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a / This sunspot is a product of the sun's magnetic fields. The darkest region in the center is about the size of our planet.

# Chapter 6. Fields

## 6.1 Farewell To the Mechanical Universe

a / A bar magnet's atoms are (partially) aligned.

b / A bar magnet interacts with our magnetic planet.

c / Magnets aligned north-south.

d / The second magnet is reversed.

e / Both magnets are reversed.

f / The wind patterns in a certain area of the ocean could be charted in a “sea of arrows” representation like this. Each arrow represents both the wind's strength and its direction at a certain location.

g / The gravitational field surrounding a clump of mass such as the earth.

As late as 1900, physicists generally conceived of the universe in mechanical terms. Newton had revealed the solar system as a collection of material objects interacting through forces that acted at a distance. By 1900, evidence began to accumulate for the existence of atoms as real things, and not just as imaginary models of reality. In this microscopic realm, the same (successful) Newtonian picture tended to be transferred over to the microscopic world. Now the actors on the stage were atoms rather than planets, and the forces were electrical rather than gravitational, but it seemed to be a variation on the same theme. Some physicists, however, began to realize that the old mechanical picture wouldn't quite work. At a deeper level, the operation of the universe came to be understood in terms of fields, the general idea being embodied fairly well in “The Force” from the Star Wars movies: “... an energy field created by all living things. It surrounds us, penetrates us, and binds the galaxy together.” Substitute “massive” for “living,” and you have a fairly good description of the gravitational field, which I first casually mentioned on page 20. Substitute “charged” instead, and it's a depiction of the electric field.

### Time delays in forces exerted at a distance

What convinced physicists that they needed this new concept of a field of force? Although we have been dealing mostly with electrical forces, let's start with a magnetic example. (In fact the main reason I've delayed a detailed discussion of magnetism for so long is that mathematical calculations of magnetic effects are handled much more easily with the concept of a field of force.) First a little background leading up to our example. A bar magnet, a, has an axis about which many of the electrons' orbits are oriented. The earth itself is also a magnet, although not a bar-shaped one. The interaction between the earth-magnet and the bar magnet, b, makes them want to line up their axes in opposing directions (in other words such that their electrons rotate in parallel planes, but with one set rotating clockwise and the other counterclockwise as seen looking along the axes). On a smaller scale, any two bar magnets placed near each other will try to align themselves head-to-tail, c.

Now we get to the relevant example. It is clear that two people separated by a paper-thin wall could use a pair of bar magnets to signal to each other. Each person would feel her own magnet trying to twist around in response to any rotation performed by the other person's magnet. The practical range of communication would be very short for this setup, but a sensitive electrical apparatus could pick up magnetic signals from much farther away. In fact, this is not so different from what a radio does: the electrons racing up and down the transmitting antenna create forces on the electrons in the distant receiving antenna. (Both magnetic and electric forces are involved in real radio signals, but we don't need to worry about that yet.)

A question now naturally arises as to whether there is any time delay in this kind of communication via magnetic (and electric) forces. Newton would have thought not, since he conceived of physics in terms of instantaneous action at a distance. We now know, however, that there is such a time delay. If you make a long-distance phone call that is routed through a communications satellite, you should easily be able to detect a delay of about half a second over the signal's round trip of 50,000 miles. Modern measurements have shown that electric, magnetic, and gravitational forces all travel at the speed of light, $$3\times10^8$$ m/s.1 (In fact, we will soon discuss how light itself is made of electricity and magnetism.)

If it takes some time for forces to be transmitted through space, then apparently there is some thing that travels through space. The fact that the phenomenon travels outward at the same speed in all directions strongly evokes wave metaphors such as ripples on a pond.

### More evidence that fields of force are real: they carry energy.

The smoking-gun argument for this strange notion of traveling force ripples comes from the fact that they carry energy.

First suppose that the person holding the bar magnet on the right decides to reverse hers, resulting in configuration d. She had to do mechanical work to twist it, and if she releases the magnet, energy will be released as it flips back to c. She has apparently stored energy by going from c to d. So far everything is easily explained without the concept of a field of force.

But now imagine that the two people start in position c and then simultaneously flip their magnets extremely quickly to position e, keeping them lined up with each other the whole time. Imagine, for the sake of argument, that they can do this so quickly that each magnet is reversed while the force signal from the other is still in transit. (For a more realistic example, we'd have to have two radio antennas, not two magnets, but the magnets are easier to visualize.) During the flipping, each magnet is still feeling the forces arising from the way the other magnet used to be oriented. Even though the two magnets stay aligned during the flip, the time delay causes each person to feel resistance as she twists her magnet around. How can this be? Both of them are apparently doing mechanical work, so they must be storing magnetic energy somehow. But in the traditional Newtonian conception of matter interacting via instantaneous forces at a distance, interaction energy arises from the relative positions of objects that are interacting via forces. If the magnets never changed their orientations relative to each other, how can any magnetic energy have been stored?

The only possible answer is that the energy must have gone into the magnetic force ripples crisscrossing the space between the magnets. Fields of force apparently carry energy across space, which is strong evidence that they are real things.

This is perhaps not as radical an idea to us as it was to our ancestors. We are used to the idea that a radio transmitting antenna consumes a great deal of power, and somehow spews it out into the universe. A person working around such an antenna needs to be careful not to get too close to it, since all that energy can easily cook flesh (a painful phenomenon known as an “RF burn”).

### The gravitational field

Given that fields of force are real, how do we define, measure, and calculate them? A fruitful metaphor will be the wind patterns experienced by a sailing ship. Wherever the ship goes, it will feel a certain amount of force from the wind, and that force will be in a certain direction. The weather is ever-changing, of course, but for now let's just imagine steady wind patterns. Definitions in physics are operational, i.e., they describe how to measure the thing being defined. The ship's captain can measure the wind's “field of force” by going to the location of interest and determining both the direction of the wind and the strength with which it is blowing. Charting all these measurements on a map leads to a depiction of the field of wind force like the one shown in the figure. This is known as the “sea of arrows” method of visualizing a field.

Now let's see how these concepts are applied to the fundamental force fields of the universe. We'll start with the gravitational field, which is the easiest to understand. We've already encountered the gravitational field, $$g$$, which we defined in terms of energy. Essentially, $$g$$ was defined as the number that would make the equation $$GE=mgh$$ give the right answer. However, we intuitively feel that the gravitational field has a direction associated with it: down! This can be more easily expressed via the following definition:

##### definition of the gravitational field

The gravitational field, $$g$$, at any location in space is found by placing a test mass $$m$$ at that point. The field is then given by $$g=F/m$$, where $$F$$ is the gravitational force on the test mass.

With this new definition, we get units of N/kg, rather then J/kg/m. These are in fact equivalent units.

The most subtle point about all this is that the gravitational field tells us about what forces would be exerted on a test mass by the earth, sun, moon, and the rest of the universe, if we inserted a test mass at the point in question. The field still exists at all the places where we didn't measure it.

### Sources and sinks

If we make a sea-of-arrows picture of the gravitational fields surrounding the earth, g, the result is evocative of water going down a drain. For this reason, anything that creates an inward-pointing field around itself is called a sink. The earth is a gravitational sink. The term “source” can refer specifically to things that make outward fields, or it can be used as a more general term for both “outies” and “innies.” However confusing the terminology, we know that gravitational fields are only attractive, so we will never find a region of space with an outward-pointing field pattern.

Knowledge of the field is interchangeable with knowledge of its sources (at least in the case of a static, unchanging field). If aliens saw the earth's gravitational field pattern they could immediately infer the existence of the planet, and conversely if they knew the mass of the earth they could predict its influence on the surrounding gravitational field.

### The electric field

The definition of the electric field is directly analogous to, and has the same motivation as, the definition of the gravitational field:

##### definition of the electric field

The electric field, $$E$$, at any location in space is found by placing a test charge $$q$$ at that point. The electric field vector is then given by $$E=F/q$$, where $$F$$ is the electric force on the test charge.

Charges are what create electric fields. Unlike gravity, which is always attractive, electricity displays both attraction and repulsion. A positive charge is a source of electric fields, and a negative one is a sink.

## 6.2 Electromagnetism

h / 1. When the circuit is incomplete, no current flows through the wire, and the magnet is unaffected. It points in the direction of the Earth's magnetic field. 2. The circuit is completed, and current flows through the wire. The wire has a strong effect on the magnet, which turns almost perpendicular to it. If the earth's field could be removed entirely, the compass would point exactly perpendicular to the wire; this is the direction of the wire's field.

i / A schematic representation of an unmagnetized material, 1, and a magnetized one, 2.

j / Magnetism is an interaction between moving charges and moving charges. The moving charges in the wire attract the moving charges in the beam of charged particles in the vacuum tube.

k / One observer sees an electric field, while the other sees both an electric field and a magnetic one.

l / A model of a charged particle and a current-carrying wire, seen in two different frames of reference. The relativistic length contraction is highly exaggerated. The force on the lone particle is purely magnetic in 1, and purely electric in 2.

m / Magnetic interactions involving only two particles at a time. In these figures, unlike figure l/1, there are electrical forces as well as magnetic ones. The electrical forces are not shown here. Don't memorize these rules!

n / Example 1

o / The magnetic field curls around the wire in circles. At each point in space, the magnetic compass shows the direction of the field.

r / A beam of electrons circles around the magnetic field arrows.

Think not that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfill. -- Matthew 5:17

### Magnetic interactions

At this stage, you understand roughly as much about the classification of interactions as physicists understood around the year 1800. There appear to be three fundamentally different types of interactions: gravitational, electrical, and magnetic. Many types of interactions that appear superficially to be distinct --- stickiness, chemical interactions, the energy an archer stores in a bow --- are really the same: they're manifestations of electrical interactions between atoms. Is there any way to shorten the list any further? The prospects seem dim at first. For instance, we find that if we rub a piece of fur on a rubber rod, the fur does not attract or repel a magnet. The fur has an electric field, and the magnet has a magnetic field. The two are completely separate, and don't seem to affect one another. Likewise we can test whether magnetizing a piece of iron changes its weight. The weight doesn't seem to change by any measurable amount, so magnetism and gravity seem to be unrelated.

That was where things stood until 1820, when the Danish physicist Hans Christian Oersted was delivering a lecture at the University of Copenhagen, and he wanted to give his students a demonstration that would illustrate the cutting edge of research. He generated a current in a wire by making a short circuit across a battery, and held the wire near a magnetic compass. The ideas was to give an example of how one could search for a previously undiscovered link between electricity (the electric current in the wire) and magnetism. One never knows how much to believe from these dramatic legends, but the story is2 that the experiment he'd expected to turn out negative instead turned out positive: when he held the wire near the compass, the current in the wire caused the compass to twist!

People had tried similar experiments before, but only with static electricity, not with a moving electric current. For instance, they had hung batteries so that they were free to rotate in the earth's magnetic field, and found no effect; since the battery was not connected to a complete circuit, there was no current flowing. With Oersted's own setup, h, the effect was only produced when the “circuit was closed, but not when open, as certain very celebrated physicists in vain attempted several years ago.”3

Oersted was eventually led to the conclusion that magnetism was an interaction between moving charges and other moving charges, i.e., between one current and another. A permanent magnet, he inferred, contained currents on a microscopic scale that simply weren't practical to measure with an ammeter. Today this seems natural to us, since we're accustomed to picturing an atom as a tiny solar system, with the electrons whizzing around the nucleus in circles. As shown in figure i, a magnetized piece of iron is different from an unmagnetized piece because the atoms in the unmagnetized piece are jumbled in random orientations, whereas the atoms in the magnetized piece are at least partially organized to face in a certain direction.

Figure j shows an example that is conceptually simple, but not very practical. If you try this with a typical vacuum tube, like a TV or computer monitor, the current in the wire probably won't be enough to produce a visible effect. A more practical method is to hold a magnet near the screen. We still have an interaction between moving charges and moving charges, but the swirling electrons in the atoms in the magnet are now playing the role played by the moving charges in the wire in figure j. Warning: if you do this, make sure your monitor has a demagnetizing button! If not, then your monitor may be permanently ruined.

### Relativity requires magnetism

So magnetism is an interaction between moving charges and moving charges. But how can that be? Relativity tells us that motion is a matter of opinion. Consider figure k. In this figure and in figure l, the dark and light coloring of the particles represents the fact that one particle has one type of charge and the other particle has the other type. Observer k/2 sees the two particles as flying through space side by side, so they would interact both electrically (simply because they're charged) and magnetically (because they're charges in motion). But an observer moving along with them, k/1, would say they were both at rest, and would expect only an electrical interaction. This seems like a paradox. Magnetism, however, comes not to destroy relativity but to fulfill it. Magnetic interactions must exist according to the theory of relativity. To understand how this can be, consider how time and space behave in relativity. Observers in different frames of reference disagree about the lengths of measuring sticks and the speeds of clocks, but the laws of physics are valid and self-consistent in either frame of reference. Similarly, observers in different frames of reference disagree about what electric and magnetic fields there are, but they agree about concrete physical events. An observer in frame of reference k/1 says there are electric fields around the particles, and predicts that as time goes on, the particles will begin to accelerate towards one another, eventually colliding. She explains the collision as being due to the electrical attraction between the particles. A different observer, k/2, says the particles are moving. This observer also predicts that the particles will collide, but explains their motion in terms of both an electric field and a magnetic field. As we'll see shortly, the magnetic field is required in order to maintain consistency between the predictions made in the two frames of reference.

To see how this really works out, we need to find a nice simple example. An example like figure k is not easy to handle, because in the second frame of reference, the moving charges create fields that change over time at any given location, like when the V-shaped wake of a speedboat washes over a buoy. Examples like figure j are easier, because there is a steady flow of charges, and all the fields stay the same over time. Figure l/1 shows a simplified and idealized model of figure j. The charge by itself is like one of the charged particles in the vacuum tube beam of figure j, and instead of the wire, we have two long lines of charges moving in opposite directions. Note that, as discussed in discussion question C on page 100, the currents of the two lines of charges do not cancel out. The dark balls represent particles with one type of charge, and the light balls have the other type. Because of this, the total current in the “wire” is double what it would be if we took away one line.

As a model of figure j, figure l/1 is partly realistic and partly unrealistic. In a real piece of copper wire, there are indeed charged particles of both types, but it turns out that the particles of one type (the protons) are locked in place, while only some of the other type (the electrons) are free to move. The model also shows the particles moving in a simple and orderly way, like cars on a two-lane road, whereas in reality most of the particles are organized into copper atoms, and there is also a great deal of random thermal motion. The model's unrealistic features aren't a problem, because the point of this exercise is only to find one particular situation that shows magnetic effects must exist based on relativity.

What electrical force does the lone particle in figure l/1 feel? Since the density of “traffic” on the two sides of the “road” is equal, there is zero overall electrical force on the lone particle. Each “car” that attracts the lone particle is paired with a partner on the other side of the road that repels it. If we didn't know about magnetism, we'd think this was the whole story: the lone particle feels no force at all from the wire.

Figure l/2 shows what we'd see if we were observing all this from a frame of reference moving along with the lone charge. Here's where the relativity comes in. Relativity tells us that moving objects appear contracted to an observer who is not moving along with them. Both lines of charge are in motion in both frames of reference, but in frame 1 they were moving at equal speeds, so their contractions were equal. In frame 2, however, their speeds are unequal. The dark charges are moving more slowly than in frame 1, so in frame 2 they are less contracted. The light-colored charges are moving more quickly, so their contraction is greater now. The “cars” on the two sides of the “road” are no longer paired off, so the electrical forces on the lone particle no longer cancel out as they did in l/1. The lone particle is attracted to the wire, because the particles attracting it are more dense than the ones repelling it. Furthermore, the attraction felt by the lone charge must be purely electrical, since the lone charge is at rest in this frame of reference, and magnetic effects occur only between moving charges and other moving charges.

Now observers in frames 1 and 2 disagree about many things, but they do agree on concrete events. Observer 2 is going to see the lone particle drift toward the wire due to the wire's electrical attraction, gradually speeding up, and eventually hit the wire. If 2 sees this collision, then 1 must as well. But 1 knows that the total electrical force on the lone particle is exactly zero. There must be some new type of force. She invents a name for this new type of force: magnetism. This was a particularly simple example, because the force was purely magnetic in one frame of reference, and purely electrical in another. In general, an observer in a certain frame of reference will measure a mixture of electric and magnetic fields, while an observer in another frame, in motion with respect to the first, says that the same volume of space contains a different mixture.

We therefore arrive at the conclusion that electric and magnetic phenomena aren't separate. They're different sides of the same coin. We refer to electric and magnetic interactions collectively as electromagnetic interactions. Our list of the fundamental interactions of nature now has two items on it instead of three: gravity and electromagnetism.

The basic rules for magnetic attractions and repulsions, shown in figure m, aren't quite as simple as the ones for gravity and electricity. Rules m/1 and m/2 follow directly from our previous analysis of figure l. Rules 3 and 4 are obtained by flipping the type of charge that the bottom particle has. For instance, rule 3 is like rule 1, except that the bottom charge is now the opposite type. This turns the attraction into a repulsion. (We know that flipping the charge reverses the interaction, because that's the way it works for electric forces, and magnetic forces are just electric forces viewed in a different frame of reference.)

##### Example 1: A magnetic weathervane placed near a current.
Figure n shows a magnetic weathervane, consisting of two charges that spin in circles around the axis of the arrow. (The magnetic field doesn't cause them to spin; a motor is needed to get them to spin in the first place.) Just like the magnetic compass in figure h, the weathervane's arrow tends to align itself in the direction perpendicular to the wire. This is its preferred orientation because the charge close to the wire is attracted to the wire, while the charge far from the wire is repelled by it.

### Magnetic fields

How should we define the magnetic field? When two objects attract each other gravitationally, their gravitational energy depends only on the distance between them, and it seems intuitively reasonable that we define the gravitational field arrows like a street sign that says “this way to lower gravitational energy.” The same idea works fine for the electric field. But what if two charged particles are interacting magnetically? Their interaction doesn't just depend on the distance, but also on their motions.

We need some way to pick out some direction in space, so we can say, “this is the direction of the magnetic field around here.” A natural and simple method is to define the magnetic field's direction according to the direction a compass points. Starting from this definition we can, for example, do experiments to show that the magnetic field of a current-carrying wire forms a circular pattern, o.

But is this the right definition? Unlike the definitions of the gravitational and electric fields' directions, it involves a particular human-constructed tool. However, compare figure h on page 111 with figure n on page 115. Note that both of these tools line themselves up along a line that's perpendicular to the wire. In fact, no matter how hard you try, you will never be able to invent any other electromagnetic device that will align itself with any other line. All you can do is make one that points in exactly the opposite direction, but along the same line. For instance, you could use paint to reverse the colors that label the ends of the magnetic compass needle, or you could build a weathervane just like figure n, but spinning like a left-handed screw instead of a right-handed one. The weathervane and the compass aren't even as different as they appear. Figure p shows their hidden similarities.

p / 1. The needle of a magnetic compass is nothing more than a bar magnet that is free to rotate in response to the earth's magnetic field. 2. A cartoon of the bar magnet's structure at the atomic level. Each atom is very much like the weathervane of figure n.

Nature is trying to tell us something: there really is something special about the direction the compass points. Defining the direction of the magnetic field in terms of this particular device isn't as arbitrary as it seems. The only arbitrariness is that we could have built up a whole self-consistent set of definitions that started by defining the magnetic field as being in the opposite direction.

##### Example 2: Head-to-tail alignment of bar magnets
$$\triangleright$$ If you let two bar magnets like the one in figure p interact, which way do they want to line up, head-to-head or head-to-tail?

$$\triangleright$$ Each bar magnet contains a huge number of atoms, but that won't matter for our result; we can imagine this as an interaction between two individual atoms. For that matter, let's model the atoms as weathervanes like the one in figure n. Suppose we put two such weather vanes side by side, with their arrows both pointing away from us. From our point of view, they're both spinning clockwise. As one of the charges in the left-hand weather vane comes down on the right side, one of the charges in the right-hand vane comes up on the left side. These two charges are close together, so their magnetic interaction is very strong at this moment. Their interaction is repulsive, so this is an unstable arrangement of the two weathervanes.

On the other hand, suppose the left-hand weathervane is pointing away from is, while its partner on the right is pointing toward us. From our point of view, we see the one on the right spinning counterclockwise. At the moment when their charges come as close as possible, they're both on the way up. Their interaction is attractive, so this is a stable arrangement.

Translating back from our model to the original question about bar magnets, we find that bar magnets will tend to align themselves head-to-tail. This is easily verified by experiment.

q / The force on a charged particle moving through a magnetic field is perpendicular to both the field and its direction of motion. The relationship is right-handed for one type of charge, and left-handed for the other type.

If you go back and apply this definition to all the examples we've encountered so far, you'll find that there's a general rule: the force on a charged particle moving through a magnetic field is perpendicular to both the field and its direction of motion. A force perpendicular to the direction of motion is exactly what is required for circular motion, so we find that a charged particle in a vacuum will go in a circle around the magnetic field arrows (or perhaps a corkscrew pattern, if it also has some motion along the direction of the field). That means that magnetic fields tend to trap charged particles.

Figure r shows this principle in action. A beam of electrons is created in a vacuum tube, in which a small amount of hydrogen gas has been left. A few of the electrons strike hydrogen molecules, creating light and letting us see the path of the beam. A magnetic field is produced by passing a current (meter) through the circular coils of wire in front of and behind the tube. In the bottom figure, with the magnetic field turned on, the force perpendicular to the electrons' direction of motion causes them to move in a circle.

##### Example 3: Sunspots

Sunspots, like the one shown in the photo on page 107, are places where the sun's magnetic field is unusually strong. Charged particles are trapped there for months at a time. This is enough time for the sunspot to cool down significantly, and it doesn't get heated back up because the hotter surrounding material is kept out by the same magnetic forces.

##### Example 4: The aurora and life on earth's surface

A strong magnetic field seems to be one of the prerequisites for the existence of life on the surface of a planet. Energetic charged particles from the sun are trapped by our planet's magnetic field, and harmlessly spiral down to the earth's surface at the poles. In addition to protecting us, this creates the aurora, or “northern lights.”

The astronauts who went to the moon were outside of the earth's protective field for about a week, and suffered significant doses of radiation during that time. The problem would be much more serious for astronauts on a voyage to Mars, which would take at least a couple of years. They would be subjected to intense radiation while in interplanetary space, and also while on Mars's surface, since Mars lacks a strong magnetic field.

Features in one Martian rock have been interpreted by some scientists as fossilized bacteria. If single-celled life evolved on Mars, it has presumably been forced to stay below the surface. (Life on Earth probably evolved deep in the oceans, and most of the Earth's biomass consists of single-celled organisms in the oceans and deep underground.)

## 6.3 Induction

s / Faraday on a British banknote.

t / Faraday's experiment, simplified and shown with modern equipment.

u / The geometry of induced fields. The induced field tends to form a whirlpool pattern around the change in the field producing it. The notation $$\Delta$$ (Greek letter delta) stands for “change in.” Note how the induced fields circulate in opposite directions.

v / A generator.

w / A transformer.

x / Observer A sees a positively charged particle moves through a region of upward magnetic field, which we assume to be uniform, between the poles of two magnets. The resulting force along the $$z$$ axis causes the particle's path to curve toward us.

y / James Clerk Maxwell (1831-1879)

We've already seen that the electric and magnetic fields are closely related, since what one observer sees as one type of field, another observer in a different frame of reference sees as a mixture of both. The relationship goes even deeper than that, however. Figure t shows an example that doesn't even involve two different frames of reference. This phenomenon of induced electric fields --- fields that are not due to charges --- was a purely experimental accomplishment by Michael Faraday (1791-1867), the son of a blacksmith who had to struggle against the rigid class structure of 19th century England. Faraday, working in 1831, had only a vague and general idea that electricity and magnetism were related to each other, based on Oersted's demonstration, a decade before, that magnetic fields were caused by electric currents.

Figure t is a simplified drawing of the experiment, as described in Faraday's original paper: “Two hundred and three feet of copper wire ... were passed round a large block of wood; [another] two hundred and three feet of similar wire were interposed as a spiral between the turns of the first, and metallic contact everywhere prevented by twine [insulation]. One of these [coils] was connected with a galvanometer [voltmeter], and the other with a battery... When the contact was made, there was a sudden and very slight effect at the galvanometer, and there was also a similar slight effect when the contact with the battery was broken. But whilst the ... current was continuing to pass through the one [coil], no ... effect ... upon the other [coil] could be perceived, although the active power of the battery was proved to be great, by its heating the whole of its own coil [through ordinary resistive heating] ...”

From Faraday's notes and publications, it appears that the situation in figure t/3 was a surprise to him, and he probably thought it would be a surprise to his readers, as well. That's why he offered evidence that the current was still flowing: to show that the battery hadn't just died. The induction effect occurred during the short time it took for the black coil's magnetic field to be established, t/2. Even more counterintuitively, we get an effect, equally strong but in the opposite direction, when the circuit is broken, t/4. The effect occurs only when the magnetic field is changing: either ramping up or ramping down.

What are we really measuring here with the voltmeter? A voltmeter is nothing more than a resistor with an attachment for measuring the current through it. A current will not flow through a resistor unless there is some electric field pushing the electrons, so we conclude that the changing magnetic field has produced an electric field in the surrounding space. Since the white wire is not a perfect conductor, there must be electric fields in it as well. The remarkable thing about the circuit formed by the white wire is that as the electrons travel around and around, they are always being pushed forward by electric fields. That is, the electric field seems to form a curly pattern, like a whirlpool.

What Faraday observed was an example of the following principle:

##### the principle of induction

Any magnetic field that changes over time will create an electric field. The induced electric field is perpendicular to the magnetic field, and forms a curly pattern around it.
Any electric field that changes over time will create a magnetic field. The induced magnetic field is perpendicular to the electric field, and forms a curly pattern around it.

The first part was the one Faraday had seen in his experiment. The geometrical relationships are illustrated in figure u. In Faraday's setup, the magnetic field was pointing along the axis of the coil of wire, so the induced electric field made a curly pattern that circled around the circumference of the block.

##### Example 5: The generator

A basic generator, v, 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. This changing magnetic field results in an electric field, which has a curly 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.

If the magnet was on a frictionless bearing, could we light the bulb for free indefinitely, thus violating conservation of energy? No. It's hard work to crank the magnet, and that's where the energy comes from. If we break the light-bulb circuit, it suddenly gets easier to crank the magnet! This is because the current in the coil sets up its own magnetic field, and that field exerts a torque on the magnet. If we stopped cranking, this torque would quickly make the magnet stop turning.

self-check:

When you're driving your car, the engine recharges the battery continuously using a device called an alternator, which is really just a generator. Why can't you use the alternator to start the engine if your car's battery is dead?

##### Example 6: The transformer

It's more efficient for the electric company to transmit 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, w, to convert everything to lower voltages and higher currents inside your house. The coil on the input side creates a magnetic field. Transformers work with alternating current (currents that reverses its direction many times a second), so the magnetic field surrounding the input coil is always changing. This induces an electric field, which drives a current around the output coil.

Since the electric field is curly, an electron can keep gaining more and more energy by circling through it again and again. Thus the output voltage can be controlled by changing the number of turns of wire on the output side. In any case, conservation of energy guarantees that the amount of power on the output side must equal the amount put in originally,

$\begin{equation*} (\text{input current})\times(\text{input voltage}) = (\text{output current})\times(\text{output voltage}) \end{equation*}$

so no matter what factor the voltage is reduced by, the current is increased by the same factor. This is analogous to a lever. A crowbar allows you to lift a heavy boulder, but to move the boulder a centimeter, you may have to move your end of the lever a meter. The advantage in force comes with a disadvantage in distance. It's as though you were allowed to lift a small weight through a large height rather than a large weight through a small height. Either way, the energy you expend is the same.

##### Example 7: Fun with sparks

Unplug a lamp while it's turned on, and watch the area around the wall outlet. You should see a blue spark in the air at the moment when the prongs of the plug lose contact with the electrical contacts inside the socket.

This is evidence that, as discussed on page 109, fields contain energy. Somewhere on your street is a transformer, one side of which is connected to the lamp's circuit. When the lamp is plugged in and turned on, there's a complete circuit, and current flows. As current flows through the coils in the transformer, a magnetic field is formed --- remember, any time there's moving charge, there will be magnetic fields. Because there is a large number turns in the coils, these fields are fairly strong, and store quite a bit of energy.

When you pull the plug, the circuit is no longer complete, and the current stops. Once the current has disappeared, there's no more magnetic field, which means that some energy has disappeared. Conservation of energy tells us that if a certain amount of energy disappears, an equal amount must reappear somewhere else. That energy goes into making the spark. (Once the spark is gone, its energy remains in the form of heat in the air.)

We now have two connections between electric and magnetic fields. One is the principle of induction, and the other is the idea that according to relativity, observers in different frames of reference must perceive different mixtures of magnetic and electric fields. At the time Faraday was working, relativity was still 70 years in the future, so the relativistic concepts weren't available --- to him, his observations were just surprising empirical facts. But in fact, the relativistic idea about frames of reference has a logical connection to the idea of induction.

Figure x is a nice example that can be interpreted either way. Observer A is at rest with respect to the bar magnets, and sees the particle swerving off in the $$z$$ direction, as it should according to the right-hand rule. Suppose observer B, on the other hand, is moving to the right along the $$x$$ axis, initially at the same speed as the particle. B sees the bar magnets moving to the left and the particle initially at rest but then accelerating along the $$z$$ axis in a straight line. It is not possible for a magnetic field to start a particle moving if it is initially at rest, since magnetism is an interaction of moving charges with moving charges. B is thus led to the inescapable conclusion that there is an electric field in this region of space, which points along the $$z$$ axis. In other words, what A perceives as a pure magnetic field, B sees as a mixture of electric and magnetic fields. This is what we expect based on the relativistic arguments, but it's also what's required by the principle of induction. In B's frame of reference, there's initially no magnetic field, but then a couple of bar magnets come barging in and create one. This is a change in the magnetic field, so the principle of induction predicts that there must be an electric field as well.

### Electromagnetic waves

Theorist James Clerk Maxwell was the first to work out the principle of induction (including the detailed numerical and geometric relationships, which we won't go into here). Legend has it that it was on a starry night that he first realized the most important implication of his equations: light itself is an electromagnetic wave, a ripple spreading outward from a disturbance in the electric and magnetic fields. He went for a walk with his wife, and told her she was the only other person in the world who really knew what starlight was.

The principle of induction tells us that there can be no such thing as a purely electric or purely magnetic wave. As an electric wave washes over you, you feel an electric field that changes over time. By the principle of induction, there must also be a magnetic field accompanying it. It works the other way, too. It may seem a little spooky that the electric field causes the magnetic field while the magnetic field causes the electric field, but the waves themselves don't seem to worry about it.

The distance from one ripple to the next is called the wavelength of the light. Light with a certain wavelength (about quarter a millionth of a meter) is at the violet end of the rainbow spectrum, while light with a somewhat longer wavelength (about twice as long) is red. Figure z/1 shows the complete spectrum of light waves. Maxwell's equations predict that all light waves have the same structure, regardless of wavelength and frequency, so even though radio and x-rays, for example, hadn't been discovered, Maxwell predicted that such waves would have to exist. Maxwell's 1865 prediction passed an important test in 1888, when Heinrich Hertz published the results of experiments in which he showed that radio waves could be manipulated in the same ways as visible light waves. Hertz showed, for example, that radio waves could be reflected from a flat surface, and that the directions of the reflected and incoming waves were related in the same way as with light waves, forming equal angles with the normal. Likewise, light waves can be focused with a curved, dish-shaped mirror, and Hertz demonstrated the same thing with a dish-shaped radio antenna.

z / Panel 1 shows the electromagnetic spectrum. Panel 2 shows how an electromagnetic wave is put together. Imagine that this is a radio wave, with a wavelength of a few meters. If you were standing inside the wave as it passed through you, you could theoretically hold a compass in your hand, and it would wiggle back and forth as the magnetic field pattern (white arrows) washed over you. (The vibration would actually be much to rapid to detect this way.) Similarly, you'd experience an electric field alternating between up and down. Panel 3 shows how this relates to the principle of induction. The changing electric field (black arrows) should create a curly magnetic field (white). Is it really curly? Yes, because if we inserted a paddlewheel that responded to electric fields, the field would make the paddlewheel spin counterclockwise as seen from above. Similarly, the changing magnetic field (white) makes an electric field (black) that curls in the clockwise direction as seen from the front.

## Homework Problems

aa / Stationary wave patterns on a clothesline (problem 6).

1. Albert Einstein wrote, “What really interests me is whether God had any choice in the creation of the world.” What he meant by this is that if you randomly try to imagine a set of rules --- the laws of physics --- by which the universe works, you'll almost certainly come up with rules that don't make sense. For instance, we've seen that if you tried to omit magnetism from the laws of physics, electrical interactions wouldn't make sense as seen by observers in different frames of reference; magnetism is required by relativity.

The magnetic interaction rules in figure m are consistent with the time-reversal symmetry of the laws of physics. In other words, the rules still work correctly if you reverse the particles' directions of motion. Now you get to play God (and fail). Suppose you're going to make an alternative version of the laws of physics by reversing the direction of motion of only one of the eight particles. You have eight choices, and each of these eight choices would result in a new set of physical laws. We can imagine eight alternate universes, each governed by one of these eight sets. Prove that all of these modified sets of physical laws are impossible, either because the are self-contradictory, or because they violate time-reversal symmetry.

2. The purpose of this problem is to show that the magnetic interaction rules shown in figure m can be simplified by stating them in terms of current. Recall that, as discussed in discussion question C on page 100, one type of charge moving in a particular direction produces the same current as the other type of charge moving in the opposite direction. Let's say arbitrarily that the current made by the dark type of charged particle is in the direction it's moving, while a light-colored particle produces a current in the direction opposite to its motion. Redraw all four panels of figure m, replacing each picture of a moving light or dark particle with an arrow showing the direction of the current it makes. Show that the rules for attraction and repulsion can now be made much simpler, and state the simplified rules explicitly.

3. Physicist Richard Feynman originated a new way of thinking about charge: a charge of a certain type is equivalent to a charge of the opposite type that happens to be moving backward in time! An electron moving backward in time is an antielectron --- a particle that has the same mass as an electron, but whose charge is opposite. Likewise we have antiprotons, and antimatter made from antiprotons and antielectrons. Antielectrons occur naturally everywhere around you due to natural radioactive decay and radiation from outer space. A small number of antihydrogen atoms has even been created in particle accelerators!

Show that, for each rule for magnetic interactions shown in m, the rule is still valid if you replace one of the charges with an opposite charge moving in the opposite direction (i.e., backward in time).

4. Refer to figure r on page 117. Electrons have the type of charge I've been representing with light-colored spheres.
(a) As the electrons in the beam pass over the top of the circle, what is the direction of the force on them? Use what you know about circular motion.
(b) From this information, use figure q on page 117 to determine the direction of the magnetic field (left, right, up, down, into the page, or out of the page).

5. You can't use a light wave to see things that are smaller than the wavelength of the light.
(a) Referring to figure z on page 123, what color of light do you think would be the best to use for microscopy?
(b) The size of an atom is about $$10^{-10}$$ meters. Can visible light be used to make images of individual atoms?

6. You know how a microwave gets some parts of your food hot, but leaves other parts cold? Suppose someone is trying to convince you of the following explanation for this fact: The microwaves inside the oven form a stationary wave pattern, like the vibrations of a clothesline or a guitar string. The food is heated unevenly because the wave crests are a certain distance apart, and the parts of the food that get heated the most are the ones where there's a crest in the wave pattern. Use the wavelength scale in figure z on page 123 as a way of checking numerically whether this is a reasonable explanation.

7. This book begins and ends with the topic of light. Give an example of how the correspondence principle applies here, referring to a concrete observation from a lab.

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

##### Footnotes
[1] As discussed in chapter 4, one consequence of Einstein's theory of relativity is that material objects can never move faster than the speed of light. It can also be shown that signals or information are subject to the same limit.
[2] Oersted's paper describing the phenomenon says that “The first experiments on the subject ... were set on foot in the classes for electricity, galvanism, and magnetism, which were held by me in the winter just past,” but that doesn't tell us whether the result was really a surprise that occurred in front of his students.
[3] All quotes are from the 1876 translation by J.E. Kempe.