Download Electromagnetic Waves

Lecture 5.1
Electromagnetic Waves
During our study of classical mechanics we discussed mechanical waves. The
most important everyday phenomenon to which you can apply your knowledge of
mechanical waves is distribution of sound. We learned that harmonic waves give an
example of periodic process in both time and in space. Mechanical wave is not the only
example of the wave motion in nature. Another phenomenon which has remarkably
similar behavior is light. Light, of course, is not a mechanical wave. In order for
mechanical wave to propagate, it is necessary to have some sort of medium. We know,
however, that we can see light coming from distant stars. They are separated from us by
an essentially empty space, which can not transmit any mechanical waves. So, the nature
of light must be different. The first answer which comes in mind is that maybe light is not
a wave at all. To see if this is true or not, we have to study properties of light in details.
First of all let us try to understand what electromagnetic wave is. We already
know the basic properties of electric and magnetic fields. The first condition for electric
field to exist is the presence of electric charges. We know that a stationary electric charge
produces electrostatic field. At the same time moving electric charge produces magnetic
field. If this charge is not just moving with constant speed but is accelerating then we
have an example of alternating electric current. This current will be a source of changing
magnetic field. So, if one considers magnetic flux of this changing magnetic field through
the surface of some closed loop then, according to Faraday’s law, the electromotive force
or induced electric field will appear in this loop. So we see a pattern: changing electric
field causes appearance of the changing magnetic field which in its turn causes the
appearance of the changing electric field and so on. This is how electromagnetic waves
are formed.
This process can be described mathematically with the help of Maxwell’s
equations. Theoretical predictions made by Maxwell on the basis of these equations were
proven in real experiments.
Solving the equations in general, Maxwell obtained the solution in the form of the
electromagnetic wave which, according to his theory, should propagate in vacuum with
the speed of light. Since electromagnetic wave, as any other wave, is the process of
propagation of energy in space then this electromagnetic wave can be detected by placing
the source of the wave (transmitting antenna) in one point and detecting the wave by
means of the receiving antenna in the other point in space. Those experiments were first
performed by Hertz.
Electromagnetic (EM) wave is a transverse wave, where both electric and
magnetic fields are in the direction perpendicular to the direction of the wave’s
transmission. At the same time electric field is perpendicular to magnetic field. If we
have harmonic sin-like electromagnetic wave, which is the simplest example of such a
wave, both electric and magnetic fields are changing with time according to the sin-like
law. It turns out that those changes for both fields are in phase, so both electric and
magnetic fields are reaching their maximum and minimum values simultaneously. The
maximum values of the fields are related as
E  cB
(5.1.1)
Direction of propagation of EM wave is related to directions of electric and magnetic
fields by means of the same right-hand rule: Point the fingers of your right hand in the
direction of electric field, curl your fingers towards direction of magnetic field and your
thumb will point in the direction of propagation of the wave.
Since electromagnetic waves are related to behavior of both types of fields,
Maxwell was able to calculate the speed of those waves from his equations in terms of
both electric and magnetic constants, which is c 
1
 0 0
 3*108
m
. On the other hand
s
one can measure the speed of light using Fizeau’s experiment with rotating wheel. This
experiment predicts the same value as the speed of electromagnetic wave in vacuum from
Maxwell’s equations. The exact value of the speed of light, based on the modern
measurements is c  2.99792458  108 m s .
Even though all electromagnetic waves have the same speed in vacuum (about the
same as in air) but they can have different wavelengths and frequencies. We know how
wavelength, frequency and speed are related, which is
cf .
(5.1.2)
So, the higher the frequency is the lower wavelength will be. You can estimate
wavelengths of typical radio-waves by knowing frequency of radio station. Visible light
has frequencies much higher and wavelengths much shorter than radio waves. Figure
33.1 in the book shows values of wavelengths and frequencies for different parts of
electromagnetic spectrum. You can see that visible light of different colors has different
wavelengths. The entire visible spectrum is located in the range of wavelengths from
3.7  107 m for violet color to 7.5  107 m for red color. The waves with lower
wavelengths are called ultraviolet light; the waves with higher wave lengths are called
infrared light.
Even though all electromagnetic waves have the same speed in vacuum, but they
may have very different speeds in different media. For instance, we know that walls of
the building do not allow passing of visible light but radio waves can penetrate into the
building. In any material the speed v of electromagnetic wave is going to be less
compared to its value c in vacuum, which is
n
c
v
(5.1.3)
index of refraction for a given substance is always bigger than 1. Not only this index of
refraction depends on the type of material, but it also depends on frequency of EM wave.
In the case of visible light, the light of different colors will have different speeds in the
same medium. This phenomenon is known as dispersion. As light passes from one
substance to another substance it should have the same frequency, because atoms of the
new substance are going to oscillate at the same frequency as the frequency of EM wave,
which causes these oscillations. This means that having the same frequency but different
speed, light should have different values of wavelengths in different substances. As light
travels from vacuum into some other medium, its wavelength decreases by the value of
the index of refraction for this medium.
If one directs a beam of white light to the glass prism, it will be bended by this
prism. Because wavelengths of different colors have different speeds in the medium of
the prism, they will be banded by different angles. As a result the white light will be split
into a spectrum of different colors. This shows that white light is not a pure light but
rather a mixture of different colors. This experiment was first performed by Newton.
The human eye is the most sensitive for wavelengths of three primary colors red,
blue and green. All other colors which we can see are the results of combination of these
3 primary colors. However, the things we see in most part of cases do not produce their
own light waves. Instead they reflect light coming from other sources of light. The fact
that we see most of the things in different colors means that they reflect different
wavelengths differently. The light of some color (or colors) may be reflected by the
object and so we see it appearing in this color, while other colors are absorbed by this
object.
Since electromagnetic wave as any other wave is a process of propagation of
energy in space, we need to figure out what is the energy transmitted by EM waves. This
energy is stored in the form of both electric and magnetic energy. At every point the
density of electric energy is exactly the same as the density of magnetic energy and each
of them compose the half of the wave’s energy. During our study of capacitors and
inductors, we have discussed both these types of energy and proved that the volume
density of electric energy is
1
uE   0 E 2
2
(5.1.4)
and the volume density of magnetic energy is
uB 
1 2
B ,
2 0
(5.1.5)
so the total energy density of EM wave is
1
1 2
1 2
u  0E2 
B  0E2 
B .
2
2 0
0
(5.1.6)
One can also introduce the Poynting vector, which is in the direction of propagation of
wave and has the dimension of the energy transferred by this wave across a unit area at
any instant, which is
S
1
EB ,
0
(5.1.7)
The absolute value of this vector is then
S
1
c 0
E2 ,
(5.1.8)
Since both fields are oscillating with time, we can also introduce rms and average values
of those quantities, which is
Erms 
E max
Brms 
Bmax
2
2
,
,
2
uavg   0 Erms

(5.1.9)
1
0
2
Brms
In a same way as for any other type of waves, we can talk about wave’s intensity, which
is the average energy transmitted by wave per unit of time through the unit of area
perpendicular to the direction of the wave’s propagation.
I
P uavg Act
c 2

 uavg c   0cErms 2 
B  Savg
A
At
0 rms
(5.1.10)
Another property of light which we shall discuss is called polarization. We have
already mentioned that light is a transverse wave, where directions of both electric and
magnetic fields are perpendicular to the direction of the wave’s propagation. To describe
this wave, we can specify the direction in which electric field oscillates. If electric field
oscillates in one plane only then this will be an example of completely polarized light. In
reality, however, regular light is a mixture of waves with different directions of
polarization. This is called unpolarized light. The most common way to produce
polarized light from unpolarized light is by using polarization filter which only allows
waves with certain polarization to pass. Such filters can be made from diachronic crystals
or from polymer films (for instance polymer films are placed on glasses and photo
objectives). In order to see if the light is polarized or not one can put two filters with
different orientations of the polar axes. The first filter will produce the light polarized in
certain direction. If the filter’s transmission axis makes angle  with the direction of
electric field in EM wave, then the only component of electric field, which can pass
through the filter is the component of the field parallel to its axis, which is E  E cos  .
Remembering that intensity of light is proportional to the square of electric field, we can
conclude that the intensity of light passing through this filter is going to be
I  I 0 cos2  ,
(5.1.11)
where I 0 is the original intensity of unpolarized light. This equation is known as Malus’
law. If original light was completely unpolarized (with equal probability of having any
orientation of the electric field) then it will loose half of its intensity passing through a
polarizer.
I
1
I0
2
(5.1.12)
If one uses 2 filters then rotating the second filter relative to the first filter we can
achieve the situation when no light is passing through this combination at all. This
happens if the axis of the second filter is perpendicular to the axis of the first filter.
The fact that light is a wave was proven by studying different phenomena such as
interference, diffraction and polarization. We will talk about interference and diffraction
later but we have just seen that polarization does take place for light as well as for any
other EM wave. This phenomenon can only be explained and understood if we admit
wave nature of light.
However, there are many other things we know about light and use in our
everyday life, which can be explained without looking into its wave nature. In fact, most
of these phenomena were described before people even realized that light is a wave. This
refers to things such as formation of images in mirrors and simple lenses. As long as the
size of the object is large enough compared to the wavelength of light, it is not necessary
to account for its wave properties. The part of physics which studies light from this
standpoint is known as Geometrical Optics. The only assumption about light that we will
have to make in order to describe these phenomena is that light propagates along the
straight lines.
If we want we can look at this from the wave-theory standpoint. We know that
light waves have wave fronts. The wave fronts are the surfaces where electromagnetic
wave has a constant phase. In the case of the point-like source of light, we have a
spherical wave propagating in all directions from the source. So, the wave fronts are
spheres and light’s rays are the straight lines in the radial directions from the source. If
the distance from the source is relatively large, we can neglect curvature of the wave
front and consider light as a plane wave or as a straight ray going in horizontal direction.
This is the only assumption we need to study the processes related to reflection and
refraction of light rays.
Talking about image formation in general, we know, of course, that not all the
objects are sources of light. For instance, if you turn off the lights in the room without
windows, you will hardly be able to see anything. So our vision becomes possible only
because we receive secondary light rays. These rays are first emitted by a source of light
and then they are reflected from the objects we see. This means that in order to
understand image formation, we first have to understand how reflection takes place. The
reason for this reflection is the fact that every point on the surface of the object, which
reflects light rays, becomes a source of the secondary light rays. Those rays are also
emitted in all radial directions as it was in the case of primary light’s source.
The simplest example is the reflection from the perfect plane mirror. Suppose the
ray of light hits the mirror at some angle relative to the normal. This angle between the
direction of the light ray and the normal line to the mirror is called the angle of incidence.
Since, in general, this angle is not zero then different parts of the wave front will hit the
mirror at different moments. By the time when the last part of the wave front reaches the
mirror, the first part of the wave front is already reflected. It only becomes possible if the
angle which the reflected ray makes with the normal to the mirror (called the angle of
reflection) is the same as angle of incidence. This is known as law of reflection, which is
that for a smooth reflecting surface, the angle of incidence is equal to the angle of
reflection
i   r
(5.1.13)
Also the reflected ray is located in the same plane as the incident ray. This law allows us
to explain how the images are formed by a plane mirror.
Reflection is not the only physical phenomenon, which may occur at the boundary
between the two different substances. If the substance is a conductor (some metallic
substance) then electric field cannot exist inside of this conductor and reflection occurs.
However, if the substance is not a conductor but some other material, which allows
transmitting of light (for instance glass), part of light will be reflected but another part
will penetrate into the new medium. Let us consider a planar border between the air and a
glass. The light ray travels in the air and hits the glass surface at some non zero angle of
incidence, then it will continue to propagate in glass. However, the speed of light in glass
is less than the speed of light in the air. Since the ray comes to the glass surface at some
angle, different parts of the wave front will hit the surface at different moments, so some
part of the wave front will be already inside of glass and travel slowly, while the other
part is still in the air and travels relatively fast. As a result, the light ray will turn to the
certain angle. This phenomenon is known as refraction. The magnitude of refraction
angle depends on how many times the speed of light in the medium is different compared
to the speed of light in the air, which is (according to equation 5.1.3) the index of
refraction.
When the ray passes into glass, it makes angle  2 (called the angle of refraction)
relative to the normal to the boundary between the two media. This angle is now different
compared to the angle of incidence 1 . Those two angles are related according to
sin 1 1 v1 f v1 n2


  ,
sin  2 2 v2 f v2 n1
(5.1.14)
which is also known as the Shell’ s law
n1 sin 1  n2 sin 2 ,
(5.1.15)
where n1 , n2 are standing for indices of refraction of each of the two materials (air and
glass in my example). More dense substances usually have larger indices of refraction, so
2  1 then light passes from the less dense substance into more dense substance and
1  2 if light passes from more dense substance into less dense substance. Once again
the incident ray, the normal to the boundary surface and the reflected ray are in the same
plane. You are to verify this law in the lab tomorrow.
Using this law one can explain why the objects appear closer to the surface of
water, when seen from above this surface, than they actually are.
The interesting phenomenon occurs than the light travels from the denser medium
with higher index of refraction into less dense medium with lower index of refraction. In
this situation 2  1 and at some point angle 1 reaches its critical value  c
sin  c 
sin 90o n2 n2

,
n1
n1
(5.1.16)
such that refracted ray just skims the surface of the second medium. If the angle of
incidence is larger than this critical value, the total internal reflection occurs and the ray
comes back to the original denser medium. In the case of the glass with index of
refraction 1.5, this critical angle of incidence is 42 degrees, so the glass-air boundary will
be a perfect mirror for rays coming from glass at angles of incidence higher than 42
degrees.
The phenomenon of total internal reflection is used in different optical devices,
such as binoculars, where prisms are used to “fold” light by means of total internal
refractions. Another application is the optical fiber used for transmission of light signals.
Another interesting fact is that polarization of light occurs during total or partial
reflection from the dense nonmetallic substance. There is one case, when the reflected
light is completely polarized. That takes place if the angle between refracted and reflected
rays is 90 degrees then the direction of total polarization is parallel to the reflecting
surface. The incidence angle, when this phenomenon occurs is called Brewster’s angle.
For this angle we have
n1 sin  B  n2 sin  2 ,
 2   B  90o ,
n1 sin  B  n2 sin  90o   B   n2 cos  B ,
tan  B 
(5.1.17)
n2
n1
For the typical situation, when the first medium is air and the second medium is glass
with index of refraction 1.5, the Brewster’s angle of incidence is about 56 degrees.