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The only two fundamental forces of nature that human beings can
directly experience through their five senses
are gravity and light. The other sensations such
as smell, heat, sound and so on detect macroscopic properties of
matter, and not its fundamental forces or constituents. Electricity
is deeply linked with light. It is the accelerated movement of electric
charge that is responsible for the generation of light: an
accelerating charge emits radiation. That part of radiation that
is visible to us is called light.
Famous physicists such as
Michael Faraday, James Clerk Maxwell, H. Coulomb, and so on
century figures who were crucial in reaching our current understanding of
electricity and magnetism. There
is an anecdote
that when Faraday had announced his discovery, he was asked as to
what was its commercial value, to which he replied "Nothing".
Today, industries based on his discoveries are valued at over
Light is a phenomenon that has fascinated human minds for
and a satisfactory explanation of its properties and underlying
principles has been possible only in the last century. Besides
being essential for vision (the eye ``sees'' objects by receiving
light that has bounced off the object in question), light has
almost limitless applications in daily life. On a more general level, all
life on earth is powered by the light energy that is received from
the Sun, and it through the process of photosynthesis that all
the energy of timber, fossil fuels and so on has become available
The most important property of light that distinguishes
it from a particle is that light is spread over a finite volume of
space, be it in a lighted room at night, or the light coming from the Sun. A
particle in contrast occupies a definite volume in space, and in
Newtonian mechanics a particle is taken to be at a single point, that
is, occupying zero volume in space.
Light is a special case of the more general phenomenon of
waves, and the discussion of light is made more coherent by first analyzing
the general properties of waves. We will then go on to discuss the
special case of electromagnetic waves - also called radiation - and show that
light in fact is a form of electromagnetic wave.
For a long time, physicists
were of the view that - similar to all the other waves seen in nature
-electromagnetic waves, or radiation in short, was also
the result of the oscillations of an underlying medium, which was called the ether.
It was only in the early twentieth century that precise observations
finally ruled out the existence of an ether, and the theory of special relativity
provided a more accurate explanation of the nature of radiation. We now know that
radiation is a special case of
transverse wave in that there is no medium undergoing transverse
oscillations, as in the case of a string. Instead, radiation is the
result of electric and magnetic fields simultaneously propagating in
Similar to all wave phenomena, radiation is characterized by its
wavelength and its frequency . Electromagnetic
radiation propagates at the velocity of light . We have from
Electromagnetic radiation; light
From the special theory of
relativity, it is known that the velocity of light is a
universal constant. Hence, in describing radiation, we can dispense
with frequency of radiation
, and use only its wavelength to fully describe its properties.
The wavelength of radiation has a vast range,
from waves with infinitely long wavelengths to waves of infinitely
short wavelength. Starting from long wavelengths, we have radio
waves which have a wavelength of about 500 m, followed by short
wave radio waves, and so on down to around 1 m wavelength for TV
waves. We then have microwave radiation with wavelength around 1
cm, followed by infrared radiation down to m.
Visible light consists of radiation with wavelength from
m to m. As the wavelength of light decreases, the
human eye perceives it as light changing its colour from red to
violet. Radiation of wavelength less than m is invisible
and is called ultraviolet radiation. Radiation with wavelength
from m to m are called X-rays, and radiation
shorter than m all the way to vanishingly small
wavelength are called gamma-rays. The electromagnetic spectrum is
given in Figure 6.1.
One can easily see from the list
of radiation that the various forms of radiation have unique
properties that have been put to society's use in varied manners, from
X-rays in medical science to TV, radios, mobile phones, microwaves, communications
and so on.
The most accurate description of radiation is provided by the quantum
theory of radiation,
which is briefly discussed in Section 11.3. In the quantum description
electromagnetic waves are composed out of quantum packets of
energy and momentum called photons. At very
small wavelengths of radiation, the description based on photons is the
only accurate one,
and no classical approximation is adequate. Also, the concept of photons is
required for understanding very low intensity radiation since this
phenomenon involves only a few photons. However,
photons of wavelength greater than X-rays can be approximately treated as
a classical transverse wave, which is what is called radiation.
As the wavelength varies, different
descriptions of radiation become more convenient, of which
descriptions (a) and (b) are only approximately correct.
(a) Geometrical Optics. The wavelength of radiation
is much smaller than the size of the equipment or probe being used
to study light, and the energy of the photon is smaller than the
sensitivity of the apparatus. This is the case for visible light.
(b)Classical electromagnetic theory. The wavelength of radiation
is comparable to the size of
the equipment, as is the case with radio waves. (c) Photons.
The need for the quantum theory of photons becomes
necessary when the energy of the photon is larger than the sensitivity of the
apparatus. This is the case with human vision, with the human eye
being able to detect incoming light consisting of as few as 4-5
So far, we have considered light to be a special case of a (transverse) wave,
namely an electromagnetic wave. Recall in Section 6.3,
we discussed a
regime of radiation for which its wavelength
is much smaller than the size of the equipment or probe being
used. In the case of light this regime can be
approximated by what has been earlier called Geometrical Optics.
Note the important fact that since we are taking the wavelength
to be very small the relation
The Electromagnetic Spectrum
implies that we simultaneously need to make the frequency of
light very large. Fortunately, this is the case with visible
light, and hence the possibility of making a consistent
approximation of photons.
In geometrical optics
light behaves like a ray that is specified by its direction of
propagation. The fundamental postulate of geometrical optics is
that light energy travels in rays. The ray
travels in a straight line from point to
point; that is, in going from point to point , light rays will
follow a straight line. As shown in Figure 6.2, light energy which is
converging (diverging) is represented by a set of converging (diverging)
set of rays.
The question naturally arises, how do light rays propagate in the
presence of material bodies. In particular, two well known cases
are the reflection of light off polished surfaces (mirrors) as
well as the refraction of light in passing from one medium
Although the speed of light in vacuum is known to be a
universal constant, the speed of light varies in different media.
The refractive index of a medium,
which allows for the propagation of light, is defined by how much
light slows down in that medium compared to vacuum. The refractive index
in general depends on the wavelength of light, since different wavelengths
behave in a different manner; we will ignore this subtlety in our discussion
on refraction. If the speed of light in the medium is , the definition of
refractive index is given by
Fermat's Principle provides an answer to the question of
propagation of light in the presence of different media. The first
guess one might have is that light should take the path that
minimizes the distance covered in going from point to point, as
this would be a
straight forward generalization of light rays moving in a straight
line in empty space. Take the example of refraction, in which a light ray goes
from say point in air to point in glass, as in Figure 6.3.
Clearly, the straight line between and is the shortest
distance. But light taking this path would contradict
the observed fact that light ``bends'' in going from air to glass.
This phenomenon can easily be seen by inserting one's hand
into water, which looks bent.
The hint for Fermat's principle can
be seen from eq.(6.3); since a light ray is traveling slower in a
more dense medium, it could choose to travel in a manner which
minimizes the time it takes from going from point to
point , with the points being in any arbitrary media.
We hence have Fermat's Principle which states:
Light rays follow
the path of least time in going from point to point.
For light rays traversing a
single medium, a path with minimum distance covered is equivalent to path with
minimum time taken of travel. Hence Fermat's principle of minimizing time of travel
reproduces the result that light, in vacuum, travels in a straight line.
The power of Fermat's principle lies in the fact that it can also account
for the behaviour of light rays in arbitrary and multiple media.
We restate Fermat's principle in a more mathematical form. Let be time
taken for the light ray to go from point to point by taking some path.
Then Fermat's principle states that
The symbol acting on and resulting in zero means
that a small variation in the path of light near its path
of minimum time yields approximately no change in the time taken to go
from initial to final point.
Light ``chooses'' a path for which the time taken by it is a
minimum; or equivalently, from the definition of what a minimum
means, light ``chooses'' a minimal path such that there are many other nearby
paths which take almost the same time.
Consider the reflection of light off a mirror. As shown in Figure xxx, the light
ray starts at point , is incident at point of the mirror at an angle
to the perpendicular to the mirrored surface, and then is reflected through point
, making a reflected angle . Given the angle of incidence , we
apply Fermat's principle of least time to
determine the angle of reflection .
The path which goes directly from point
to point is a trivial solution; we are interested only in paths that reflect off
the mirror. Since the entire path of the ray is in a single medium, the path with a minimum
time is equivalent to a path with minimum distance. The distance
traversed by the light ray in going from to is given by . This distance
determined by reflecting point about the mirror, with reflected point denoted by
as shown in Figure6.4.
The distance is
equal to the distance . As shown in Figure 6.4, consider a light ray incident
at point , and suppose the reflected ray goes through points , or through point
, and so on. We need to determine what is the correct direction of the reflected ray.
Consider the reflections, namely and . Clearly, the distance
between and , and so on is a minimum if the line connecting them
is a straight
line. This in turn implies from elementary geometry shown in Figure 6.4 that
Hence, Fermat's principle yields that the angle of incidence is equal
to the angle of reflection, a well known result of geometrical
Refraction is the phenomenon of light propagating through several media with
different indices of refraction. Consider light rays propagating in air with
refractive index , and
then refracting (propagating) through a piece of glass with refractive
index , and follows a path shown in Figure 6.3.
Recall the index of refraction has been defined in
eq.(6.3). From its definition, we see that is dimensionless. Table
6.1 gives the indices of refraction for some typical
transparent materials at a particular frequency of light.
Indices of Refraction for Light with
The relation of the angles made by light in the two media
goes under the name of
We now derive what the value of angle should be,
given the inputs of and .
is more difficult than the case of reflection because, due to the varying
velocity of light in the two different media, the path of minimum distance is no
longer the same as the path of minimum time.
Instead of plunging into the
details of the derivation, let us illustrate the problem with an
analogy. Since the velocity of light is faster in air than in
glass, an analogy of the propagation of light in air is a man
running. The analogy of light propagating in glass can be taken to
be that of a man swimming. Suppose a
lifeguard standing on the ground
is trying to reach a buoy floating in a swimming pool in the shortest
possible time. If he goes in a straight line, he will spend a
certain fraction of time
running on the ground, and the remaining fraction in swimming.
Since he swims slower than he runs, he should try run for a longer
time on the ground before diving into the pool. On the other hand, if he runs too
far on the ground, he will end up swimming more than he should. The exact point
where he should dive into the pool for minimizing the
time he takes to reach the buoy is the analogy of how light rays should bend as they
leave one medium and enter the other.
The result of minimizing the time taken by light yields Snell's law of refraction, namely
Lifeguard Choosing Optimum Path, with Dashed Paths being Non-Optimal
Recall in deriving the law of reflection, we found the shortest path
by a geometrical method, and this was sufficient. In the
derivation of Snell's law we need to have a deeper understanding
of Fermat's principle, in particular, what does it mean to say
that the time taken for a light ray is a minimum?
To implement Fermat's principle, in analogy with the discussion on how to find
the minima of a function, we need to compare two paths
close to each other. We will try and fix the value of
by the requirement that the time
taken by these paths is equal. If we can carry out this procedure
successfully, we can then conclude that we have
found the path which takes light the minimum time to go from to .
Consider, as shown is Figure 6.6, Path 1 along AOB and Path 2
along APB. We compare the time taken by light, along
Path 1 and Path 2, in traveling from point to point . Path 1 and Path 2 hit
the boundary of the two media at points and respectively.
A shown in Figure 6.3, Path 1 has as its angle of incidence and
as its angle of refraction. To determine the difference in the
distance traveled by light for the two paths, draw
perpendicular to and perpendicular to . See Figure 6.6.
travels an extra distance of in medium 1 compared to Path 2, and
in medium 2 Path 2 travels an extra distance of compared
to Path 1.
Recall velocity of light in air is
, and in glass is
. The extra time spent by light
in traversing Paths 1 and 2 is, consequently, given by
respectively. We can choose either Path 1 or
Path 2 to be the path of minimum time, as they are taken to be very close
to each other.
Path of Minimum Time
For Path 1 to be the path of minimum time, Fermat's
principle requires that the
difference in time taken by light for Paths 1 and 2 should be zero, that
From Figure 6.6 we have that
Hence, from equations above we have the condition for minimum time given by
Question: In our derivation
of Snell's law, where did we use the condition that Path 1 and Path 2 are
close to each other?
If light travels from a more dense to a less dense medium, as in the
case of light traveling from water into air, we have the phenomenon of
total internal reflection in which light is refracted back
into the more dense medium as shown in Figure 6.7.
If the incident angle is equal to a critical angle, called
, the refracted angle becomes , and is given by Snell's law by
For all incident angles greater than , light undergoes
total internal reflection and is reflected back into the more dense medium.
One important application of this phenomenon is the
entire field of fibre optics. Fibre optic cables are
made out of dense optically transparent fibres, and
light propagates inside these light pipes due to total internal
reflection with almost no loss of energy. Fibre optics have
many uses from transmitting high volumes of telecommunication
signals, as well as in medical laproscopic procedures.
What are electromagnetic waves made out of? We have mentioned a
few times that, unlike ordinary sound or water waves,
electromagnetic waves are not the result of the oscillations of
any underlying medium. So what is the entity that is undergoing wave
motion? To answer this question, we need to introduce a major new
concept, which is the concept of a field.
To understand the concept of a field, we need to first re-examine
the conception in physics as to what is a material entity. From the
time of Newton until the mid-nineteenth century, all material
entities were thought to be similar to a solid body. That is,a
solid body has a definite position in space, and with the body ``carrying'' its
energy and momentum. Recall that in Section 5.1.4,
in our discussion on the
propagation of energy for a taut string, we
constructed the continuous
string out of infinitesimal point particles. All solid bodies in turn can
thought of as being composed out of small point-like constituents,
and hence all physical entities can be thought of as being composed out of a
collection of point particles. This view has had great success,
including the formulation of fluid mechanics and of aerodynamics.
In the nineteenth century, a major conceptual revolution was wrought by
physicists such as Michael Faraday and James Clerk Maxwell. In
their study of electricity and magnetism, they discovered that the
effects of, for example an electric charge could be best understood by
postulating a physical entity that exists in its own right, and
carries the effects of the charge. This entity was called a
field. So, for example, if we introduce an electric charge into a
space, the electric charge generates an electric field which
then acts on say other charges. If the charge
introduced is a positive charge, the field will act on other
neighboring charges and create an attractive force on negative charges
and a repulsive force on positive charges and so on.
The fundamental and unique feature of a field, in contrast with a particle,
is that the field is spread all over space, whereas the particle
occupies a definite position in space. A particle is described by
specifying its position in space, and in three dimensions, this means
specifying three numbers, namely the particles and coordinates.
In contrast, the electric field, for example,
is described by assigning numerical value for the field at
every point in space. In other words, a field is a function of space (and of time),
and has much more information than say a particle. The field is a material entity
that is as physical and as ``substantial''
as a particle. Just like a particle, a field has energy,
momentum, angular momentum, entropy and so on.
The behaviour of a particle is typically as shown in Figures
3.2 and fig1b. A wave is an example of a field, and as shown in
Figure 5.2, is spread over space.
In addition to the electric field, we also have the magnetic field,
denoted by . The magnetic field, for example, describe how two magnets
affect each other.
The great discovery of Faraday
and Maxwell was the understanding that electric and magnetic fields
are really the same
entity, and appear to be different only in certain special
circumstances. A charge moving with a constant
velocity, for example, generates a combination of both electric and magnetic fields.
Furthermore, by transforming from one inertial frame to another,
the values of the electric and magnetic fields change, showing
that these are frame dependent, and not physical, quantities.
The electric and magnetic fields
and are generated by arbitrary collections of moving charges.
and (t,x) are vectors that depend on time, and are
spread all over space;
what this means is that at every instant , at every point in
space there are three real numbers that specify the electric field, and similarly
for the magnetic field. In other words, we need an infinite collection of vectors,
one for each instant and each point to specify the electric
and magnetic fields.
An accelerating charge generates electromagnetic waves.
Electromagnetic fields are a special case of
transverse waves. As is the case for any wave propagation, energy
has to be constantly pumped into the system, in this case, the work done to
accelerate a charged particle for sustaining
Intuitively speaking, if one attaches an electric charge
to the end of a stick, and shakes the stick back and forth with a
frequency , the charge will ``radiate off'' electromagnetic
waves of frequency . The energy in the waves comes from the
energy being expended in shaking the charge. The only reason that the
the radiation that we create by shaking a piece of stick
is not easily observable is because the
amplitude of the radiation is very low, making detection difficult.
A modern antennae is not much different from our intuitive picture, since in an
antennae charge is made to flow back and forth - with some frequency - along
the length of an antennae, and results in the emission of electromagnetic
radiation. Radio waves are generated in this manner, and
constitute the signals that are up picked by a radio receiver.
Electromagnetic radiation is the result of the simultaneous
oscillations of electric and magnetic fields. A changing electric
field creates a magnetic field, and in turn, a changing magnetic
field creates an electric field. It is this positive feedback
mechanism that sustains the propagation of electromagnetic wave over
billions of light years of distance. Since the
electric field and magnetic field are
three-dimensional vectors, the propagation of radiation is only
possible in three-dimensions. For simplicity, we study the electric
and magnetic fields far from the charges that have generated the
Suppose the electromagnetic wave is propagating in vacuum along
the -direction. Since the electromagnetic radiation is a
transverse wave, the oscillations of the electric and magnetic
field are in the directions perpendicular to the direction of
propagation, and hence lie in the -plane. Let us denote the
unit vector in the -directions by
respectively. The simplest example of radiation is
given by the electric field always lying along the
-axis, and the magnetic field always lying along the
-axis. In other words, the and fields for radiation
are similar to the transverse wave given in
eq.(5.22), with the added property that all electromagnetic waves
propagate in vacuum at the speed of light. From eq.(6.1), we have
Hence, from eqs.(5.22) and (6.23), the electric and magnetic fields
that constitute electromagnetic wave propagation are given by
The propagation of the radiation given by eqs. (6.24) and (6.25)
is shown in the Figure 6.8, with the direction of propagation
being in the -direction, and the sinusodial oscillations of the
and field drawn in the -planes.
The energy, per unit volume, of the electric and magnetic fields at
point , denoted by , is given by
To obtain the total energy contained in the electromagnetic field,
we have to sum the contribution to energy from all the volume of
space. To develop a physical sense of the energy of a field, one
can imagine breaking up space into volume elements, say into little cubes of
with elemental volume
. In every volume of , one can think of the
field in that volume as being a tiny harmonic oscillator that
can each carry a tiny amount of energy equal to
The total energy of the field is then obtained by summing over
the energy of all the oscillators in the volume elements, and we
Energy propagates along the direction of motion of the wave, which
in the case of the electromagnetic wave being considered is in the
-direction. The intensity of light is a measure of how
much radiant energy is received at a point, and as can be seen by
the formula above, is proportional to .
To understand the dimensions of the and , recall
is the permittivity constant
defined in eq.(3.98), and is a measure of the strength of electromagnetic
fields in vacuum. In a medium other than vacuum, the permittivity constant of
the medium would appear in the energy equation eq.(6.26). From eqs.(3.97)
. Since is the
energy per unit volume,
, and hence from eq.(6.26)
From a simple exercise in dimensional analysis, we have obtained an important
result that the
electric field is proportional to a force. The SI unit
for electric field
From eq.(6.26) it also follows that
and hence the SI unit of is given by
Recall that an accelerated charge
generate electromagnetic radiation; we see from equation (6.37) that
electric and magnetic in turn react back on charged particles, and
exert a force on them. We will study these effects in some detail
in the chapter on electric and magnetic fields.
The momentum of the electromagnetic field is along its direction of propagation,
and the field momentum per unit volume is given by
Light, as has been mentioned in our earlier discussion on the
electromagnetic spectrum, is a special case of electromagnetic
waves when its wavelength is in the range of m to m, and
which is visible to the human eye.
The phenomena of interference and diffraction of light are essentially
similar, and both are special cases of the superposition of waves.
When we studied the superposition of waves, we considered the
simple example of a waves propagating along a fixed direction,
which was taken to be along the x-axis. We now need to consider a more
general case of wave propagation in two dimensions.
Consider the famous experiment of Thomas Young in 1801 which conclusively
showed that light is a wave. Consider a bright source of light that goes
through an opaque screen which has two slits. These slits let through light,
and act as two sources of light. We want to ascertain the intensity of
light on a screen as some fixed distance from the two slits.
As shown in Figure 6.9,the distance between the two slits and is
denoted by , and the distance of the point from and is
and respectively. We assume . Since we want to determine the
intensity of light being received at point at a distance along the screen, we
need to know
the electric field at point due to to light arriving at
from and . Since the intensity of light is proportional to , we will, for simplicity, ignore the vector nature of the
electric field and only examine the magnitude of due
to light emanating from the two slits. Let the electric fields at point due
to and be given, respectively, by
Although the electric field is in phase at the slits (since light is from a common
source), the electric field at from and has a phase difference
since light from and have traveled different distances to arrive at .
The superposition of waves yields the total electric field at to be
and eq.(6.42) follows from eq.(5.61).
The intensity of light received at the screen is given in Figure 6.10.
We analyze phase to determine the pattern of intensity shown in Figure 6.10.
Note that the phase is a result of the path
difference in the distance from and to . From the
Figure 6.9 we see that
Hence, from eq.(6.43)
The intensity at is given by
The maxima of intensity is at
Hence, from eqs.(6.46) and (6.48), the maxima's of
intensity are given by
If light waves from and arrive at
with a path difference of that is a multiple of
wavelength ,then they are in phase leading to
constructive interference. Note the important point that the
intensity of light is the same at each of the maxima's. Similarly, the minima's in
intensity are the result light arriving at out of phase
and resulting in destructive interference. Similar to above derivation, it
can be shown that
As one moves the point on the screen, the angle changes;
only for certain values of will the equation
be satisfied for some , and at these points there will be maximum intensity.
At other points, the equation for the minima's will be satisfied,
leading to minimum intensity. Hence there will be alternating
dark and bright bands of light arriving at the screen, and is
shown in Figure 6.10.
Young's experiment found the maxima's and minima's for the intensity of light
at exactly the points that have been derived above, and
was a major landmark experiment in
proving that light is a wave phenomenon. Two hundred years later,
a similar result for electrons verified the wave description
of particles in quantum theory.
Consider a source of light passing through a circular aperture, and being received
on a screen. The pattern of intensity received at the screen will
show a sharp maxima at the point of the screen closest to the
aperture, and then have a series of maxima's (separated by minima's) of lower
and lower intensity. See Figure 6.11. This phenomenon is called diffraction,
and is unlike interference patterns where all the maxima's have
the same intensity.
To understand the physics of diffraction, a N-slit experiment in which light from a source
passes through equally spaced N-slits and is then received on a screen.
Radiation of light from a source moving at
a velocity is more complicated than the Doppler effect
discussed for sound waves in Section 5.6. The reason being that
in theory of special relativity, the frame
in which the source emits the light is different from the frame in which the emitted
light is absorbed, since the two frames are moving with a relative velocity given
by . Due to there being
two different frames of reference, the time
variable needs to be transformed from one frame to the other according to
laws of special relativity. The result we have derived in eq.(5.91)for
sound waves gets
modified. The speed of light replaces the velocity of sound . and
there is a factor to reflect the difference in the flow of time in the two frames;
this yields, similar to eq.(5.91), the following.
Due to laws of special relativity, the velocity of the light source can
never exceed velocity of light in vacuum, namely . However, in a
medium light travels at a speed less than , and a source can then exceed light's velocity
leading to a phenomenon similar to shock waves for sound, and is
called Crenenkov radiation.
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