Chapter 24 Electromagnetic Waves - Physics@ ? Chapter 24 Electromagnetic Waves Wednesday, March 24,
Post on 08-Aug-2018
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
Chapter 24 Electromagnetic WavesWednesday, March 24, 2010 3:16 PM
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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,
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."
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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
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.
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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 time-varying 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 time-varying 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:
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
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 0.) 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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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._________________________________________________________
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?
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:
Let's calculate the value based on currently accepted constants:
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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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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 particlestransitions 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
conceptual unification of electrical, magnetic, and optical phenomena.____________________________________________________
"From a long view of the history of mankindseen from, say, ten thousand years from nowthere can be little doubt that the most significant event of the 19th century will be judged as Maxwells discovery of the laws of electrodynamics." Richard Feynman, Feynman Lectures on Physics,
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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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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
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.
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.
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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.
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 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. ________________________________________________________
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).
The range of wavelengths is between 2.78 m and 3.41 m.________________________________________________
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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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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
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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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 rotating-mirror 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:
Problem: Determine the speed of light in vacuum in units of m/s and ft/ns.
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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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.
familiar with that non-SI unit for distance, the foot._______________________________________________________
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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. ___________________________________________________________
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.
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Problem: How long does it take for electromagnetic waves to travel one-way 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.
(b) The distance between the Sun and the Earth is 1.5 1011 m.
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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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.
(c) The distance between the Sun and Jupiter is 7.8 1011 m.
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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 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.
the same as the speed of light in vacuum:
The speed of the car is therefore
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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 uof a region of vacuum in which there is an electric field is
The analogous expression for magnetic fields 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,
Thus, the total energy density carried by an electromagnetic wave is
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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. __________________________________________________________
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
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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?
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 m3 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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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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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._____________________________________________________
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.
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.
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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 extremelysmall 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
Alternative Solution: From the (*) equation above, you can also continue as follows:
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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
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.
The remainder of the solution proceeds as above.
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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._______________________________________________________
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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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 perpendicularto 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 ) = (48)(1/2)(3/4) = 18 W/m2._________________________________________________________
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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Additional Problems and Solutions
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
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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