Lecture Notes and Solved Problems

Chapter 24 Electromagnetic Waves
Wednesday, March 24, 2010
3:16 PM
Maxwell's Equations
According to Freeman Dyson, the two greatest advances in science in
the 19th century were Charles Darwin's The Origin of Species (1859) and
James Clerk Maxwell's A Dynamical Theory of the Electromagnetic Field
(1865). In his 1999 essay, Why is Maxwell's Theory so Hard to
Understand?, Dyson continues:
"But the importance of Maxwell's work was not obvious to his
contemporaries. For more than twenty years, his theory of
electromagnetism was largely ignored. Physicists found it hard to
understand because the equations were complicated.
Mathematicians found it hard to understand because Maxwell
used physical language to explain it. It was regarded as an obscure
speculation with not much experimental evidence to support it."
Dyson goes on to explain that although Maxwell was at the level of
Newton, he was a very modest and nice person, in contrast with
Newton, who had a giant ego and was a difficult person to deal with.
Dyson claims that Maxwell delayed progress in physics by 20 years
because he did not aggressively promote his own theory, as others
might have.
(It's interesting to note that Dyson just missed out on the 1965 Nobel
prize in physics, which was awarded to Feynman, Schwinger, and
Tomonaga. Dyson is 91 years old and still very much alive as of February
2015, and is said to be shy and modest himself.)
Who knows if Dyson's claim about Maxwell's modesty delaying progress
in physics is correct? Dyson himself mentions another important reason:
"There were other reasons, besides Maxwell's modesty, why his
theory was hard to understand. He replaced the Newtonian
universe of tangible objects interacting with one another at a
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universe of tangible objects interacting with one another at a
distance by a universe of fields extending through space and only
interacting locally with tangible objects. The notion of a field was
hard to grasp because fields are intangible. The scientists of that
time, including Maxwell himself tried to picture fields as
mechanical structures composed of a multitude of little wheels
and vortices extending throughout space. These structures were
supposed to carry the mechanical stresses that electric and
magnetic fields transmitted between electric charges and currents.
To make the fields satisfy Maxwell's equations, the system of
wheels and vortices had to be extremely complicated. If you try to
visualise the Maxwell theory with such mechanical models, it looks
like a throwback to Ptolemaic astronomy with planets riding on
cycles and epicycles in the sky. It does not look like the elegant
astronomy of Newton. Maxwell's equations, written in the clumsy
notations that Maxwell used, were forbiddingly complicated, and
his mechanical models were even worse. To his contemporaries.
Maxwell's theory was only one of many theories of electricity and
magnetism. It was difficult to visualise, and it did not have any
clear advantage over other theories that described electric and
magnetic forces in Newtonian style as direct action at a distance
between charges and magnets. It is no wonder that few of
Maxwell's contemporaries made the effort to learn it."
Nearly two centuries of the spectacular successes of Newtonian
mechanics had trained physicists to conceive of the universe in
mechanical terms. Although Faraday introduced the field concept, it
took a few generations for physicists to get the hang of the field
concept, and begin to look at the world from a non-mechanical
perspective. Certainly Einstein had fully absorbed the field concept by
the first decade of the 20th century, and his special and general
theories of relativity helped other physicists to appreciate the field
concept.
In many ways Maxwell's theory of electromagnetism is a paradigm for
modern theories in physics. For example, when Einstein's special theory
of relativity showed that Newtonian mechanics was in need of revision,
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of relativity showed that Newtonian mechanics was in need of revision,
an assessment of Maxwell's theory showed that it was perfectly
consistent with special relativity and needed no revision.
As another example, Maxwell's theory provides, in a way, a model for
our approach to quantum mechanics, which we will begin discussing
next week. Maxwell's theory has "two layers," in Dyson's words; there is
the primary layer, which consists of fields that satisfy partial differential
equations, and which can be calculated but not measured. They give
rise to tangible, measurable quantities, such as forces and energies,
which form the secondary layer, and which are typically quadratic (or
bilinear) functions of primary quantities. Quantities in the secondary
layer are more directly experienced and can be directly measured, but
they do not satisfy simple equations, and therefore are not as
fundamental in the theory. Quantum mechanics is similar in that there
is a primary layer consisting of what are called wave functions, which
satisfy a partial differential equation, and which are not measurable,
and a secondary layer of measurable quantities that are quadratic (or
bilinear) functions of the primary quantities.
As a third example, Maxwell's theory is now understood as a gauge
theory, and all modern quantum field theories are gauge theories. As
such, Maxwell's theory is a paradigm for all modern quantum field
theories, and progress in the development of quantum field theories
was aided by parallel studies and deepening understanding of Maxwell's
theory of electromagnetism.
Returning to a previous train of thought, Maxwell's equations were
complicated in his time because Maxwell wrote them out in component
form. Nowadays we use vector notation, which makes his equations
easier to write, easier to understand, and easier to work with. This may
be the true reason for the delay that Dyson mentions. Although
Maxwell himself became a champion of vectors, and Clifford (starting in
1878), Gibbs (starting in 1881), and Heaviside (starting in 1883) worked
hard to promote them, others, such as Tait (since 1867), promoted
quaternions as a better alternative to vectors. Maxwell died young (only
47 years old) in 1879, so he didn't see the resolution of the battle
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47 years old) in 1879, so he didn't see the resolution of the battle
between vectors and quaternions, which by about 1910 had turned
decisively in favour of vectors thanks to the promotional work of Gibbs
and Heaviside.
One of the main reasons for vectors winning out over quaternions is the
heavy use by Gibbs, and especially Heaviside, in applying vectors to the
teaching of Maxwell's equations. So let's get back to Maxwell's
equations, which did so much to unify electrical and magnetic
phenomena. Maxwell's equations are a system of four partial
differential equations (or eight, actually, if you write them in
component form). What do Maxwell's equations look like in modern
notation? Behold:
Here are Maxwell's equations written out in Cartesian components:
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Cartesian coordinates are not always the sensible ones to use; indeed,
an important part of problem-solving in physics is to choose a
coordinate system wisely, and in this case wisdom often amounts to
choosing a coordinate system that is adapted to any symmetry present.
In any case, one can see how much more complicated Maxwell's
equations appear in component form than in the compact vector form.
It makes Maxwell's achievements (the formulation of the equations,
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It makes Maxwell's achievements (the formulation of the equations,
and the truly remarkable conclusions he was able to deduce from them,
as we'll see shortly) all the more impressive.
Part of what Maxwell did in formulating his equations was just to take
all of the existing electrical and magnetic relationships discovered by
others and either placing them in mathematical form, and/or extracting
a logically minimal subset of them from which all the others can be
derived, and finally correcting one of the equations with an ingenious
discovery. Equation 1 is Gauss's law, Equation 2 is Gauss's law for
magnetic fields, Equation 3 is Faraday's law of induction (placed in
mathematical form by Maxwell), and Equation 4 is Ampere's law
corrected/completed by Maxwell.
And what do Maxwell's equations mean? Can we understand them in
broad outline, intuitively? Yes we can; let's tackle them one by one:
1: Gauss's law for electric fields says, in essence, that the source of an
electric field (OK, let's say electrostatic field to be more precise) is
electric charge. The equation gives a lot more specific mathematical
detail: The left side of the equation describes the rate of change of the
electric field as you move in space, and the relationship tells us that this
rate of change of the electric field is proportional to the electric charge
density.
You can get a complementary interpretation of Gauss's law by
considering an imaginary closed surface S in space and integrating both
sides of the equation over the volume V enclosed by S. The integral of
the right side is proportional to the total electric charge within S. The
integral of the left side represents the total electric flux leaving V, the
region enclosed by S. In other words, the density of electric field lines
on S is proportional to the total charge enclosed by S. In this sense, each
of Maxwell's equations is a precise mathematical expression of what we
have already learned in less precise English phrases.
2: Gauss's law for magnetic fields has a zero on its right side, which
expresses the fact that there is no magnetic analogue of electric charge.
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expresses the fact that there is no magnetic analogue of electric charge.
There are no microscopic little bits of magnetism; that is, there are no
magnetic monopoles. Otherwise, the same interpretations as above for
electric fields also apply to Gauss's law for magnetic fields.
3: Faraday's law of induction; the right side of the equation is the rate of
change of the magnetic field with respect to time, and the left side says
something about the rate of change of the electric field with respect to
changes in spatial coordinates. In other words, a magnetic field that
changes in time induces an electric field that varies in space according
to the left side of the equation.
So there are two sources of electric fields: Electric charges and timevarying magnetic fields.
4: Ampere's law, as corrected by Maxwell; if we ignore the second term
on the right side of the equation, we have the original Ampere's law,
which states that electric current is the source of magnetic fields. The
equation makes the connection between the electric current density
and the spatial variation of the magnetic field mathematically precise.
The second term on the right side of the equation is Maxwell's
correction to Ampere's law, and in a way completes his unification of
electric and magnetic phenomena (Einstein pushed it further with
special relativity in 1905). The second term on the right side of the
equation is analogous to Faraday's law of induction, but with the roles
of electric and magnetic field interchanged: A time-varying electric field
induces a magnetic field.
So there are two sources of magnetic fields: Electric currents and timevarying electric fields.
By integrating each side of the equation, in a certain way, we obtain a
complementary interpretation of Ampere's law, which is much closer to
the interpretation we discussed in lectures. Select an imaginary simple
closed loop and then select an imaginary surface that is bounded by the
loop and "pierced" by the current, as in the figure:
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loop and "pierced" by the current, as in the figure:
In words, the sum of the parallel components of the magnetic field around
the closed simple loop is proportional to the current that pierces any
surface bounded by the loop. Contemplating deeply on the figure,
Maxwell figured out that Ampere's law is not complete, and he figured out
how to correct it. This was a tremendous achievement, and required real
ingenuity and insight. However, it does make one wonder: Why didn't the
numerous investigators of the early 19th century, upon whose strong
foundations Maxwell built, notice the error in Ampere's law? Why didn't
discrepancies show up in their measurements?
The answer to this question is that for the typical experiments of the early
19th century, the magnitude of the missing term in Ampere's law is much
too small to measure. (Look at Maxwell's fourth equation and consider the
magnitude of ε0.) It's only in situations where the electric field oscillates
extremely rapidly in time does the additional term become measurable. As
we shall see, for electromagnetic radiation the electric and magnetic fields
can oscillate extremely rapidly.
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Do you need to know any of the previous discussion for tests and
exams? No. But I hope that the previous and following discussion will
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exams? No. But I hope that the previous and following discussion will
successfully communicate something about the excitement of Maxwell's
equations, their importance in physics (more about this later), and why
they are right near the top of the list of outstanding scientific
achievements of the 19th century.
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Once Maxwell's equations were formulated, this was not the end of the
story, but the beginning of a flood of new research. Besides numerous
experiments performed to test Maxwell's equations, they were used by
many scientists, engineers, and inventers to design new equipment,
new devices, and perfect existing ones. The intensive flowering of
electrification in the late 19th century and throughout the 20th century
is a testament to the importance of Maxwell's equations. The design
and construction of radio antennas (both transmitters and receivers;
Marconi sent radio waves across the Atlantic ocean in the first few years
of the 20th century), electrical generation and transmission systems, all
sorts of electrical devices such as motors and generators, and many
other types of electrical machinery, were all made possible thanks to
Maxwell's equations.
Besides these practical, technological advances made possible by
Maxwell's equations, we obtained deeper insights into the universe and
how it works thanks to Maxwell's equations. One of these advances was
made by Maxwell himself: By playing with his equations, he was able to
combine them in such a way to derive the following two equations,
which disentangle the electric and magnetic fields:
Maxwell immediately recognized these two partial differential
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Maxwell immediately recognized these two partial differential
equations as having the same structure as the partial differential
equations describing the motion of mechanical waves on a string,
which had been developed by d'Alembert as early as 1746 for one
dimension (the three-dimensional version was discovered by Euler soon
after). These are wave equations! Could there be waves involving
electric or magnetic phenomena?
Let's calculate the value based on currently accepted constants:
First, note that the units come out right; because of the formula for
the force exerted by a magnetic field on a moving charged particle,
we conclude that the tesla unit is related to other SI units as follows:
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Thus, the units of
are:
This is the speed of light! Imagine the excitement Maxwell must
have felt when he made this observation! Based on these
calculations, Maxwell boldly hypothesized that light is an
electromagnetic wave. That is, light is a wave formed of an
oscillating electric field and an oscillating magnetic field.
Experiments by Hertz in 1887 provided strong support for
Maxwell's hypothesis.
Maxwell extended the work of Ampere, Faraday, Oersted, and
others, to show that electrical, magnetic, and optical
phenomena were all part of the same fundamental interaction.
Einstein's special theory of relativity (1905) completed the
conceptual unification of electrical, magnetic, and optical
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conceptual unification of electrical, magnetic, and optical
phenomena.
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"From a long view of the history of mankind—seen from, say, ten
thousand years from now—there can be little doubt that the most
significant event of the 19th century will be judged as Maxwell’s
discovery of the laws of electrodynamics." — Richard Feynman,
Feynman Lectures on Physics,
<http://www.feynmanlectures.caltech.edu/II_01.html#Ch1-S6>
Electromagnetic Waves and their Properties
We've discussed the fact that there are wavelike solutions to
Maxwell's equations, but in practice how can one create
electromagnetic waves? Well, nature takes care of it quite nicely in
just two different ways (ignoring some minor, exotic, unexplained
methods, such as parametric down-conversion):
• acceleration of charged particles
• transitions of particles from one quantum state to a lower-energy
quantum state, such as happens in an atom or in an atomic
nucleus (we'll learn about this starting next week)
Note that we aren't talking about reflection or scattering, which
happen frequently, but don't bring anything essentially new to the
table; we're talking about how electromagnetic waves are created
initially.
And what exactly are electromagnetic waves? Do we encounter
them in every-day life? Yes. Light, microwaves, radio waves, X-rays,
and other kinds of waves, are all examples of electromagnetic
waves. We'll further discuss the different types of electromagnetic
radiation in the next section.
Notice that we used the phrases "electromagnetic waves" and
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Notice that we used the phrases "electromagnetic waves" and
"electromagnetic radiation" in the previous paragraph; they mean
exactly the same thing and can be used interchangeably. Also note
that "electromagnetic radiation" is not the same as "nuclear
radiation," so don't confuse the two; the latter is more general, and
includes other types of radiation as well, as we'll learn later in the
course.
It might be fun for you to think about all the sources of
electromagnetic radiation that you can think of, such as light bulbs,
the Sun, sparks, flames, microwave ovens, radio antennas, etc., and
classify them according to which of the two processes is used to
produce electromagnetic radiation.
What is an electromagnetic wave like? Well, it is a combination of an
oscillating electric field and an oscillating magnetic field; that is, it's
something like a combination of two transverse waves. (A transverse
wave is one in which the oscillation of the wave in space is
perpendicular to the motion of the wave in time. A longitudinal wave
is one in which the oscillation of the wave in space is parallel to the
motion of the wave in time. Sound waves are longitudinal.)
Consider the following diagrams, which represent a "snapshot" of an
electromagnetic wave at a particular time. As time passes, the
electromagnetic wave moves in the direction of the green arrow.
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- the electric and magnetic field vectors oscillate in phase
and with the same frequency; if the electromagnetic
wave was formed by oscillating charged particles, then
the frequency of the electromagnetic wave is the same
as the frequency of oscillation of the charged particles
changes in the electric field generate the magnetic field,
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- changes in the electric field generate the magnetic field,
and changes in the magnetic field generate the electric
field, consistent with Maxwell's equations; the changing
electric and magnetic fields re-generate each other as
the wave moves
- the wave in the diagram is a "plane wave;" that is, it is
what the wave looks like far from its source; closer to
the source, the wave looks more complicated
- electromagnetic waves propagate without the need for a
material medium; the electromagnetic field itself is the
medium that supports an electromagnetic wave; recall
our discussion about the "luminiferous ether" when we
studied special relativity
- all electromagnetic waves travelling in vacuum have the
same speed, which we call c, "the speed of light"
- the way the diagrams are drawn you will naturally
assume that the magnitudes of the electric and magnetic
fields in an electromagnetic wave are equal; this is not
the case, as they don't even have the same units, and as
we'll see later, E = cB.
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The Electromagnetic Spectrum
The range of frequencies (and therefore wavelengths) of
electromagnetic waves is enormous. The frequency and wavelength of
an electromagnetic wave travelling in vacuum are related by
so the frequency and wavelength of an electromagnetic wave are
inversely proportional. Different parts of the electromagnetic spectrum
are typically labelled in certain ways, although there are no definite
boundaries between, say, X-rays and gamma rays.
Notice that the diagram below has a logarithmic scale (equal adjacent
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Notice that the diagram below has a logarithmic scale (equal adjacent
intervals on the scale differ by factors of ten). Also notice how extremely
tiny the visible part of the spectrum is.
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Problem: In a dentist's office an X-ray of a tooth is taken using X-rays
that have a frequency of 6.05 × 1018 Hz. What is the wavelength in
vacuum of these X-rays?
Solution: Although the X-rays are travelling through air (and for a small
distance through human tissue, bone, and teeth), where their speed and
wavelength will be different from their wavelength in vacuum, we are
asked to calculate what their wavelength would be if they were
travelling in vacuum, where there speed is c.
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travelling in vacuum, where there speed is c.
The wavelength of visible light in vacuum is approximately in the 400
nm to 700 nm range, so the wavelength of X-rays is much, much
smaller than the wavelength of visible light, by a factor of about
10,000.
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Problem: FM radio waves have frequencies between 88.0 MHz and
108.0 MHz. Determine the range of wavelengths for these waves.
Solution: Assume that the radio waves are travelling through vacuum
instead of air; the difference in the calculated wavelength is not much.
The range of wavelengths is between 2.78 m and 3.41 m.
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The range of wavelengths is between 2.78 m and 3.41 m.
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Problem: A certain type of laser emits light that has a frequency of
5.2 × 1014 Hz. The light, however, occurs as a series of short pulses,
each lasting for a time of 2.7 × 10-11 s.
(a) How many wavelengths are there in one pulse?
(b) The light enters a pool of water. The frequency of the light remains
the same, but the speed of the light slows down to 2.3 × 108 m/s. How
many wavelengths are there now in one pulse?
Strategy: (a) Determine the wavelength of the light, then determine
how far the light travels during one pulse, then divide.
(b) Calculate the new wavelength of the light, then repeat the strategy
of Part (a).
Solution:
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The results for Parts (a) and (b) are exactly the same. It's worth
thinking about this to understand why. The speed decreases in Part
(b), but the wavelength decreases by exactly the same factor, so the
two changes cancel.
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Another interesting point contained in this problem is that if
electromagnetic waves slow down in crossing the boundary from one
medium to another, their wavelength decreases but their frequency
remains the same.
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The Speed of Light in Vacuum
The speed of light in vacuum is extremely large, and determining its value
was quite a challenging task for physicists over the centuries. Ancient
philosophers debated whether the speed was finite or infinite, with no
agreement. Part of the problem was that some philosophers figured that
light was emitted by the eyes, and the fact that distant stars appeared
instantly upon opening one's eyes led to the conclusion that light travelled
instantly.
An early attempt by Galileo in 1638 involved measuring the time difference
between unveiling a lantern on one mountainside and seeing the light on
another mountainside a kilometre or two distant. Conclusion: If not
infinite, the speed of light is very, very fast.
The Danish scientist Ole Romer was the first to demonstrate convincingly
that the speed of light is not infinite. He noted that the apparent orbital
period of one of Jupiter's moons, Io, was different depending on how close
Jupiter is to Earth. He reasoned that the difference is due to the extra
distance that light must travel. He measured the time difference for two
positions of the Earth relative to Jupiter, one when Earth is as close to
Jupiter as possible, and one when Earth is as far from Jupiter as possible.
The difference in distance is the diameter of Earth's orbit, and from this
one can determine the speed of light. Romer's value was about 35% too
small, but was very important nevertheless because it was the first
convincing demonstration that the speed of light is not infinite.
James Bradley discovered stellar aberration in the early 1700s. He
observed that stars seemed to move in very minute circles, and he
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observed that stars seemed to move in very minute circles, and he
attributed this to the motion of the Earth. He was able to use this
observation to deduce the speed of light in 1729 (see the following
diagrams), and his calculated value for the speed of light is remarkably
close to the current value.
(Source: Introduction to Special Relativity, Robert Resnick, Wiley, 1968, page
29.)
The first effective terrestrial determination of the speed of light was by
Hippolyte Fizeau in 1849. He sent light from a source to a mirror 8 km away,
and then detected the reflected light. However, before detection the
transmitted and reflected beams of light had to pass through the gaps in the
teeth of a rotating wheel. Only for certain rotation speeds would the light
pass through the gaps, and this allowed Fizeau to calculate light's speed.
Fizeau's result was not quite as good as Bradley's, but he was still within 5%
of the true value.
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Leon Foucault replaced Fizeau's toothed wheel with a rotating mirror and
improved the accuracy to within a fraction of a percent in measurements
performed in 1862. By 1926, Albert Michelson had developed this rotatingmirror method to a very high degree, and obtained extremely accurate
results for the speed of light.
Being aware of the speed of light is important for observations at great
distances. For example, if you observe a very distant galaxy, from which
it takes light a few billion years to reach us, you have to be aware that
you are looking into the past, and that what you observe now is not the
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you are looking into the past, and that what you observe now is not the
present for the galaxy, but an image of the galaxy as it was a few billion
years ago. In this sense, all of our observations are of the past, just
further into the past the farther away we observe.
Even big stock brokerages are conscious of the non-infinite speed of
light, and build server stations as close as possible to stock exchanges to
get an edge of a few microseconds over the other guy in their drive to
maximize profits. However, faster computer algorithms and faster
trading strategies can also lead to disasters, as the following 2012 story
explains:
http://www.wired.com/2012/08/ff_wallstreet_trading/all/
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Problem: Determine the speed of light in vacuum in units of m/s and
ft/ns.
Solution:
This provides a (possibly) useful approximation for small-distance
applications (in the home or laboratory), at least for those who are
familiar with that non-SI unit for distance, the foot.
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familiar with that non-SI unit for distance, the foot.
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Problem: Determine the number of kilometres in
(a) one light-second.
(b) one light-minute.
(c) one light-hour.
(d) one light-year.
Solution: Remember that all of these quantities are distances.
(a) One light-second is the distance that light travels in one second.
(b) One light-minute is the distance that light travels in one minute.
(c) One light-hour is the distance that light travels in one hour.
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(d) One light-year is the distance that light travels in one year.
Astronomical distances are enormous, which means that the light-year is
a useful distance unit for quoting astronomical distances.
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The light-year is a unit of distance, but sometimes it's used mistakenly as
a unit of time. For example, in the beautiful song Diamond and Rust, by
Joan Baez, the song-writer uses the light-year to mean a long time ago.
We can forgive Ms. Baez and say she is using poetic license, but
remember that the light-year is a unit of distance.
https://www.youtube.com/watch?v=dcaZi_G3xVs
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Problem: How long does it take for electromagnetic waves to travel oneway from
(a) the Earth to the Moon.
(b) the Sun to the Earth.
(c) the Sun to Jupiter.
Solution: (a) The distance between the Earth and the Moon is 384,400 km.
The time delay seems perhaps slight, but it was quite noticeable when
humans first went to the moon starting in 1969 and their
communications back to Earth were broadcast live on television.
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(b) The distance between the Sun and the Earth is 1.5 × 1011 m.
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(c) The distance between the Sun and Jupiter is 7.8 × 1011 m.
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Problem: The figure below illustrates Michelson's setup for measuring
the speed of light with mirrors placed on Mt. San Antonio and Mt. Wilson
in California, which are 35 km apart. Using a value of 3.00 × 108 m/s for
the speed of light, determine the minimum angular speed (in rev/s) for
the rotating mirror.
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Solution: Light must make a round-trip of 70 km in the same time that
the mirror rotates through k/8th of a revolution, where k = 1, 2, 3, … .
The time needed for light to make this round trip is
Thus, the minimum angular speed of the mirror is
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Problem: A lidar (laser radar) gun is an alternative to the standard
radar gun that uses the Doppler effect to catch speeders. A lidar gun
uses an infrared laser and emits a precisely timed series of pulses of
infrared electromagnetic waves. The time for each pulse to travel to
the speeding vehicle and return to the gun is measured. In one
situation a lidar gun in a stationary police car observes a difference
of 1.27 × 10-7 s in round-trip travel times for two pulses that are
emitted 0.450 s apart. Assuming that the speeding vehicle is
approaching the police car essentially head-on, determine the speed
of the vehicle.
Solution: Draw a diagram to illustrate the relevant distances:
The car moves a distance d in the time between the emission of the
two pulses, assuming that the time needed for the pulses to travel
can be neglected as very small compared to 0.450 s. (Remember that
light travels at about 1 foot per nanosecond, so the time needed to
travel 1000 feet, which is almost 300 m, is 1 microsecond.) In other
words, the time interval between the arrival of the two pulses at the
car can be approximated to be the same as the time interval
between the emission of the two pulses.
The first pulse travels a distance 2d more than the second pulse, in a
time that is 1.27 × 10-7 s more than the first pulse. This allows us to
calculate d, assuming that the speed of light in air is approximately
the same as the speed of light in vacuum:
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the same as the speed of light in vacuum:
The speed of the car is therefore
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The Energy Carried by Electromagnetic Waves
Experiments show that electromagnetic waves carry both energy
and momentum. Some of these experiments will be familiar to you;
for example, if you lie outside on a sunny day, you will get warmer.
Going into a closed car on a sunny day is also good evidence that
light carries energy (this is the greenhouse effect). Microwave
ovens also provide good evidence.
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ovens also provide good evidence.
In this section we'll examine the energy carried by electromagnetic
waves; as for momentum, we'll leave that for another time (see the
Compton effect a few chapters down the road).
As you'll recall from an earlier chapter, the electric energy density u
of a region of vacuum in which there is an electric field is
The analogous expression for magnetic fields is
Thus, the total energy density carried by an electromagnetic wave is
It is an experimental fact that in an electromagnetic wave, the
electric field and magnetic field carry equal amounts of energy.
Thus, for an electromagnetic wave,
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We can also write the total energy density in an electromagnetic wave as
It often occurs that an electromagnetic wave is created by a compact
source (perhaps a "point source") and then spreads out over larger and
larger surfaces as it travels outwards. The energy of the wave thereby
spreads over larger and larger surface areas, and thus becomes less
intense. A useful measure of this quantity is the intensity S of an
electromagnetic wave, which is defined as
where P is the total power in the wave, and A is the surface area over
which the wave is spreading.
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Problem: A laser emits a narrow beam of light. The radius of the beam is
1.0 × 10-3 m, and the power is 1.2 × 10-3 W. What is the intensity of the
laser beam?
Solution:
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Problem: The maximum strength of the magnetic field in an
electromagnetic wave is 3.3 × 10-6 T. What is the maximum
strength of the wave's electric field?
Solution:
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Problem: The microwave radiation left over from the Big Bang explosion
of the universe has an average energy density of 4 × 10-14 J/m3. What is
the rms value of the electric field of this radiation?
Solution: The rms value of a quantity that varies sinusoidally is a type of
average; using calculus, you can show that this kind of average is
times the amplitude of the quantity.
The relation between the average value of the electric field and the
average energy density is
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Problem: On a cloudless day, the sunlight that reaches the surface of
the Earth has an intensity of about 1.0 × 103 W/m2. What is the
electromagnetic energy contained in 5.5 m3 of space just above the
Earth's surface?
Solution: Check the derivation on Page 749 of the textbook, which
relates the intensity of an electromagnetic wave to its energy
density: S = cu. (You might also guess this relation using dimensional
analysis.) Using this relation, the energy density of the
electromagnetic wave is
Thus, the total electromagnetic energy contained in a volume of 5.5 m 3 is
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Problem: A light bulb emits light uniformly in all directions. The
average emitted power is 150.0 W. At a distance of 5.00 m from the
bulb, determine
(a) the average intensity of the light,
(b) the rms value of the electric field, and
(c) the peak value of the electric field.
Solution: Assume that the power is spread uniformly over a spherical
shell of radius r.
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The Doppler Effect
In PHYS 1P23/1P93, one develops a formula for the Doppler effect for
sound waves:
where fs is the frequency emitted by the sound source, fo is the observed
frequency of the sound waves, vo is the speed of the observer, vs is the
speed of the sound source, c is the speed of sound, and all speeds are
measured relative to the sound medium. The top signs are used if the
distance between the sound source and the observer is decreasing
("approaching") and the bottom signs are used if the distance between
the sound source and the observer is increasing ("receding"). This simple
formula is valid if the relative motion between the source and observer
is along the line joining their positions; otherwise, the formula is more
complicated.
To get a qualitative understanding of the Doppler effect, consider the
following diagrams, the first two for a moving source and the third for a
moving observer:
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moving observer:
You've undoubtedly heard this many times, but here's another example
if you need it: https://www.youtube.com/watch?v=imoxDcn2Sgo
If the speeds of the observer and source are small relative to the wave
speed, then we can approximate the Doppler formula as
where, as before, fs is the frequency emitted by the source, fo is the
observed frequency of the waves, vrel is the relative speed of the observer
and source, and c is the wave speed. The top sign is used if the distance
between the sound source and the observer is decreasing ("approaching")
and the bottom sign is used if the distance between the sound source and
the observer is increasing ("receding"). This is typically an excellent
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the observer is increasing ("receding"). This is typically an excellent
approximation for light, because typically sources and observers have
relative speeds that are small with respect to the speed of light.
There are numerous interesting astronomical applications of the
Doppler effect. For example, one can deduce rotational speeds of stars
and galaxies and identify supermassive black holes (see Example 10 on
page 141 of the textbook). Most spectacular is Hubble's work in the
1920s showing that the universe is expanding, which he published in
1929. There are also all kinds of terrestrial applications of the Doppler
effect, useful when you need to measure speeds, ranging all the way
from medical applications (measuring the speed of blood flow), to the
measurement of wind speeds by meteorologists ("Doppler radar"), all
the way to police radar (hello, speed traps); the same method is used to
measure the speeds of baseball pitches (radar gun = Doppler shift
detector). In all of these terrestrial applications, electromagnetic waves
are emitted by the "radar gun" and then detected again after they
reflect from the moving object. The electronics in the radar gun then
compares the frequency of the emitted waves and the detected waves
and calculates the speed of the moving object based on the frequency
difference.
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Problem: A distant galaxy emits light that has a wavelength of 434.1 nm.
On earth, the wavelength of this light is measured to be 438.6 nm.
(a) Decide whether this galaxy is approaching or receding from the
earth. Give your reasoning.
(b) Find the speed of the galaxy relative to the earth.
Solution: (a) The observed wavelength is redshifted; that is, it is shifted
to longer wavelengths relative to the emitted wavelength. Thus, the
galaxy is receding.
(b) Use the lower sign in the Doppler shift formula, because the galaxy
is receding.
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Problem: A speeder is pulling directly away and increasing his distance
from a police car that is moving at 25 m/s with respect to the ground.
The radar gun in the police car emits an electromagnetic wave with a
frequency of 7.0 × 109 Hz. The wave reflects from the speeder's car and
returns to the police car, where its frequency is measured to be 320 Hz
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returns to the police car, where its frequency is measured to be 320 Hz
less than the emitted frequency. Find the speeder's speed with respect
to the ground.
Solution: Use the lower sign in the Doppler shift formula, because the
car is receding. However, note that there is an original emission
frequency, a second frequency that is received by the car and then
reflected, and finally a third frequency that is received back at the police
car.
(The diagram from the textbook is for a different situation, where the cars
approach, but is reproduced here just to focus our attention on the three
different relevant frequencies.)
Start with the basic formula:
Now write specific formulas relating f1, f2, and f3:
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This is a quadratic equation for the relative speed, and we can use
the quadratic formula to solve for the relative speed. However, note
that in the second factor on the left, the second term is extremely
small relative to the first term, and so can be ignored. This simplifies
the solution considerably.
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Thus, the speed of the car relative to the ground is
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Alternative Solution: From the (*) equation above, you can also
continue as follows:
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The remainder of the solution proceeds as above.
The result in the second solution is slightly more accurate, and
therefore a better solution. For the science major it's worth
reflecting on these two solutions and trying to extract general
principles. In practice, one wants to minimize calculation errors in
general and rounding errors in particular. In this case it seems that
throwing away the term in the first solution leads to an error of
about 3%, which could be significant enough to make one prefer
the second solution.
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Polarization
Typically light emitted from a source is unpolarized. This means that
the electric vectors for the individual light rays are oriented in random
directions. However, under some circumstances it's possible that all
the light rays from some source have their electric vectors oriented in
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the light rays from some source have their electric vectors oriented in
the same plane. This is called polarized light, and the plane containing
the electric field vectors is called the plane of polarization. (Technically
this is called linearly polarized light; it is also possible for the direction
of the electric field vector to rotate in a circle in what is called circularly
polarized light.)
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Experiments show that if the light incident on a polarizing filter is
unpolarized, the transmitted light is polarized and has intensity
1/2 of the incident intensity.
If polarized light is incident on a polarizing filter, the transmitted
light is polarized along a new axis (parallel to the filter's polarizing
axis) and the transmitted intensity is reduced according to Malus's
law:
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Thus, if polarized light incident on a polarizing filter has its axis
perpendicular to the polarizer's axis, then none of the light gets
through the polarizer. No real polarizer is perfect, but in practice
the transmitted intensity is much reduced. This is the idea behind
polarizing sunglasses.
As we saw earlier, one way to produce polarized light is to pass
unpolarized light through a polarizing filter, which results in light
that is polarized parallel to the filter's polarizing axis. Another way
to produce polarized light is to reflect unpolarized light. Light
reflected from lakes, wet spots on roads, etc., is often polarized,
but polarized sunglasses are designed to have their polarizing
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but polarized sunglasses are designed to have their polarizing
axes perpendicular to the axis of polarization of the reflected
light, and absorbs almost all of the "reflected glare."
Further applications of polarization: IMAX theatres, liquid-crystal
displays. Some species of animals can detect polarized light and use it
for various purposes; for example, bees use polarized light for
navigation and some butterflies detect polarized light reflected by the
wings of other butterflies to facilitate finding mates.
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Problem: The drawing shows three polarizer/analyzer pairs. The
incident light beam for each pair is unpolarized and has the same
average intensity of 48 W/m2. Find the average intensity of the
transmitted beam for each of the three cases (A, B, and C) shown in
the drawing.
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Solution:
A: The average intensity transmitted by the first filter is half of the
incident average intensity, and is polarized parallel to the first filter's
polarization axis. The second filter has polarization axis parallel to the
first filter's polarization axis, so the average intensity transmitted by
the second filter is equal to the average intensity incident on the
second filter. Thus, the average intensity transmitted through the
second filter is (48)(1/2)(1) = 24 W/m2.
B: The average intensity transmitted by the first filter is half of the
incident average intensity, and is polarized parallel to the first filter's
polarization axis. The second filter has polarization axis perpendicular
to the first filter's polarization axis, so the average intensity transmitted
by the second filter is zero. Thus, the average intensity transmitted
through the second filter is (48)(1/2)(0) = 0 W/m2.
C: The average intensity transmitted by the first filter is half of the
incident average intensity, and is polarized parallel to the first filter's
polarization axis. The second filter has polarization axis inclined at an
angle of 30 degrees relative to the first filter's polarization axis, so the
average intensity transmitted by the second filter is
(48)(1/2)(cos2 [30]) = (48)(1/2)(3/4) = 18 W/m2.
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Problem: Light that is polarized along the vertical direction is incident on a
sheet of polarizing material. Only 94% of the intensity of the light passes
through the sheet and strikes a second sheet of polarizing material. No
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through the sheet and strikes a second sheet of polarizing material. No
light passes through the second sheet. What angle does the transmission
axis of the second sheet make with the vertical?
Solution: The angle of the first filter's polarization axis with respect to the
vertical is theta, where
The second filter has polarization axis perpendicular to the first filter's
polarization axis, so the angle of the second filter's polarization axis with
respect to the vertical is
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Additional Problems and Solutions
Problem: A microwave oven operates at 2.4 GHz with an
intensity inside the oven of 2500 W/m2. Determine the
amplitudes of the oscillating electric and magnetic fields.
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Problem: At what distance from a 10 W point source of
electromagnetic waves is the electric field amplitude
(a) 100 V/m, and (b) 0.010 V/m.
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Problem: Only 25% of the intensity of a polarized light wave
passes through a polarizing filter. What is the angle between
the electric field and the axis of the filter?
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