creating
an oscillating electric or
magnetic field.
Charges oscillating in a
linear antenna will produce
an oscillating electric field.
An oscillating electric field
will produce an oscillating
magnetic field.
This produces a wave which
travels away from the source
with the speed of light.
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There are other types of
antenna. For example the loop
antenna, uses an oscillating
current to produce a oscillating
magnetic field.