You are viewing the html version of Light and Matter, by Benjamin Crowell. This version is only designed for casual browsing, and may have some formatting problems. For serious reading, you want the Adobe Acrobat version. (c) 1998-2011 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version. |
a / Breaking a bar magnet in half doesn't create two monopoles, it creates two smaller dipoles.
b / An explanation at the atomic level.
The unit of magnetic field, the tesla, is named after Serbian-American inventor Nikola Tesla.
A standard dipole made from a square loop of wire shorting across a battery. It acts very much like a bar magnet, but its strength is more easily quantified.
A dipole tends to align itself to the surrounding magnetic field.
c / The magnetic field pattern of a bar magnet. This picture was made by putting iron filings on a piece of paper, and bringing a bar magnet up underneath it. Note how the field pattern passes across the body of the magnet, forming closed loops, as in figure d/2. There are no sources or sinks.
d / Electric fields, 1, have sources and sinks, but magnetic fields, 2, don't.
If you could play with a handful of electric dipoles and a handful of bar magnets, they would appear very similar. For instance, a pair of bar magnets wants to align themselves head-to-tail, and a pair of electric dipoles does the same thing. (It is unfortunately not that easy to make a permanent electric dipole that can be handled like this, since the charge tends to leak.)
You would eventually notice an important difference between the two types of objects, however. The electric dipoles can be broken apart to form isolated positive charges and negative charges. The two-ended device can be broken into parts that are not two-ended. But if you break a bar magnet in half, a, you will find that you have simply made two smaller two-ended objects.
The reason for this behavior is not hard to divine from our microscopic picture of permanent iron magnets. An electric dipole has extra positive “stuff” concentrated in one end and extra negative in the other. The bar magnet, on the other hand, gets its magnetic properties not from an imbalance of magnetic “stuff” at the two ends but from the orientation of the rotation of its electrons. One end is the one from which we could look down the axis and see the electrons rotating clockwise, and the other is the one from which they would appear to go counterclockwise. There is no difference between the “stuff” in one end of the magnet and the other, b.
Nobody has ever succeeded in isolating a single magnetic pole. In technical language, we say that magnetic monopoles do not seem to exist. Electric monopoles do exist --- that's what charges are.
Electric and magnetic forces seem similar in many ways. Both act at a distance, both can be either attractive or repulsive, and both are intimately related to the property of matter called charge. (Recall that magnetism is an interaction between moving charges.) Physicists's aesthetic senses have been offended for a long time because this seeming symmetry is broken by the existence of electric monopoles and the absence of magnetic ones. Perhaps some exotic form of matter exists, composed of particles that are magnetic monopoles. If such particles could be found in cosmic rays or moon rocks, it would be evidence that the apparent asymmetry was only an asymmetry in the composition of the universe, not in the laws of physics. For these admittedly subjective reasons, there have been several searches for magnetic monopoles. Experiments have been performed, with negative results, to look for magnetic monopoles embedded in ordinary matter. Soviet physicists in the 1960s made exciting claims that they had created and detected magnetic monopoles in particle accelerators, but there was no success in attempts to reproduce the results there or at other accelerators. The most recent search for magnetic monopoles, done by reanalyzing data from the search for the top quark at Fermilab, turned up no candidates, which shows that either monopoles don't exist in nature or they are extremely massive and thus hard to create in accelerators.
Since magnetic monopoles don't seem to exist, it would not make much sense to define a magnetic field in terms of the force on a test monopole. Instead, we follow the philosophy of the alternative definition of the electric field, and define the field in terms of the torque on a magnetic test dipole. This is exactly what a magnetic compass does: the needle is a little iron magnet which acts like a magnetic dipole and shows us the direction of the earth's magnetic field.
To define the strength of a magnetic field, however, we need some way of defining the strength of a test dipole, i.e., we need a definition of the magnetic dipole moment. We could use an iron permanent magnet constructed according to certain specifications, but such an object is really an extremely complex system consisting of many iron atoms, only some of which are aligned. A more fundamental standard dipole is a square current loop. This could be little resistive circuit consisting of a square of wire shorting across a battery.
We will find that such a loop, when placed in a magnetic field, experiences a torque that tends to align plane so that its face points in a certain direction. (Since the loop is symmetric, it doesn't care if we rotate it like a wheel without changing the plane in which it lies.) It is this preferred facing direction that we will end up defining as the direction of the magnetic field.
Experiments show if the loop is out of alignment with the field, the torque on it is proportional to the amount of current, and also to the interior area of the loop. The proportionality to current makes sense, since magnetic forces are interactions between moving charges, and current is a measure of the motion of charge. The proportionality to the loop's area is also not hard to understand, because increasing the length of the sides of the square increases both the amount of charge contained in this circular “river” and the amount of leverage supplied for making torque. Two separate physical reasons for a proportionality to length result in an overall proportionality to length squared, which is the same as the area of the loop. For these reasons, we define the magnetic dipole moment of a square current loop as
![text{ dipole moment of a square current loop]}](math/eq_db98c826.png)
We now define the magnetic field in a manner entirely analogous to the second definition of the electric field:
The magnetic field vector, B, at any location in space is
defined by observing the torque exerted on a magnetic test
dipole Dmt consisting of a square current loop. The
field's magnitude is 
, where
θ is the angle by which the loop is misaligned. The
direction of the field is perpendicular to the loop; of the
two perpendiculars, we choose the one such that if we look
along it, the loop's current is counterclockwise.
We find from this definition that the magnetic field has
units of 
. This unwieldy combination of
units is abbreviated as the tesla, 1
. Refrain from memorizing the part about the
counterclockwise direction at the end; in section 24.5 we'll
see how to understand this in terms of more basic principles.
The nonexistence of magnetic monopoles means that unlike an electric field, d/1, a magnetic one, d/2, can never have sources or sinks. The magnetic field vectors lead in paths that loop back on themselves, without ever converging or diverging at a point.
Our study of the electric field built on our previous understanding of electric forces, which was ultimately based on Coulomb's law for the electric force between two point charges. Since magnetism is ultimately an interaction between currents, i.e., between moving charges, it is reasonable to wish for a magnetic analog of Coulomb's law, an equation that would tell us the magnetic force between any two moving point charges.
Such a law, unfortunately, does not exist. Coulomb's law describes the special case of electrostatics: if a set of charges is sitting around and not moving, it tells us the interactions among them. Coulomb's law fails if the charges are in motion, since it does not incorporate any allowance for the time delay in the outward propagation of a change in the locations of the charges.
A pair of moving point charges will certainly exert magnetic forces on one another, but their magnetic fields are like the v-shaped bow waves left by boats. Each point charge experiences a magnetic field that originated from the other charge when it was at some previous position. There is no way to construct a force law that tells us the force between them based only on their current positions in space.
There is, however, a science of magnetostatics that covers a great many important cases. Magnetostatics describes magnetic forces among currents in the special case where the currents are steady and continuous, leading to magnetic fields throughout space that do not change over time.
The magnetic field of a long, straight wire is one example that we can say something about without resorting to fancy mathematics. We saw in examples 4 on p. 607 and 14 on p. 619 that the electric field of a uniform line of charge is E=2kq/Lr, where r is the distance from the line and q/L is the charge per unit length. In a frame of reference moving at velocity v parallel to the line, this electric field will be observed as a combination of electric and magnetic fields. It therefore follows that the magnetic field of a long, straight, current-carrying wire must be proportional to 1/r. We also expect that it will be proportional to the Coulomb constant, which sets the strength of electric and magnetic interactions, and to the current I in the wire. The complete expression turns out to be B=(k/c2)(2I/r). This is identical to the expression for E except for replacement of q/L with I and an additional factor of 1/c2. The latter occurs because magnetism is a purely relativistic effect, and the relativistic length contraction depends on v2/c2.
Figure e shows the equations for some of the more commonly
encountered configurations, with illustrations of their
field patterns. They all have a factor of k/c2 in front, which shows that magnetism
is just electricity (k) seen through the lens of relativity (1/c2). A convenient
feature of SI units is that k/c2 has a numerical value of exactly 10-7, with units of 
.
Field created by a long, straight wire carrying current I:
Here r is the distance from the center of the wire. The field vectors trace circles in planes perpendicular to the wire, going clockwise when viewed from along the direction of the current.
Field created by a single circular loop of current:
The field vectors form a dipole-like pattern, coming through the loop and back
around on the outside. Each oval path traced out by the field vectors appears clockwise
if viewed from along the direction the current is going when it punches through it.
There is no simple equation for a field at an arbitrary point in space, but
for a point lying along the central axis perpendicular to the loop,
the field is
where b is the radius of the loop and z is the distance of the point from the plane of the loop.
Field created by a solenoid (cylindrical coil):
The field pattern is similar to that of a single loop, but for a long solenoid
the paths of the field vectors become very straight on the inside of the coil
and on the outside immediately next to the coil. For a sufficiently long solenoid,
the interior field also becomes very nearly uniform, with a magnitude of
where N is the number of turns of wire and ℓ is the length of the solenoid. The field near the mouths or outside the coil is not constant, and is more difficult to calculate. For a long solenoid, the exterior field is much smaller than the interior field.
Don't memorize the equations!
We now know how to calculate magnetic fields in some typical situations, but one might also like to be able to calculate magnetic forces, such as the force of a solenoid on a moving charged particle, or the force between two parallel current-carrying wires.
We will restrict ourselves to the case of the force on a charged particle moving through a magnetic field, which allows us to calculate the force between two objects when one is a moving charged particle and the other is one whose magnetic field we know how to find. An example is the use of solenoids inside a TV tube to guide the electron beam as it paints a picture.
Experiments show that the magnetic force on a moving charged particle has a magnitude given by
where v is the velocity vector of the particle, and θ is the angle between the v and B vectors. Unlike electric and gravitational forces, magnetic forces do not lie along the same line as the field vector. The force is always perpendicular to both v and B. Given two vectors, there is only one line perpendicular to both of them, so the force vector points in one of the two possible directions along this line. For a positively charged particle, the direction of the force vector can be found as follows. First, position the v and B vectors with their tails together. The direction of F is such that if you sight along it, the B vector is clockwise from the v vector; for a negatively charged particle the direction of the force is reversed. Note that since the force is perpendicular to the particle's motion, the magnetic field never does work on it.
From the photo, it's possible to find the direction of the magnetic field made by the permanent magnet. The electrons in the copper wire are negatively charged, so they flow from the negative (flat) terminal of the battery to the positive terminal (the one with the bump, in front). As the electrons pass by the permanent magnet, we can imagine that they would experience a field either toward the magnet, or away from it, depending on which way the magnet was flipped when it was stuck onto the battery. Imagine sighting along the upward force vector, which you could do if you were a tiny bug lying on your back underneath the wire. Since the electrons are negatively charged, the B vector must be counterclockwise from the v vector, which means toward the magnet.
During an MRI scan of the head, the patient's nervous system is exposed to intense magnetic fields. The average velocities of the charge-carrying ions in the nerve cells is fairly low, but if the patient moves her head suddenly, the velocity can be high enough that the magnetic field makes significant forces on the ions. This can result in visual and auditory hallucinations, e.g., frying bacon sounds.
On p. 611 I gave equations for the energy stored in the gravitational and electric fields. Since a magnetic field is essentially an electric field seen in a different frame of reference, we expect the magnetic-field equation to be closely analogous to the electric version, and it is:
The idea here is that k/c2 is the magnetic version of the electric quantity k, the 1/c2 representing the fact that magnetism is a relativistic effect.
Solenoids are very common electrical devices, but they can be a hazard to someone who is working on them. Imagine a solenoid that initially has a DC current passing through it. The current creates a magnetic field inside and around it, which contains energy. Now suppose that we break the circuit. Since there is no longer a complete circuit, current will quickly stop flowing, and the magnetic field will collapse very quickly. The field had energy stored in it, and even a small amount of energy can create a dangerous power surge if released over a short enough time interval. It is prudent not to fiddle with a solenoid that has current flowing through it, since breaking the circuit could be hazardous to your health.
As a typical numerical estimate, let's assume a 40 cm × 40 cm × 40 cm solenoid with an interior magnetic field of 1.0 T (quite a strong field). For the sake of this rough estimate, we ignore the exterior field, which is weak, and assume that the solenoid is cubical in shape. The energy stored in the field is
That's a lot of energy!
f / A proof that causality imposes a universal speed limit. In the original frame of reference, represented by the square, event A happens a little before event B. In the new frame, shown by the parallelogram, A happens after t=0, but B happens before t=0; that is, B happens before A. The time ordering of the two events has been reversed. This can only happen because events A and B are very close together in time and fairly far apart in space. The line segment connecting A and B has a slope greater than 1, meaning that if we wanted to be present at both events, we would have to travel at a speed greater than c (which equals 1 in the units used on this graph). You will find that if you pick any two points for which the slope of the line segment connecting them is less than 1, you can never get them to straddle the new t=0 line in this funny, time-reversed way. Since different observers disagree on the time order of events like A and B, causality requires that information never travel from A to B or from B to A; if it did, then we would have time-travel paradoxes. The conclusion is that c is the maximum speed of cause and effect in relativity.
Let's think a little more about the role of the 45-degree diagonal in the Lorentz transformation. Slopes on these graphs are interpreted as velocities. This line has a slope of 1 in relativistic units, but that slope corresponds to c in ordinary metric units. We already know that the relativistic distance unit must be extremely large compared to the relativistic time unit, so c must be extremely large. Now note what happens when we perform a Lorentz transformation: this particular line gets stretched, but the new version of the line lies right on top of the old one, and its slope stays the same. In other words, if one observer says that something has a velocity equal to c, every other observer will agree on that velocity as well. (The same thing happens with -c.)
This is counterintuitive, since we expect velocities to add and subtract in relative motion. If a dog is running away from me at 5 m/s relative to the sidewalk, and I run after it at 3 m/s, the dog's velocity in my frame of reference is 2 m/s. According to everything we have learned about motion (p. 81), the dog must have different speeds in the two frames: 5 m/s in the sidewalk's frame and 2 m/s in mine. But velocities are measured by dividing a distance by a time, and both distance and time are distorted by relativistic effects, so we actually shouldn't expect the ordinary arithmetic addition of velocities to hold in relativity; it's an approximation that's valid at velocities that are small compared to c.
For example, suppose Janet takes a trip in a spaceship, and accelerates until she is moving at 0.6c relative to the earth. She then launches a space probe in the forward direction at a speed relative to her ship of 0.6c. We might think that the probe was then moving at a velocity of 1.2c, but in fact the answer is still less than c (problem 1, page 675). This is an example of a more general fact about relativity, which is that c represents a universal speed limit. This is required by causality, as shown in figure f (but see section 26.7, p. 754).
Now consider a beam of light. We're used to talking casually about the “speed of light,” but what does that really mean? Motion is relative, so normally if we want to talk about a velocity, we have to specify what it's measured relative to. A sound wave has a certain speed relative to the air, and a water wave has its own speed relative to the water. If we want to measure the speed of an ocean wave, for example, we should make sure to measure it in a frame of reference at rest relative to the water. But light isn't a vibration of a physical medium; it can propagate through the near-perfect vacuum of outer space, as when rays of sunlight travel to earth. This seems like a paradox: light is supposed to have a specific speed, but there is no way to decide what frame of reference to measure it in. The way out of the paradox is that light must travel at a velocity equal to c. Since all observers agree on a velocity of c, regardless of their frame of reference, everything is consistent.
The constancy of the speed of light had in fact already been observed when Einstein was an 8-year-old boy, but because nobody could figure out how to interpret it, the result was largely ignored. In 1887 Michelson and Morley set up a clever apparatus to measure any difference in the speed of light beams traveling east-west and north-south. The motion of the earth around the sun at 110,000 km/hour (about 0.01% of the speed of light) is to our west during the day. Michelson and Morley believed that light was a vibration of a mysterious medium called the ether, so they expected that the speed of light would be a fixed value relative to the ether. As the earth moved through the ether, they thought they would observe an effect on the velocity of light along an east-west line. For instance, if they released a beam of light in a westward direction during the day, they expected that it would move away from them at less than the normal speed because the earth was chasing it through the ether. They were surprised when they found that the expected 0.01% change in the speed of light did not occur.
The Michelson-Morley experiment, shown in photographs, and drawings from the original 1887 paper. 1. A simplified drawing of the apparatus. A beam of light from the source, s, is partially reflected and partially transmitted by the half-silvered mirror h1. The two half-intensity parts of the beam are reflected by the mirrors at a and b, reunited, and observed in the telescope, t. If the earth's surface was supposed to be moving through the ether, then the times taken by the two light waves to pass through the moving ether would be unequal, and the resulting time lag would be detectable by observing the interference between the waves when they were reunited. 2. In the real apparatus, the light beams were reflected multiple times. The effective length of each arm was increased to 11 meters, which greatly improved its sensitivity to the small expected difference in the speed of light. 3. In an earlier version of the experiment, they had run into problems with its “extreme sensitiveness to vibration,” which was “so great that it was impossible to see the interference fringes except at brief intervals ... even at two o'clock in the morning.” They therefore mounted the whole thing on a massive stone floating in a pool of mercury, which also made it possible to rotate it easily. 4. A photo of the apparatus.
The figure shows a famous thought experiment devised by Einstein. A train is moving at constant velocity to the right when bolts of lightning strike the ground near its front and back. Alice, standing on the dirt at the midpoint of the flashes, observes that the light from the two flashes arrives simultaneously, so she says the two strikes must have occurred simultaneously. Bob, meanwhile, is sitting aboard the train, at its middle. At the moment when Alice sees the two flashes arrive, Bob is right alongside her, inside the train, so he also sees them. But Bob reasons that when the flashes were emitted, he was closer to flash 1 than to flash 2. Therefore if light from both flashes is reaching him simultaneously, and both traveled at c, flash 2 must have had more time to travel. How can this be reconciled with Alice's belief that the flashes were simultaneous?
g / The geometry of induced fields. The induced field tends to form a whirlpool pattern around the change in the vector producing it. Note how they circulate in opposite directions.
j / A generator.
k / A transformer.
Physicists of Michelson and Morley's generation thought that light was a mechanical vibration of the ether, but we now know that it is a ripple in the electric and magnetic fields. With hindsight, relativity essentially requires this:
What is less obvious is that there are not two separate kinds of waves, electric and magnetic. In fact an electric wave can't exist without a magnetic one, or a magnetic one without an electric one. This new fact follows from the principle of induction, which was discovered experimentally by Faraday in 1831, seventy-five years before Einstein. Let's state Faraday's idea first, and then see how something like it must follow inevitably from relativity:
Any electric field that changes over time will produce a magnetic field in the space around it.
Any magnetic field that changes over time will produce an electric field in the space around it.
The induced field tends to have a whirlpool pattern, as shown in figure i, but the whirlpool image is not to be taken too literally; the principle of induction really just requires a field pattern such that, if one inserted a paddlewheel in it, the paddlewheel would spin. All of the field patterns shown in figure j are ones that could be created by induction; all have a counterclockwise “curl” to them.
h / Three fields with counterclockwise “curls.”
i / Observer 1 is at rest with respect to the bar magnet, and observes magnetic fields that have different strengths at different distances from the magnet. Observer 2, hanging out in the region to the left of the magnet, sees the magnet moving toward her, and detects that the magnetic field in that region is getting stronger as time passes.
Figure k shows an example of the fundamental reason why a changing B field must create an E field. In section 23.2 we established that according to relativity, what one observer describes as a purely magnetic field, an observer in a different state of motion describes as a mixture of magnetic and electric fields. This is why there must be both an E and a B in observer 2's frame. Observer 2 cannot explain the electric field as coming from any charges. In frame 2, the E can only be explained as an effect caused by the changing B.
Observer 1 says, “2 feels a changing B field because he's moving through a static field.” Observer 2 says, “I feel a changing B because the magnet is getting closer.”
Although this argument doesn't prove the “whirlpool” geometry, we can
verify that the fields I've drawn in figure k are consistent
with it.
The 
vector is upward,
and the electric field has a curliness to it: a paddlewheel inserted
in the electric field would spin clockwise as seen from above, since
the clockwise torque made by the strong electric field on the right is
greater than the counterclockwise torque made by the weaker electric
field on the left.
A generator, l, consists of a permanent magnet that rotates within a coil of wire. The magnet is turned by a motor or crank, (not shown). As it spins, the nearby magnetic field changes. According to the principle of induction, this changing magnetic field results in an electric field, which has a whirlpool pattern. This electric field pattern creates a current that whips around the coils of wire, and we can tap this current to light the lightbulb.
When you're driving a car, the engine recharges the battery continuously using a device called an alternator, which is really just a generator like the one shown on the previous page, except that the coil rotates while the permanent magnet is fixed in place. Why can't you use the alternator to start the engine if your car's battery is dead?
(answer in the back of the PDF version of the book)In example 18 on p. 572 we discussed the advantages of transmitting power over electrical lines using high voltages and low currents. However, we don't want our wall sockets to operate at 10000 volts! For this reason, the electric company uses a device called a transformer, m, to convert to lower voltages and higher currents inside your house. The coil on the input side creates a magnetic field. Transformers work with alternating current, so the magnetic field surrounding the input coil is always changing. This induces an electric field, which drives a current around the output coil.
If both coils were the same, the arrangement would be symmetric, and the output would be the same as the input, but an output coil with a smaller number of coils gives the electric forces a smaller distance through which to push the electrons. Less mechanical work per unit charge means a lower voltage. Conservation of energy, however, guarantees that the amount of power on the output side must equal the amount put in originally, IinVin = IoutVout, so this reduced voltage must be accompanied by an increased current.
Heinrich Hertz (1857-1894).
l / An electromagnetic wave strikes an ohmic surface. The wave's electric field causes currents to flow up and down. The wave's magnetic field then acts on these currents, producing a force in the direction of the wave's propagation. This is a pre-relativistic argument that light must possess inertia.
The most important consequence of induction is the existence of electromagnetic waves. Whereas a gravitational wave would consist of nothing more than a rippling of gravitational fields, the principle of induction tells us that there can be no purely electrical or purely magnetic waves. Instead, we have waves in which there are both electric and magnetic fields, such as the sinusoidal one shown in the figure. Maxwell proved that such waves were a direct consequence of his equations, and derived their properties mathematically. The derivation would be beyond the mathematical level of this book, so we will just state the results.
An electromagnetic wave.
A sinusoidal electromagnetic wave has the geometry shown above.
The E and B fields are perpendicular
to the direction of motion, and are also perpendicular to
each other. If you look along the direction of motion of the
wave, the B vector is always 90 degrees clockwise from the
E vector. The magnitudes of the two fields are related by
the equation 
.
How is an electromagnetic wave created? It could be emitted, for example, by an electron orbiting an atom or currents going back and forth in a transmitting antenna. In general any accelerating charge will create an electromagnetic wave, although only a current that varies sinusoidally with time will create a sinusoidal wave. Once created, the wave spreads out through space without any need for charges or currents along the way to keep it going. As the electric field oscillates back and forth, it induces the magnetic field, and the oscillating magnetic field in turn creates the electric field. The whole wave pattern propagates through empty space at the velocity c.
The resolution of the paradox is that c is a universal speed limit, so the motorcycle can't be accelerated to c. Observers can never be at rest relative to a light wave, so no observer can have a frame of reference in which a light wave is observed to be at rest.
Two electromagnetic waves traveling in the same direction through space can differ by having their electric and magnetic fields in different directions, a property of the wave called its polarization.
Once Maxwell had derived the existence of electromagnetic waves, he became certain that they were the same phenomenon as light. Both are transverse waves (i.e., the vibration is perpendicular to the direction the wave is moving), and the velocity is the same.
Heinrich Hertz (for whom the unit of frequency is named) verified Maxwell's ideas experimentally. Hertz was the first to succeed in producing, detecting, and studying electromagnetic waves in detail using antennas and electric circuits. To produce the waves, he had to make electric currents oscillate very rapidly in a circuit. In fact, there was really no hope of making the current reverse directions at the frequencies of 1015 Hz possessed by visible light. The fastest electrical oscillations he could produce were 109 Hz, which would give a wavelength of about 30 cm. He succeeded in showing that, just like light, the waves he produced were polarizable, and could be reflected and refracted (i.e., bent, as by a lens), and he built devices such as parabolic mirrors that worked according to the same optical principles as those employing light. Hertz's results were convincing evidence that light and electromagnetic waves were one and the same.
Today, electromagnetic waves with frequencies in the range employed by Hertz are known as radio waves. Any remaining doubts that the “Hertzian waves,” as they were then called, were the same type of wave as light waves were soon dispelled by experiments in the whole range of frequencies in between, as well as the frequencies outside that range. In analogy to the spectrum of visible light, we speak of the entire electromagnetic spectrum, of which the visible spectrum is one segment.
The terminology for the various parts of the spectrum is worth memorizing, and is most easily learned by recognizing the logical relationships between the wavelengths and the properties of the waves with which you are already familiar. Radio waves have wavelengths that are comparable to the size of a radio antenna, i.e., meters to tens of meters. Microwaves were named that because they have much shorter wavelengths than radio waves; when food heats unevenly in a microwave oven, the small distances between neighboring hot and cold spots is half of one wavelength of the standing wave the oven creates. The infrared, visible, and ultraviolet obviously have much shorter wavelengths, because otherwise the wave nature of light would have been as obvious to humans as the wave nature of ocean waves. To remember that ultraviolet, x-rays, and gamma rays all lie on the short-wavelength side of visible, recall that all three of these can cause cancer. (As we'll discuss later in the course, there is a basic physical reason why the cancer-causing disruption of DNA can only be caused by very short-wavelength electromagnetic waves. Contrary to popular belief, microwaves cannot cause cancer, which is why we have microwave ovens and not x-ray ovens!)
When sunlight enters the upper atmosphere, a particular air molecule finds itself being washed over by an electromagnetic wave of frequency f. The molecule's charged particles (nuclei and electrons) act like oscillators being driven by an oscillating force, and respond by vibrating at the same frequency f. Energy is sucked out of the incoming beam of sunlight and converted into the kinetic energy of the oscillating particles. However, these particles are accelerating, so they act like little radio antennas that put the energy back out as spherical waves of light that spread out in all directions. An object oscillating at a frequency f has an acceleration proportional to f2, and an accelerating charged particle creates an electromagnetic wave whose fields are proportional to its acceleration, so the field of the reradiated spherical wave is proportional to f2. The energy of a field is proportional to the square of the field, so the energy of the reradiated wave is proportional to f4. Since blue light has about twice the frequency of red light, this process is about 24=16 times as strong for blue as for red, and that's why the sky is blue.
Newton defined momentum as mv, and that would lead us to believe that light, which has no mass, should have no momentum. However, Newton's laws only work at velocities that are small compared to the speed of light, and light travels at the speed of light, so there is no reason to trust Newton here. In fact, it's straightforward to show that electromagnetic waves have momentum. If a light wave strikes an ohmic surface, as in figure o, the wave's electric field causes charges to vibrate back and forth in the surface. These currents then experience a magnetic force from the wave's magnetic field, and application of the geometrical rule on p. 658 shows that the resulting force is in the direction of propagation of the wave. Thus the light wave acts as if it has momentum and inertia. This is explored further in problem 12 on p. 759.
The physicist Richard Feynman helped to get me hooked on physics with an educational film containing the following puzzle. Imagine that you establish radio contact with an alien on another planet. Neither of you even knows where the other one's planet is, and you aren't able to establish any landmarks that you both recognize. You manage to learn quite a bit of each other's languages, but you're stumped when you try to establish the definitions of left and right (or, equivalently, clockwise and counterclockwise). Is there any way to do it?
If there was any way to do it without reference to external landmarks, then it would imply that the laws of physics themselves were asymmetric, which would be strange. Why should they distinguish left from right? The gravitational field pattern surrounding a star or planet looks the same in a mirror, and the same goes for electric fields. However, the field patterns shown in section 24.2 seem to violate this principle, but do they really? Could you use these patterns to explain left and right to the alien? In fact, the answer is no. If you look back at the definition of the magnetic field in section 24.1, it also contains a reference to handedness: the counterclockwise direction of the loop's current as viewed along the magnetic field. The aliens might have reversed their definition of the magnetic field, in which case their drawings of field patterns would look like mirror images of ours.
Until the middle of the twentieth century, physicists assumed that any reasonable set of physical laws would have to have this kind of symmetry between left and right. An asymmetry would be grotesque. Whatever their aesthetic feelings, they had to change their opinions about reality when experiments showed that the weak nuclear force (section 22.4) violates right-left symmetry! It is still a mystery why right-left symmetry is observed so scrupulously in general, but is violated by one particular type of physical process.
m / A graphical representation of the Lorentz transformation for a velocity of (3/5)c. The long diagonal is stretched by a factor of two, the short one is half its former length, and the area is the same as before.
n / The pattern of waves made by a point source moving to the right across the water. Note the shorter wavelength of the forward-emitted waves and the longer wavelength of the backward-going ones.
o / At event O, the source and the receiver are on top of each other, so as the source emits a wave crest, it is received without any time delay. At P, the source emits another wave crest, and at Q the receiver receives it.
Figure p shows our now-familiar method of visualizing a Lorentz transformation, in a case where the numbers come out to be particularly simple. This diagram has two geometrical features that we have referred to before without digging into their physical significance: the stretch factor of the diagonals, and the area. In this section we'll see that the former can be related to the Doppler effect, and the latter to clock time.
When Doppler shifts happen to ripples on a pond or the sound waves from an airplane, they can depend on the relative motion of three different objects: the source, the receiver, and the medium. But light waves don't have a medium. Therefore Doppler shifts of light can only depend on the relative motion of the source and observer.
One simple case is the one in which the relative motion of the source and the receiver is perpendicular to the line connecting them. That is, the motion is transverse. Nonrelativistic Doppler shifts happen because the distance between the source and receiver is changing, so in nonrelativistic physics we don't expect any Doppler shift at all when the motion is transverse, and this is what is in fact observed to high precision. For example, the photo shows shortened and lengthened wavelengths to the right and left, along the source's line of motion, but an observer above or below the source measures just the normal, unshifted wavelength and frequency. But relativistically, we have a time dilation effect, so for light waves emitted transversely, there is a Doppler shift of 1/γ in frequency (or γ in wavelength).
The other simple case is the one in which the relative motion of the source and receiver is longitudinal, i.e., they are either approaching or receding from one another. For example, distant galaxies are receding from our galaxy due to the expansion of the universe, and this expansion was originally detected because Doppler shifts toward the red (low-frequency) end of the spectrum were observed.
Nonrelativistically, we would expect the light from such a galaxy to be Doppler shifted down in frequency by some factor, which would depend on the relative velocities of three different objects: the source, the wave's medium, and the receiver. Relativistically, things get simpler, because light isn't a vibration of a physical medium, so the Doppler shift can only depend on a single velocity v, which is the rate at which the separation between the source and the receiver is increasing.
The square in figure r is the “graph paper” used by someone who considers the source to be at rest, while the parallelogram plays a similar role for the receiver. The figure is drawn for the case where v=3/5 (in units where c=1), and in this case the stretch factor of the long diagonal is 2. To keep the area the same, the short diagonal has to be squished to half its original size. But now it's a matter of simple geometry to show that OP equals half the width of the square, and this tells us that the Doppler shift is a factor of 1/2 in frequency. That is, the squish factor of the short diagonal is interpreted as the Doppler shift. To get this as a general equation for velocities other than 3/5, one can show by straightforward fiddling with the result of part c of problem 2 on p. 651 that the Doppler shift is
Here v>0 is the case where the source and receiver are getting farther apart, v<0 the case where they are approaching. (This is the opposite of the sign convention used in section 19.5. It is convenient to change conventions here so that we can use positive values of v in the case of cosmological red-shifts, which are the most important application.)
Suppose that Alice stays at home on earth while her twin Betty takes off in her rocket ship at 3/5 of the speed of light. When I first learned relativity, the thing that caused me the most pain was understanding how each observer could say that the other was the one whose time was slow. It seemed to me that if I could take a pill that would speed up my mind and my body, then naturally I would see everybody else as being slow. Shouldn't the same apply to relativity? But suppose Alice and Betty get on the radio and try to settle who is the fast one and who is the slow one. Each twin's voice sounds slooooowed doooowwwwn to the other. If Alice claps her hands twice, at a time interval of one second by her clock, Betty hears the hand-claps coming over the radio two seconds apart, but the situation is exactly symmetric, and Alice hears the same thing if Betty claps. Each twin analyzes the situation using a diagram identical to r, and attributes her sister's observations to a complicated combination of time distortion, the time taken by the radio signals to propagate, and the motion of her twin relative to her.
Turn your book upside-down and reinterpret figure r.
(answer in the back of the PDF version of the book)The result of example 9 was the basis of one of the earliest laboratory tests of special relativity, by Ives and Stilwell in 1938. They observed the light emitted by excited by a beam of H2+ and H3+ ions with speeds of a few tenths of a percent of c. Measuring the light from both ahead of and behind the beams, they found that the product of the Doppler shifts D(v)D(-v) was equal to 1, as predicted by relativity. If relativity had been false, then one would have expected the product to differ from 1 by an amount that would have been detectable in their experiment. In 2003, Saathoff et al. carried out an extremely precise version of the Ives-Stilwell technique with Li+ ions moving at 6.4% of c. The frequencies observed, in units of MHz, were:
| ftextupo | = 546466918.8±0.4 |
| (unshifted frequency) | |
| ftextupoDv | = 582490203.44±.09 |
| (shifted frequency, forward) | |
| ftextupo Dv | = 512671442.9±0.5 |
| (shifted frequency, backward) | |
| sqrtftextupoDvcdot ftextupo Dv | =546466918.6±0.3 |
The results show incredibly precise agreement between fo and 
, as expected
relativistically because D(v)D(-v) is supposed to equal 1. The agreement extends to 9 significant figures, whereas
if relativity had been false there should have been a relative disagreement of about v2=.004, i.e., a discrepancy in the third significant figure.
The spectacular agreement with theory has made this experiment a lightning rod for
anti-relativity kooks.
We saw on p. 660 that relativistic velocities should not be expected to be exactly additive, and problem 1 on p. 675 verifies this in the special case where A moves relative to B at 0.6c and B relative to C at 0.6c --- the result not being 1.2c. The relativistic Doppler shift provides a simple way of deriving a general equation for the relativistic combination of velocities; problem 21 on p. 680 guides you through the steps of this derivation.
On p. 638 we proved that the Lorentz transformation doesn't change the area of a shape in the x-t plane. We used this only as a stepping stone toward the Lorentz transformation, but it is natural to wonder whether this kind of area has any physical interest of its own.
The equal-area result is not relativistic, since the proof never appeals to property 5 on page 634. Cases I and II on page 637 also have the equal-area property. We can see this clearly in a Galilean transformation like figure g on p. 635, where the distortion of the rectangle could be accomplished by cutting it into vertical slices and then displacing the slices upward without changing their areas.
But the area does have a nice interpretation in the relativistic case. Suppose that we have events A (Charles VII is restored to the throne) and B (Joan of Arc is executed). Now imagine that technologically advanced aliens want to be present at both A and B, but in the interim they wish to fly away in their spaceship, be present at some other event P (perhaps a news conference at which they give an update on the events taking place on earth), but get back in time for B. Since nothing can go faster than c (which we take to equal 1 in appropriate units), P cannot be too far away. The set of all possible events P forms a rectangle, figure s/1, in the x-t plane that has A and B at opposite corners and whose edges have slopes equal to ± 1. We call this type of rectangle a light-rectangle, because its sides could represent the motion of rays of light.
p / 1. The gray light-rectangle represents the set of all events such as P that could be visited after A and before B.
2. The rectangle becomes a square in the frame in which A and B occur at the same location in space.
3. The area of the dashed square is τ2, so the area of the gray square is τ2/2.
The area of this rectangle will be the same regardless of one's frame of reference. In particular, we could choose a special frame of reference, panel 2 of the figure, such that A and B occur in the same place. (They do not occur at the same place, for example, in the sun's frame, because the earth is spinning and going around the sun.) Since the speed c, which equals 1 in our units, is the same in all frames of reference, and the sides of the rectangle had slopes ± 1 in frame 1, they must still have slopes ± 1 in frame 2. The rectangle becomes a square with its diagonals parallel to the x and t axes, and the length of these diagonals equals the time τ elapsed on a clock that is at rest in frame 2, i.e., a clock that glides through space at constant velocity from A to B, meeting up with the planet earth at the appointed time. As shown in panel 3 of the figure, the area of the gray regions can be interpreted as half the square of this gliding-clock time. If events A and B are separated by a distance x and a time t, then in general t2-x2 gives the square of the gliding-clock time.1
When |x| is greater than |t|, events A and are so far apart in space and so close together in time that it would be impossible to have a cause and effect relationship between them, since c=1 is the maximum speed of cause and effect. In this situation t2-x2 is negative and cannot be interpreted as a clock time, but it can be interpreted as minus the square of the distance bewteen A and B as measured by rulers at rest in a frame in which A and B are simultaneous.
No matter what, t2-x2 is the same as measured in all frames of reference. Geometrically, it plays the same role in the x-t plane that ruler measurements play in the Euclidean plane. In Euclidean geometry, the ruler-distance between any two points stays the same regardless of rotation, i.e., regardless of the angle from which we view the scene; according to the Pythagorean theorem, the square of this distance is x2+y2. In the x-t plane, t2-x2 stays the same regardless of the frame of reference.
magnetic field — a field of force, defined in terms of the torque exerted on a test dipole
magnetic dipole — an object, such as a current loop, an atom, or a bar magnet, that experiences torques due to magnetic forces; the strength of magnetic dipoles is measured by comparison with a standard dipole consisting of a square loop of wire of a given size and carrying a given amount of current
induction — the production of an electric field by a changing magnetic field, or vice-versa
B — the magnetic field
Dm — magnetic dipole moment
{}
The magnetic field is defined in terms of the torque on a magnetic test dipole. It has no sources or sinks; magnetic field patterns never converge on or diverge from a point.
Relativity dictates a maximum speed limit c for cause and effect. This speed is the same in all frames of reference.
Relativity requires that the magnetic and electric fields be intimately related. The principle of induction states that any changing electric field produces a magnetic field in the surrounding space, and vice-versa. These induced fields tend to form whirlpool patterns.
The most important consequence of the principle of induction is that there are no purely magnetic or purely electric waves. Electromagnetic disturbances propagate outward at c as combined magnetic and electric waves, with a well-defined relationship between the magnitudes and directions of the electric and magnetic fields. These electromagnetic waves are what light is made of, but other forms of electromagnetic waves exist besides visible light, including radio waves, x-rays, and gamma rays.
1. The figure illustrates a Lorentz transformation using the conventions employed in section 23.1. For simplicity, the transformation chosen is one that lengthens one diagonal by a factor of 2. Since Lorentz transformations preserve area, the other diagonal is shortened by a factor of 2. Let the original frame of reference, depicted with the square, be A, and the new one B. (a) By measuring with a ruler on the figure, show that the velocity of frame B relative to frame A is 0.6c. (b) Print out a copy of the page. With a ruler, draw a third parallelogram that represents a second successive Lorentz transformation, one that lengthens the long diagonal by another factor of 2. Call this third frame C. Use measurements with a ruler to determine frame C's velocity relative to frame A. Does it equal double the velocity found in part a? Explain why it should be expected to turn out the way it does.(answer check available at lightandmatter.com)
2. (a) In this chapter we've represented Lorentz transformations as distortions of a square into
various parallelograms, with the degree of distortion depending on the velocity of one
frame of reference relative to another. Suppose that one frame of reference was moving
at c relative to another. Discuss what would happen in terms of distortion of a square,
and show that this is impossible by using an argument similar to the one used to rule out
transformations like the one in figure h on page 636.
(b) Resolve the following paradox. Two pen-pointer lasers are placed side by side and aimed in parallel
directions. Their beams both travel at c relative to the hardware, but each beam has a velocity of zero relative to the neighboring beam.
But the speed of light can't be zero; it's supposed to be the same in all frames of reference.
3. Consider two solenoids, one of which is smaller so that it can be put inside the other. Assume they are long enough so that each one only contributes significantly to the field inside itself, and the interior fields are nearly uniform. Consider the configuration where the small one is inside the big one with their currents circulating in the same direction, and a second configuration in which the currents circulate in opposite directions. Compare the energies of these configurations with the energy when the solenoids are far apart. Based on this reasoning, which configuration is stable, and in which configuration will the little solenoid tend to get twisted around or spit out? [Hint: A stable system has low energy; energy would have to be added to change its configuration.]
4. The figure shows a nested pair of circular wire loops
used to create magnetic fields. (The twisting of the leads
is a practical trick for reducing the magnetic fields they
contribute, so the fields are very nearly what we would
expect for an ideal circular current loop.) The coordinate
system below is to make it easier to discuss directions in
space. One loop is in the y-z plane, the other in the
x-y plane. Each of the loops has a radius of
1.0 cm, and carries 1.0 A in the direction indicated by the arrow.
(a) Using the equation in optional section 24.2, calculate
the magnetic field that would be produced by one such
loop, at its center.(answer check available at lightandmatter.com)
(b) Describe the direction of the magnetic field that would
be produced, at its center, by the loop in the x-y plane alone.
(c) Do the same for the other loop.
(d) Calculate the magnitude of the magnetic field produced
by the two loops in combination, at their common center.
Describe its direction.(answer check available at lightandmatter.com)
5.
One model of the hydrogen atom has the electron circling
around the proton at a speed of 2.2×106 m/s, in an
orbit with a radius of 0.05 nm. (Although the electron and
proton really orbit around their common center of mass, the
center of mass is very close to the proton, since it is 2000
times more massive. For this problem, assume the proton is
stationary.) In homework problem 19 on page 584, you
calculated the electric current created.
(a) Now estimate the magnetic field created at the center
of the atom by the electron. We are treating the circling
electron as a current loop, even though it's only a single particle.(answer check available at lightandmatter.com)
(b) Does the proton experience a nonzero force from the
electron's magnetic field? Explain.
(c) Does the electron experience a magnetic field from
the proton? Explain.
(d) Does the electron experience a magnetic field created by
its own current? Explain.
(e) Is there an electric force acting between the proton
and electron? If so, calculate it.(answer check available at lightandmatter.com)
(f) Is there a gravitational force acting between the proton
and electron? If so, calculate it.
(g) An inward force is required to keep the electron in its
orbit -- otherwise it would obey Newton's first law and go
straight, leaving the atom. Based on your answers to the
previous parts, which force or forces (electric, magnetic
and gravitational) contributes significantly to this inward force?
[Based on a problem by Arnold Arons.]
6. Suppose a charged particle is moving through a region of space in which there is an electric field perpendicular to its velocity vector, and also a magnetic field perpendicular to both the particle's velocity vector and the electric field. Show that there will be one particular velocity at which the particle can be moving that results in a total force of zero on it. Relate this velocity to the magnitudes of the electric and magnetic fields. (Such an arrangement, called a velocity filter, is one way of determining the speed of an unknown particle.)
7. If you put four times more current through a solenoid, how many times more energy is stored in its magnetic field?(answer check available at lightandmatter.com)
8. Suppose we are given a permanent magnet with a complicated, asymmetric shape. Describe how a series of measurements with a magnetic compass could be used to determine the strength and direction of its magnetic field at some point of interest. Assume that you are only able to see the direction to which the compass needle settles; you cannot measure the torque acting on it.
9. Consider two solenoids, one of which is smaller so that it can be put inside the other. Assume they are long enough to act like ideal solenoids, so that each one only contributes significantly to the field inside itself, and the interior fields are nearly uniform. Consider the configuration where the small one is partly inside and partly hanging out of the big one, with their currents circulating in the same direction. Their axes are constrained to coincide.
(a) Find the magnetic potential energy as a function of the length x of the part of the small solenoid that is inside the big one. (Your equation will include other relevant variables describing the two solenoids.)
(b) Based on your answer to part (a), find the force acting between the solenoids.
r / Problem 10.
10. Four long wires are arranged, as shown, so that their cross-section forms a square, with connections at the ends so that current flows through all four before exiting. Note that the current is to the right in the two back wires, but to the left in the front wires. If the dimensions of the cross-sectional square (height and front-to-back) are b, find the magnetic field (magnitude and direction) along the long central axis.(answer check available at lightandmatter.com)
11. (solution in the pdf version of the book) The purpose of this problem is to find the force experienced by a straight, current-carrying wire running perpendicular to a uniform magnetic field. (a) Let A be the cross-sectional area of the wire, n the number of free charged particles per unit volume, q the charge per particle, and v the average velocity of the particles. Show that the current is I=Avnq. (b) Show that the magnetic force per unit length is AvnqB. (c) Combining these results, show that the force on the wire per unit length is equal to IB.
12. (solution in the pdf version of the book) Suppose two long, parallel wires are carrying current I1 and I2. The currents may be either in the same direction or in opposite directions. (a) Using the information from section 24.2, determine under what conditions the force is attractive, and under what conditions it is repulsive. Note that, because of the difficulties explored in problem 14, it's possible to get yourself tied up in knots if you use the energy approach of section 22.4. (b) Starting from the result of problem 11, calculate the force per unit length.
13. Section 24.2 states the following rule:
For a positively charged particle, the direction of the F vector is the one such that if you sight along it, the B vector is clockwise from the v vector.
Make a three-dimensional model of the three vectors using pencils or rolled-up pieces of paper to represent the vectors assembled with their tails together. Now write down every possible way in which the rule could be rewritten by scrambling up the three symbols F, B, and v. Referring to your model, which are correct and which are incorrect?
14. Prove that any two planar current loops with the same value of IA will experience the same torque in a magnetic field, regardless of their shapes. In other words, the dipole moment of a current loop can be defined as IA, regardless of whether its shape is a square.
15. A Helmholtz coil is defined as a pair of identical circular coils separated by a distance, h, equal to their radius, b. (Each coil may have more than one turn of wire.) Current circulates in the same direction in each coil, so the fields tend to reinforce each other in the interior region. This configuration has the advantage of being fairly open, so that other apparatus can be easily placed inside and subjected to the field while remaining visible from the outside. The choice of h=b results in the most uniform possible field near the center. (a) Find the percentage drop in the field at the center of one coil, compared to the full strength at the center of the whole apparatus. (b) What value of h (not equal to b) would make this percentage difference equal to zero?
16. (solution in the pdf version of the book) (a) In the photo of the vacuum tube apparatus in section 24.2, infer the direction of the magnetic field from the motion of the electron beam. (b) Based on your answer to a, find the direction of the currents in the coils. (c) What direction are the electrons in the coils going? (d) Are the currents in the coils repelling or attracting the currents consisting of the beam inside the tube? Compare with figure z on p. 648.
17. In the photo of the vacuum tube apparatus in section 24.2, an approximately uniform magnetic field caused circular motion. Is there any other possibility besides a circle? What can happen in general?
18. This problem is now problem 16 on p. 627
19. (solution in the pdf version of the book) In section 24.2 I gave an equation for the magnetic field in the interior of a solenoid, but that equation doesn't give the right answer near the mouths or on the outside. Although in general the computation of the field in these other regions is complicated, it is possible to find a precise, simple result for the field at the center of one of the mouths, using only symmetry and vector addition. What is it?
20. Prove that in an electromagnetic wave, half the energy is in the electric field and half in the magnetic field.
21. (solution in the pdf version of the book) As promised in section 24.7, this problem will lead you through the steps of finding an equation for the combination of velocities in relativity, generalizing the numerical result found in problem 1. Suppose that A moves relative to B at velocity u, and B relative to C at v. We want to find A's velocity w relative to C, in terms of u and v. Suppose that A emits light with a certain frequency. This will be observed by B with a Doppler shift D(u). C detects a further shift of D(v) relative to B. We therefore expect the Doppler shifts simply multiply, D(w)=D(u)D(v), and this provides an implicit rule for determining w if u and v are known. (a) Using the expression for D given in section 24.7.1, write down an equation relating u, v, and w. (b) Solve for w in terms of u and v. (c) Show that your answer to part b satisfies the correspondence principle.
\begin{handson}{}{Polarization}{\onecolumn}
Apparatus:
calcite (Iceland spar) crystal
polaroid film
1. Lay the crystal on a piece of paper that has print on it. You will observe a double image. See what happens if you rotate the crystal.
Evidently the crystal does something to the light that passes through it on the way from the page to your eye. One beam of light enters the crystal from underneath, but two emerge from the top; by conservation of energy the energy of the original beam must be shared between them. Consider the following three possible interpretations of what you have observed:
(a) The two new beams differ from each other, and from the original beam, only in energy. Their other properties are the same.
(b) The crystal adds to the light some mysterious new property (not energy), which comes in two flavors, X and Y. Ordinary light doesn't have any of either. One beam that emerges from the crystal has some X added to it, and the other beam has Y.
(c) There is some mysterious new property that is possessed by all light. It comes in two flavors, X and Y, and most ordinary light sources make an equal mixture of type X and type Y light. The original beam is an even mixture of both types, and this mixture is then split up by the crystal into the two purified forms.
In parts 2 and 3 you'll make observations that will allow you to figure out which of these is correct.
2. Now place a polaroid film over the crystal and see what you observe. What happens when you rotate the film in the horizontal plane? Does this observation allow you to rule out any of the three interpretations?
3. Now put the polaroid film under the crystal and try the same thing. Putting together all your observations, which interpretation do you think is correct?
4. Look at an overhead light fixture through the polaroid, and try rotating it. What do you observe? What does this tell you about the light emitted by the lightbulb?
5. Now position yourself with your head under a light fixture and directly over a shiny surface, such as a glossy tabletop. You'll see the lamp's reflection, and the light coming from the lamp to your eye will have undergone a reflection through roughly a 180-degree angle (i.e. it very nearly reversed its direction). Observe this reflection through the polaroid, and try rotating it. Finally, position yourself so that you are seeing glancing reflections, and try the same thing. Summarize what happens to light with properties X and Y when it is reflected. (This is the principle behind polarizing sunglasses.) \end{handson}
