Contents
Section 11.1 - More About the Magnetic Field
Section 11.2 - Magnetic Fields by Superposition
Section 11.3 - Magnetic Fields by Ampère's Law
Section 11.4 - Ampère's Law in Differential Form (optional)
Section 11.5 - Induced Electric Fields
Section 11.6 - Maxwell's Equations
Section 11.7 - Electromagnetic Properties of Materials

Chapter 11. Electromagnetism

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

11.1 More About the Magnetic Field

Magnetic forces

In this chapter, I assume you know a few basic ideas about Einstein's theory of relativity, as described in section 7.1. Unless your typical workday involves rocket ships or particle accelerators, all this relativity stuff might sound like a description of some bizarre futuristic world that is completely hypothetical. There is, however, a relativistic effect that occurs in everyday life, and it is obvious and dramatic: magnetism. Magnetism, as we discussed previously, is an interaction between a moving charge and another moving charge, as opposed to electric forces, which act between any pair of charges, regardless of their motion. Relativistic effects are weak for speeds that are small compared to the speed of light, and the average speed at which electrons drift through a wire is quite low (centimeters per second, typically), so how can relativity be behind an impressive effect like a car being lifted by an electromagnet hanging from a crane? The key is that matter is almost perfectly electrically neutral, and electric forces therefore cancel out almost perfectly. Magnetic forces really aren't very strong, but electric forces are even weaker.

What about the word “relativity” in the name of the theory? It would seem problematic if moving charges interact differently than stationary charges, since motion is a matter of opinion, depending on your frame of reference. 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 and forces there are, but they agree about concrete physical events. For instance, figure a/1 shows two particles, with opposite charges, which are not moving at a particular moment in time. An observer in this frame of reference 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. A different observer, a/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, E, and a magnetic field, B. 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 that is easy to calculate. An example like figure a 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. Examples like figure b are easier, because there is a steady flow of charges, and all the fields stay the same over time.¹ What is remarkable about this demonstration is that there can be no electric fields acting on the electron beam at all, since the total charge density throughout the wire is zero. Unlike figure a/2, figure b is purely magnetic.

To see why this must occur based on relativity, we make the mathematically idealized model shown in figure c. The charge by itself is like one of the electrons in the vacuum tube beam of figure b, and a pair of moving, infinitely long line charges has been substituted for the wire. The electrons in a real wire are in rapid thermal motion, and the current is created only by a slow drift superimposed on this chaos. A second deviation from reality is that in the real experiment, the protons are at rest with respect to the tabletop, and it is the electrons that are in motion, but in c/1 we have the positive charges moving in one direction and the negative ones moving the other way. If we wanted to, we could construct a third frame of reference in which the positive charges were at rest, which would be more like the frame of reference fixed to the tabletop in the real demonstration. However, as we'll see shortly, frames c/1 and c/2 are designed so that they are particularly easy to analyze. It's important to note that even though the two line charges are moving in opposite directions, their currents don't cancel. A negative charge moving to the left makes a current that goes to the right, so in frame c/1, the total current is twice that contributed by either line charge.

Frame 1 is easy to analyze because the charge densities of the two line charges cancel out, and the electric field experienced by the lone charge is therefore zero:

In frame 1, any force experienced by the lone charge must therefore be attributed solely to magnetism.

Frame 2 shows what we'd see if we were observing all this from a frame of reference moving along with the lone charge. Why don't the charge densities also cancel in this frame? 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 line charges are in motion in both frames of reference, but in frame 1, the line charges were moving at equal speeds, so their contractions were equal, and their charge densities canceled out. In frame 2, however, their speeds are unequal. The positive charges are moving more slowly than in frame 1, so in frame 2 they are less contracted. The negative charges are moving more quickly, so their contraction is greater now. Since the charge densities don't cancel, there is an electric field in frame 2, which points into the wire, attracting the lone charge. 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.²

To summarize, frame 1 displays a purely magnetic attraction, while in frame 2 it is purely electrical.

A common source of confusion in this argument is that it seems as though the excess of negative electric charge must have been stolen from some other part of the wire, leaving that part with a net positive charge. That wouldn't make sense, because there is no physical reason why one part of the wire would behave differently than any other. The flaw in this reasoning has to do with the fact that simultaneity is not well defined in relativity. In frame c/1, suppose that we label all the positive charges with integers, and likewise all the negative ones, so that the positive charge labeled 42 is on top of the negative charge labeled 42, and so on. In this frame of reference, every charge has a partner that cancels it, and the net charge everywhere is zero. If simultaneity were a valid concept in relativity, then not only would 42's pairing with 42, and 43's pairing with 43, occur simultaneously in frame c/1, but these same pairings would occur all at the same time in frame c/2. But observers in different frames of reference do not agree on simultaneity. For simplicity, let's imagine that the Lorentz contractions are such that the spacing between the negative charges in frame c/2 is exactly half as much as the spacing between the positive charges. Then we may have negative charge number 42 paired up with positive charge 21, and negative charge 44 paired with positive charge 22, while negative charge 43 has no partner.

Now we can calculate the force in frame 2, and equating it to the force in frame 1, we can find out how much magnetic force occurs. To keep the math simple, and to keep from assuming too much about your knowledge of relativity, we're going to carry out this whole calculation in the approximation where all the speeds are fairly small compared to the speed of light.³ For instance, if we find an expression such as (v/c)²+(v/c)⁴, we will assume that the fourth-order term is negligible by comparison. This is known as a calculation “to leading order in v/c.” In fact, I've already used the leading-order approximation twice without saying so! The first time I used it implicitly was in figure c, where I assumed that the velocities of the two line charges were u-v and -u-v. Relativistic velocities don't just combine by simple addition and subtraction like this, but this is an effect we can ignore in the present approximation. The second sleight of hand occurred when I stated that we could equate the forces in the two frames of reference. Force, like time and distance, is distorted relativistically when we change from one frame of reference to another. Again, however, this is an effect that we can ignore to the desired level of approximation.

Let ±λ be the charge per unit length of each line charge without relativistic contraction, i.e., in the frame moving with that line charge. Using the approximation γ=(1-v²/c²)^-1/2≈ 1+v²/2c² for v<< c, the total charge per unit length in frame 2 is

Let R be the distance from the line charge to the lone charge. Applying Gauss' law to a cylinder of radius R centered on the line charge, we find that the magnitude of the electric field experienced by the lone charge in frame 2 is

and the force acting on the lone charge q is

In frame 1, the current is I=2λ₁ u (see homework problem 5), which we can approximate as I=2λ u, since the current, unlike λ_{total, 2}, doesn't vanish completely without the relativistic effect. The magnetic force on the lone charge q due to the current I is

The magnetic field

Definition in terms of the force on a moving particle

With electricity, it turned out to be useful to define an electric field rather than always working in terms of electric forces. Likewise, we want to define a magnetic field, B. Let's look at the result of the preceding subsection for insight. The equation

shows that when we put a moving charge near other moving charges, there is an extra magnetic force on it, in addition to any electric forces that may exist. Equations for electric forces always have a factor of k in front --- the Coulomb constant k is called the coupling constant for electric forces. Since magnetic effects are relativistic in origin, they end up having a factor of k/c² instead of just k. In a world where the speed of light was infinite, relativistic effects, including magnetism, would be absent, and the coupling constant for magnetism would be zero. A cute feature of the metric system is that we have k/c²=10^-7 N⋅s²/C² exactly, as a matter of definition.

Naively, we could try to work by analogy with the electric field, and define the magnetic field as the magnetic force per unit charge. However, if we think of the lone charge in our example as the test charge, we'll find that this approach fails, because the force depends not just on the test particle's charge, but on its velocity, v, as well. Although we only carried out calculations for the case where the particle was moving parallel to the wire, in general this velocity is a vector, v, in three dimensions. We can also anticipate that the magnetic field will be a vector. The electric and gravitational fields are vectors, and we expect intuitively based on our experience with magnetic compasses that a magnetic field has a particular direction in space. Furthermore, reversing the current I in our example would have reversed the force, which would only make sense if the magnetic field had a direction in space that could be reversed. Summarizing, we think there must be a magnetic field vector B, and the force on a test particle moving through a magnetic field is proportional both to the B vector and to the particle's own v vector. In other words, the magnetic force vector F is found by some sort of vector multiplication of the vectors v and B. As proved on page 848, however, there is only one physically useful way of defining such a multiplication, which is the cross product.

From this definition, we see that the magnetic field's units are N⋅s/C⋅m, which are usually abbreviated as teslas, 1 T=1 N⋅s/C⋅m. The definition implies a right-hand-rule relationship among the vectors, figure d, if the charge q is positive, and the opposite handedness if it is negative.

This is not just a definition but a bold prediction! Is it really true that for any point in space, we can always find a vector B that successfully predicts the force on any passing particle, regardless of its charge and velocity vector? Yes --- it's not obvious that it can be done, but experiments verify that it can. How? Well for example, the cross product of parallel vectors is zero, so we can try particles moving in various directions, and hunt for the direction that produces zero force; the B vector lies along that line, in either the same direction the particle was moving, or the opposite one. We can then go back to our data from one of the other cases, where the force was nonzero, and use it to choose between these two directions and find the magnitude of the B vector. We could then verify that this vector gave correct force predictions in a variety of other cases.

Even with this empirical reassurance, the meaning of this equation is not intuitively transparent, nor is it practical in most cases to measure a magnetic field this way. For these reasons, let's look at an alternative method of defining the magnetic field which, although not as fundamental or mathematically simple, may be more appealing.

Definition in terms of the torque on a dipole

A compass needle in a magnetic field experiences a torque which tends to align it with the field. This is just like the behavior of an electric dipole in an electric field, so we consider the compass needle to be a magnetic dipole. In subsection 10.1.3 on page 520, we gave an alternative definition of the electric field in terms of the torque on an electric dipole.

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 with each other. 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, f.

Applying F=v×B, we find that such a loop, when placed in a magnetic field, g, experiences a torque that tends to align plane so that its interior “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 using as our alternative definition of the magnetic field.

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

where the direction of the vectors is defined as shown in figure h.

We can now give an alternative definition of the magnetic field:

Let's show that this is consistent with the previous definition, using the geometry shown in figure i. The velocity vector that point in and out of the page are shown using the convention defined in figure j. Let the mobile charge carriers in the wire have linear density λ, and let the sides of the loop have length h, so that we have I=λ v, and m=h²λ v. The only nonvanishing torque comes from the forces on the left and right sides. The currents in these sides are perpendicular to the field, so the magnitude of the cross product F=qv×B is simply |F|=qvB. The torque supplied by each of these forces is r×F, where the lever arm r has length h/2, and makes an angle θ with respect to the force vector. The magnitude of the total torque acting on the loop is therefore

which is consistent with the second definition of the field.

It undoubtedly seems artificial to you that we have discussed dipoles only in the form of a square loop of current. A permanent magnet, for example, is made out of atomic dipoles, and atoms aren't square! However, it turns out that the shape doesn't matter. To see why this is so, consider the additive property of areas and dipole moments, shown in figure k. Each of the square dipoles has a dipole moment that points out of the page. When they are placed side by side, the currents in the adjoining sides cancel out, so they are equivalent to a single rectangular loop with twice the area. We can break down any irregular shape into little squares, as shown in figure l, so the dipole moment of any planar current loop can be calculated based on its area, regardless of its shape.

Discussion Questions

Some applications

Homework problem 12 is a quantitative analysis of circular orbits.

No magnetic monopoles

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, p, 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, q.

Nobody has ever succeeded in isolating a single magnetic pole. In technical language, we say that magnetic monopoles 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.

The nonexistence of magnetic monopoles means that unlike an electric field, a magnetic one, 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, as in the fields shown in figure r. Gauss' law for magnetism is therefore much simpler than Gauss' law for electric fields:

The magnetic flux through any closed surface is zero.

Symmetry and handedness

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?

t / Left-handed and right-handed definitions.

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 magnetic field patterns shown in figure s seems to violate this principle. Could you use these patterns to explain left and right to the alien? No. If you look back at the definition of the magnetic field, it also contains a reference to handedness: the direction of the vector cross product. 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, as in the left panel of figure t.

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 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.

11.2 Magnetic Fields by Superposition

Superposition of straight wires

In chapter 10, one of the most important goals was to learn how to calculate the electric field for a given charge distribution. The corresponding problem for magnetism would be to calculate the magnetic field arising from a given set of currents. So far, however, we only know how to calculate the magnetic field of a long, straight wire,

with the geometry shown in figure a. Whereas a charge distribution can be broken down into individual point charges, most currents cannot be broken down into a set of straight-line currents. Nevertheless, let's see what we can do with the tools that we have.

An alarming infinity

An interesting aspect of the R^-2 dependence of the field in example 8 is the energy of the field. We've already established in the preceding chapter that the energy density of the magnetic field must be proportional to the square of the field strength, B², the same as for the gravitational and electric fields. Suppose we try to calculate the energy per unit length stored in the field of a single wire. We haven't yet found the proportionality factor that goes in front of the B², but that doesn't matter, because the energy per unit length turns out to be infinite! To see this, we can construct concentric cylindrical shells of length L, with each shell extending from R to R+d R. The volume of the shell equals its circumference times its thickness times its length, d v=(2π R)(d R)(L)=2π Ld R. For a single wire, we have B∼ R^-1, so the energy density is proportional to R^-2, and the energy contained in each shell varies as R^-2d v∼ R^-1d r. Integrating this gives a logarithm, and as we let R approach infinity, we get the logarithm of infinity, which is infinite.

Taken at face value, this result would imply that electrical currents could never exist, since establishing one would require an infinite amount of energy per unit length! In reality, however, we would be dealing with an electric circuit, which would be more like the two wires of example 8: current goes out one wire, but comes back through the other. Since the field really falls off as R^-2, we have an energy density that varies as R^-4, which does not give infinity when integrated out to infinity. (There is still an infinity at R=0, but this doesn't occur for a real wire, which has a finite diameter.)

Still, one might worry about the physical implications of the single-wire result. For instance, suppose we turn on an electron gun, like the one in a TV tube. It takes perhaps a microsecond for the beam to progress across the tube. After it hits the other side of the tube, a return current is established, but at least for the first microsecond, we have only a single current, not two. Do we have infinite energy in the resulting magnetic field? No. It takes time for electric and magnetic field disturbances to travel outward through space, so during that microsecond, the field spreads only to some finite value of R, not R=∞.

This reminds us of an important fact about our study of magnetism so far: we have only been considering situations where the currents and magnetic fields are constant over time. The equation B = 2kI/c²R was derived under this assumption. This equation is only valid if we assume the current has been established and flowing steadily for a long time, and if we are talking about the field at a point in space at which the field has been established for a long time. The generalization to time-varying fields is nontrivial, and qualitatively new effects will crop up. We have already seen one example of this on page 550, where we inferred that an inductor's time-varying magnetic field creates an electric field --- an electric field which is not created by any charges anywhere. Effects like these will be discussed in section 11.5

A sheet of current

There is a saying that in computer science, there are only three nice numbers: zero, one, and however many you please. In other words, computer software shouldn't have arbitrary limitations like a maximum of 16 open files, or 256 e-mail messages per mailbox. When superposing the fields of long, straight wires, the really interesting cases are one wire, two wires, and infinitely many wires. With an infinite number of wires, each carrying an infinitesimal current, we can create sheets of current, as in figure d. Such a sheet has a certain amount of current per unit length, which we notate η (Greek letter eta). The setup is similar to example 8, except that all the currents are in the same direction, and instead of adding up two fields, we add up an infinite number of them by doing an integral. For the y component, we have

where in the last step we have used the identity tan^-1(-x)=-tan^-1x, combined with the relation tan^-1b/z+tan^-1a/z=γ, which can be verified with a little geometry and trigonometry. The calculation of B_z is left as an exercise (problem 23). More interesting is what happens underneath the sheet: by the right-hand rule, all the currents make rightward contributions to the field there, so B_y abruptly reverses itself as we pass through the sheet.

Close to the sheet, the angle γ approaches π, so we have

Figure e shows the similarity between this result and the result for a sheet of charge. In one case the sources are charges and the field is electric; in the other case we have currents and magnetic fields. In both cases we find that the field changes suddenly when we pass through a sheet of sources, and the amount of this change doesn't depend on the size of the sheet. It was this type of reasoning that eventually led us to Gauss' law in the case of electricity, and in section 11.3 we will see that a similar approach can be used with magnetism. The difference is that, whereas Gauss' law involves the flux, a measure of how much the field spreads out, the corresponding law for magnetism will measure how much the field curls.

Is it just dumb luck that the magnetic-field case came out so similar to the electric field case? Not at all. We've already seen that what one observer perceives as an electric field, another observer may perceive as a magnetic field. An observer flying along above a charged sheet will say that the charges are in motion, and will therefore say that it is both a sheet of current and a sheet of charge. Instead of a pure electric field, this observer will experience a combination of an electric field and a magnetic one. (We could also construct an example like the one in figure c on page 598, in which the field was purely magnetic.)

Energy in the magnetic field

In section 10.4, I've already argued that the energy density of the magnetic field must be proportional to |B|², which we can write as B² for convenience. To pin down the constant of proportionality, we now need to do something like the argument on page 534: find one example where we can calculate the mechanical work done by the magnetic field, and equate that to the amount of energy lost by the field itself. The easiest example is two parallel sheets of charge, with their currents in opposite directions. Homework problem 53 is such a calculation, which gives the result

Superposition of dipoles

To understand this subsection, you'll have to have studied section 4.2.4, on iterated integrals.

The distant field of a dipole, in its midplane

Most current distributions cannot be broken down into long, straight wires, and subsection 11.2.1 has exhausted most of the interesting cases we can handle in this way. A much more useful building block is a square current loop. We have already seen how the dipole moment of an irregular current loop can be found by breaking the loop down into square dipoles (figure l on page 604), because the currents in adjoining squares cancel out on their shared edges. Likewise, as shown in figure f, if we could find the magnetic field of a square dipole, then we could find the field of any planar loop of current by adding the contributions to the field from all the squares.

The field of a square-loop dipole is very complicated close up, but luckily for us, we only need to know the current at distances that are large compared to the size of the loop, because we're free to make the squares on our grid as small as we like. The distant field of a square dipole turns out to be simple, and is no different from the distant field of any other dipole with the same dipole moment. We can also save ourselves some work if we only worry about finding the field of the dipole in its own plane, i.e., the plane perpendicular to its dipole moment. By symmetry, the field in this plane cannot have any component in the radial direction (inward toward the dipole, or outward away from it); it is perpendicular to the plane, and in the opposite direction compared to the dipole vector. (The field inside the loop is in the same direction as the dipole vector, but we're interested in the distant field.) Letting the dipole vector be along the z axis, we find that the field in the x-y plane is of the form B_z=f(r), where f(r) is some function that depends only on r, the distance from the dipole.

We can pin down the result even more without any math. We know that the magnetic field made by a current always contains a factor of k/c², which is the coupling constant for magnetism. We also know that the field must be proportional to the dipole moment, m=IA. Fields are always directly proportional to currents, and the proportionality to area follows because dipoles add according to their area. For instance, a square dipole that is 2 micrometers by 2 micrometers in size can be cut up into four dipoles that are 1 micrometer on a side. This tells us that our result must be of the form B_z=(k/c²)(IA)g(r). Now if we multiply the quantity (k/c²)(IA) by the function g(r), we have to get units of teslas, and this only works out if g(r) has units of m^-3 (homework problem 15), so our result must be of the form

where β is a unitless constant. Thus our only task is to determine β, and we will have determined the field of the dipole (in the plane of its current, i.e., the midplane with respect to its dipole moment vector).

If we wanted to, we could simply build a dipole, measure its field, and determine β empirically. Better yet, we can get an exact result if we take a current loop whose field we know exactly, break it down into infinitesimally small squares, integrate to find the total field, set this result equal to the known expression for the field of the loop, and solve for β. There's just one problem here. We don't yet know an expression for the field of any current loop of any shape --- all we know is the field of a long, straight wire. Are we out of luck? No, because, as shown in figure g, we can make a long, straight wire by putting together square dipoles! Any square dipole away from the edge has all four of its currents canceled by its neighbors. The only currents that don't cancel are the ones on the edge, so by superimposing all the square dipoles, we get a straight-line current.

This might seem strange. If the squares on the interior have all their currents canceled out by their neighbors, why do we even need them? Well, we need the squares on the edge in order to make the straight-line current. We need the second row of squares to cancel out the currents at the top of the first row of squares, and so on.

Integrating as shown in figure h, we have

This is the field of a wire, which we already know equals 2kI/c²R, so we have β=1. Remember, the point of this whole calculation was not to find the field of a wire, which we already knew, but to find the unitless constant β in the expression for the field of a dipole. The distant field of a dipole, in its midplane, is therefore B_z=β kIA/c²r³= kIA/c²r³, or, in terms of the dipole moment,

The distant field of a dipole, out of its midplane

What about the field of a magnetic dipole outside of the dipole's midplane? Let's compare with an electric dipole. An electric dipole, unlike a magnetic one, can be built out of two opposite monopoles, i.e., charges, separated by a certain distance, and it is then straightforward to show by vector addition that the field of an electric dipole is

where r is the distance from the dipole to the point of interest, θ is the angle between the dipole vector and the line connecting the dipole to this point, and E_z and E_R are, respectively, the components of the field parallel to and perpendicular to the dipole vector.

In the midplane, θ equals π/2, which produces E_z=-kDr^-3 and E_R=0. This is the same as the field of a magnetic dipole in its midplane, except that the electric coupling constant k replaces the magnetic version k/c², and the electric dipole moment D is substituted for the magnetic dipole moment m. It is therefore reasonable to conjecture that by using the same presto-change-o recipe we can find the field of a magnetic dipole outside its midplane:

This turns out to be correct. ⁵

Comparing the results of examples 9 and 10, we see that the directions of the fields are both out of the page. In example 9, the outer loop has a current in the opposite direction, so it contributes a field that is into the page. This, however, is weaker than the field due to the inner loop, which dominates because it is less distant.

The biot-savart law (optional)

In section 11.2.3 we developed a method for finding the field due to a given current distribution by tiling a plane with square dipoles. This method has several disadvantages:

• The currents all have to lie in a single plane, and the point at which we're computing the field must be in that plane as well.
• We need to do integral over an area, which means one integral inside another, e.g., . That can get messy.
• It's physically bizarre to have to construct square dipoles in places where there really aren't any currents.

Figure k shows the first step in eliminating these defects: instead of spreading our dipoles out in a plane, we bring them out along an axis. As shown in figure l, this eliminates the restriction to currents that lie in a plane. Now we have to use the general equations for a dipole field from page 618, rather than the simpler expression for the field in the midplane of a dipole. This increase in complication is more than compensated for by a fortunate feature of the new geometry, which is that the infinite tube can be broken down into strips, and we can find the field of such a strip for once and for all. This means that we no longer have to do one integral inside another. The derivation of the most general case is a little messy, so I'll just present the case shown in figure m, where the point of interest is assumed to lie in the y-z plane. Intuitively, what we're really finding is the field of the short piece of length on the end of the U; the two long parallel segments are going to be canceled out by their neighbors when we assemble all the strips to make the tube. We expect that the field of this end-piece will form a pattern that circulates around the y axis, so at the point of interest, it's really the x component of the field that we want to compute:

In the more general case, l, the current loop is not planar, the point of interest is not in the end-planes of the U's, and the U shapes have their ends staggered, so the end-piece is not the only part of each U whose current is not canceled. Without going into the gory details, the correct general result is as follows:

which is known as the Biot-Savart law. (It rhymes with “leo bazaar.” Both t's are silent.) The distances and r are now defined as vectors, and r, which point, respectively, in the direction of the current in the end-piece and the direction from the end-piece to the point of interest. The new equation looks different, but it is consistent with the old one. The vector cross product has a magnitude , which cancels one of r's in the denominator and makes the into a vector with magnitude .

Is it the field of a particle?

We have a simple equation, based on Coulomb's law, for the electric field surrounding a charged particle. Looking at figure n, we can imagine that if the current segment was very short, then it might only contain one electron. It's tempting, then, to interpret the Biot-Savart law as a similar equation for the magnetic field surrounding a moving charged particle. Tempting but wrong! Suppose you stand at a certain point in space and watch a charged particle move by. It has an electric field, and since it's moving, you will also detect a magnetic field on top of that. Both of these fields change over time, however. Not only do they change their magnitudes and directions due to your changing geometric relationship to the particle, but they are also time-delayed, because disturbances in the electromagnetic field travel at the speed of light, which is finite. The fields you detect are the ones corresponding to where the particle used to be, not where it is now. Coulomb's law and the Biot-Savart law are both false in this situation, since neither equation includes time as a variable. It's valid to think of Coulomb's law as the equation for the field of a stationary charged particle, but not a moving one. The Biot-Savart law fails completely as a description of the field of a charged particle, since stationary particles don't make magnetic fields, and the Biot-Savart law fails in the case where the particle is moving.

If you look back at the long chain of reasoning that led to the Biot-Savart law, it all started from the relativistic arguments at the beginning of this chapter, where we assumed a steady current in an infinitely long wire. Everything that came later was built on this foundation, so all our reasoning depends on the assumption that the currents are steady. In a steady current, any charge that moves away from a certain spot is replaced by more charge coming up behind it, so even though the charges are all moving, the electric and magnetic fields they produce are constant. Problems of this type are called electrostatics and magnetostatics problems, and it is only for these problems that Coulomb's law and the Biot-Savart law are valid.

You might think that we could patch up Coulomb's law and the Biot-Savart law by inserting the appropriate time delays. However, we've already seen a clear example of a phenomenon that wouldn't be fixed by this patch: on page 550, we found that a changing magnetic field creates an electric field. Induction effects like these also lead to the existence of light, which is a wave disturbance in the electric and magnetic fields. We could try to apply another band-aid fix to Coulomb's law and the Biot-Savart law to make them deal with induction, but it won't work.

So what are the fundamental equations that describe how sources give rise to electromagnetic fields? We've already encountered two of them: Gauss' law for electricity and Gauss' law for magnetism. Experiments show that these are valid in all situations, not just static ones. But Gauss' law for magnetism merely says that the magnetic flux through a closed surface is zero. It doesn't tell us how to make magnetic fields using currents. It only tells us that we can't make them using magnetic monopoles. The following section develops a new equation, called Ampère's law, which is equivalent to the Biot-Savart law for magnetostatics, but which, unlike the Biot-Savart law, can easily be extended to nonstatic situations.

11.3 Magnetic Fields by Ampère's Law

Ampère's law

As discussed at the end of subsection 11.2.4, our goal now is to find an equation for magnetism that, unlike the Biot-Savart law, will not end up being a dead end when we try to extend it to nonstatic situations.⁶ Experiments show that Gauss' law is valid in both static and nonstatic situations, so it would be reasonable to look for an approach to magnetism that is similar to the way Gauss' law deals with electricity.

How can we do this? Figure a, reproduced from page 614, is our roadmap. Electric fields spread out from charges. Magnetic fields curl around currents. In figure b/1, we define a Gaussian surface, and we define the flux in terms of the electric field pointing out through this surface. In the magnetic case, b/2, we define a surface, called an Ampèrian surface, and we define a quantity called the circulation, Γ (uppercase Greek gamma), in terms of the magnetic field that points along the edge of the Ampèrian surface, c. We break the edge into tiny parts s_j, and for each of these parts, we define a contribution to the circulation using the dot product of ds with the magnetic field:

The circulation is a measure of how curly the field is. Like a Gaussian surface, an Ampèrian surface is purely a mathematical construction. It is not a physical object.

In figure b/2, the field is perpendicular to the edges on the ends, but parallel to the top and bottom edges. A dot product is zero when the vectors are perpendicular, so only the top and bottom edges contribute to Γ. Let these edges have length s. Since the field is constant along both of these edges, we don't actually have to break them into tiny parts; we can just have s₁ on the top edge, pointing to the left, and s₂ on the bottom edge, pointing to the right. The vector s₁ is in the same direction as the field B₁, and s₂ is in the same direction as B₂, so the dot products are simply equal to the products of the vectors' magnitudes. The resulting circulation is

But η s is (current/length)(length), i.e., it is the amount of current that pierces the Ampèrian surface. We'll call this current I_through. We have found one specific example of the general law of nature known as Ampère's law:

Positive and negative signs

Figures d/1 and d/2 show what happens to the circulation when we reverse the direction of the current I_through. Reversing the current causes the magnetic field to reverse itself as well. The dot products occurring in the circulation are all negative in d/2, so the total circulation is now negative. To preserve Ampère's law, we need to define the current in d/2 as a negative number. In general, determine these plus and minus signs using the right-hand rule shown in the figure. As the fingers of your hand sweep around in the direction of the s vectors, your thumb defines the direction of current which is positive. Choosing the direction of the thumb is like choosing which way to insert an ammeter in a circuit: on a digital meter, reversing the connections gives readings which are opposite in sign.

Example 13 shows how much easier it can sometimes be to calculate a field using Ampère's law rather than the approaches developed previously in this chapter. However, if we hadn't already known something about the field, we wouldn't have been able to get started. In situations that lack symmetry, Ampère's law may make things harder, not easier. Anyhow, we will have no choice in nonstatic cases, where Ampère's law is true, and static equations like the Biot-Savart law are false.

A quick and dirty proof

Here's an informal sketch for a proof of Ampère's law, with no pretensions to rigor. Even if you don't care much for proofs, it would be a good idea to read it, because it will help to build your ability to visualize how Ampère's law works.

First we establish by a direct computation (homework problem 26) that Ampère's law holds for the geometry shown in figure f/1, a circular Ampèrian surface with a wire passing perpendicularly through its center. If we then alter the surface as in figure f/2, Ampère's law still works, because the straight segments, being perpendicular to the field, don't contribute to the circulation, and the new arc makes the same contribution to the circulation as the old one it replaced, because the weaker field is compensated for by the greater length of the arc. It is clear that by a series of such modifications, we could mold the surface into any shape, f/3.

Next we prove Ampère's law in the case shown in figure f/4: a small, square Ampèrian surface subject to the field of a distant square dipole. This part of the proof can be most easily accomplished by the methods of section 11.4. It should, for example, be plausible in the case illustrated here. The field on the left edge is stronger than the field on the right, so the overall contribution of these two edges to the circulation is slightly counterclockwise. However, the field is not quite perpendicular to the top and bottoms edges, so they both make small clockwise contributions. The clockwise and counterclockwise parts of the circulation end up canceling each other out. Once Ampère's law is established for a square surface like f/4, it follows that it is true for an irregular surface like f/5, since we can build such a shape out of squares, and the circulations are additive when we paste the surfaces together this way.

By pasting a square dipole onto the wire, f/6, like a flag attached to a flagpole, we can cancel out a segment of the wire's current and create a detour. Ampère's law is still true because, as shown in the last step, the square dipole makes zero contribution to the circulation. We can make as many detours as we like in this manner, thereby morphing the wire into an arbitrary shape like f/7.

What about a wire like f/8? It doesn't pierce the Ampèrian surface, so it doesn't add anything to I_through, and we need to show that it likewise doesn't change the circulation. This wire, however, could be built by tiling the half-plane on its right with square dipoles, and we've already established that the field of a distant dipole doesn't contribute to the circulation. (Note that we couldn't have done this with a wire like f/7, because some of the dipoles would have been right on top of the Ampèrian surface.)

If Ampère's law holds for cases like f/7 and f/8, then it holds for any complex bundle of wires, including some that pass through the Ampèrian surface and some that don't. But we can build just about any static current distribution we like using such a bundle of wires, so it follows that Ampère's law is valid for any static current distribution.

Maxwell's equations for static fields

Static electric fields don't curl the way magnetic fields do, so we can state a version of Ampère's law for electric fields, which is that the circulation of the electric field is zero. Summarizing what we know so far about static fields, we have

This set of equations is the static case of the more general relations known as Maxwell's equations. On the left side of each equation, we have information about a field. On the right is information about the field's sources.

It is vitally important to realize that these equations are only true for statics. They are incorrect if the distribution of charges or currents is changing over time. For example, we saw on page 550 that the changing magnetic field in an inductor gives rise to an electric field. Such an effect is completely inconsistent with the static version of Maxwell's equations; the equations don't even refer to time, so if the magnetic field is changing over time, they will not do anything special. The extension of Maxwell's equations to nonstatic fields is discussed in section 11.6.

Discussion Questions

11.4 Ampère's Law in Differential Form (optional)

The curl operator

The differential form of Gauss' law is more physically satisfying than the integral form, because it relates the charges that are present at some point to the properties of the electric field at the same point. Likewise, it would be more attractive to have a differential version of Ampère's law that would relate the currents to the magnetic field at a single point. intuitively, the divergence was based on the idea of the div-meter, a/1. The corresponding device for measuring the curliness of a field is the curl-meter, a/2. If the field is curly, then the torques on the charges will not cancel out, and the wheel will twist against the resistance of the spring. If your intuition tells you that the curlmeter will never do anything at all, then your intuition is doing a beautiful job on static fields; for nonstatic fields, however, it is perfectly possible to get a curly electric field.

Gauss' law in differential form relates , a scalar, to the charge density, another scalar. Ampère's law, however, deals with directions in space: if we reverse the directions of the currents, it makes a difference. We therefore expect that the differential form of Ampère's law will have vectors on both sides of the equal sign, and we should be thinking of the curl-meter's result as a vector. First we find the orientation of the curl-meter that gives the strongest torque, and then we define the direction of the curl vector using the right-hand rule shown in figure a/3.

To convert the div-meter concept to a mathematical definition, we found the infinitesimal flux, dΦ through a tiny cubical Gaussian surface containing a volume d v. By analogy, we imagine a tiny square Ampèrian surface with infinitesimal area dA. We assume this surface has been oriented in order to get the maximum circulation. The area vector dA will then be in the same direction as the one defined in figure a/3. Ampère's law is

The complete set of Maxwell's equations in differential form is collected on page 852.

Properties of the curl operator

The curl is a derivative.

As an example, let's calculate the curl of the field shown in figure c. For our present purposes, it doesn't really matter whether this is an electric or a magnetic field; we're just getting out feet wet with the curl as a mathematical definition. Applying the definition of the curl directly, we construct an Ampèrian surface in the shape of an infinitesimally small square. Actually, since the field is uniform, it doesn't even matter very much whether we make the square finite or infinitesimal. The right and left edges don't contribute to the circulation, since the field is perpendicular to these edges. The top and bottom do contribute, but the top's contribution is clockwise, i.e., into the page according to the right-hand rule, while the bottom contributes an equal amount in the counterclockwise direction, which corresponds to an out-of-the-page contribution to the curl. They cancel, and the circulation is zero. We could also have determined this by imagining a curl-meter inserted in this field: the torques on it would have canceled out.

It makes sense that the curl of a constant field is zero, because the curl is a kind of derivative. The derivative of a constant is zero.

The curl is rotationally invariant.

Figure c looks just like figure c, but rotated by 90 degrees. Physically, we could be viewing the same field from a point of view that was rotated. Since the laws of physics are the same regardless of rotation, the curl must be zero here as well. In other words, the curl is rotationally invariant. If a certain field has a certain curl vector, then viewed from some other angle, we simply see the same field and the same curl vector, viewed from a different angle. A zero vector viewed from a different angle is still a zero vector.

As a less trivial example, let's compute the curl of the field shown in figure e, at the point (x=0,y=0). The circulation around a square of side s centered on the origin can be approximated by evaluating the field at the midpoints of its sides, \begin{alignat*}{4} x=s/2 & y=0 & F=(s/2)\hat{y} & s_1\cdotF=s^2/2
x=0 & y=s/2 & F=0 & s_2\cdotF=0
x=-s/2 & y=0 & F=-(s/2)\hat{y} & s_3\cdotF=s^2/2
x=0 & y=-s/2 & F=0 & s_4\cdotF=0 ,
\end{alignat*} which gives a circulation of s², and a curl with a magnitude of s²/area=s²/s²=1. By the right-hand rule, the curl points out of the page, i.e., along the positive z axis, so we have

Now consider the field , shown in figure f. This is the same as the previous field, but with your book rotated by 90 degrees about the z axis. Rotating the result of the first calculation, , about the z axis doesn't change it, so the curl of this field is also .

Scaling

When you're taking an ordinary derivative, you have the rule

In other words, multiplying a function by a constant results in a derivative that is multiplied by that constant. The curl is a kind of derivative operator, and the same is true for a curl.

The curl is additive.

We have only calculated each field's curl at the origin, but each of these fields actually has the same curl everywhere. In example 14, for instance, it is obvious that the curl is constant along any vertical line. But even if we move along the x axis, there is still an imbalance between the torques on the left and right sides of the curl-meter. More formally, suppose we start from the origin and move to the left by one unit. We find ourselves in a region where the field is very much as it was before, except that all the field vectors have had one unit worth of added to them. But what do we get if we take the curl of ? The curl, like any god-fearing derivative operation, has the additive property

so

But the second term is zero, so we get the same result as at the origin.

The curl in component form

Now consider the field

The only terms whose curls we haven't yet explicitly computed are the a, e, and f terms, and their curls turn out to be zero (homework problem 50). Only the b and d terms have nonvanishing curls. The curl of this field is

But any field in the x-y plane can be approximated with this type of field, as long as we only need to get a good approximation within a small region. The infinitesimal Ampèrian surface occurring in the definition of the curl is tiny enough to fit in a pretty small region, so we can get away with this here. The d and b coefficients can then be associated with the partial derivatives ∂ F_y/∂ x and ∂ F_x/∂ y. We therefore have

for any field in the x-y plane. In three dimensions, we just need to generate two more equations like this by doing a cyclic permutation of the variables x, y, and z:

Figure k is a summary of the vector calculus presented in the optional sections of this book. The first column shows that one function is a related to another by a kind of differentiation. The second column states the fundamental theorem of calculus, which says that if you integrate the derivative over the interior of a region, you get some information about the original function at the boundary of that region.

k / A summary of the derivative, gradient, curl, and divergence.

11.5 Induced Electric Fields

Faraday's experiment

Nature is simple, but the simplicity may not become evident until a hundred years after the discovery of some new piece of physics. We've already seen, on page 550, that the time-varying magnetic field in an inductor causes an electric field. This electric field is not created by charges. That argument, however, only seems clear with hindsight. The discovery of 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 b is a simplified drawing of the following 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 b/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, b/2. Even more counterintuitively, we get an effect, equally strong but in the opposite direction, when the circuit is broken, b/4. The effect occurs only when the magnetic field is changing, and it appears to be proportional to the derivative ∂B/∂t, which is in one direction when the field is being established, and in the opposite direction when it collapses.

The effect is proportional to ∂B/∂t, but what is the effect? 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. This violates the loop rule, which says that when an electron makes a round trip, there is supposed to be just as much “uphill” (moving against the electric field) as “downhill” (moving with it). That's OK. The loop rule is only true for statics. Faraday's experiments show that an electron really can go around and around, and always be going “downhill,” as in the famous drawing by M.C. Escher shown in figure c. That's just what happens when you have a curly field.

When a field is curly, we can measure its curliness using a circulation. Unlike the magnetic circulation Γ_B, the electric circulation Γ_E is something we can measure directly using ordinary tools. A circulation is defined by breaking up a loop into tiny segments, ds, and adding up the dot products of these distance vectors with the field. But when we multiply electric field by distance, what we get is an indication of the amount of work per unit charge done on a test charge that has been moved through that distance. The work per unit charge has units of volts, and it can be measured using a voltmeter, as shown in figure e, where Γ_E equals the sum of the voltmeter readings. Since the electric circulation is directly measurable, most people who work with circuits are more familiar with it than they are with the magnetic circulation. They usually refer to Γ_E using the synonym “emf,” which stands for “electromotive force,” and notate it as . (This is an unfortunate piece of terminology, because its units are really volts, not newtons.) The term emf can also be used when the path is not a closed loop.

Faraday's experiment demonstrates a new relationship

where the negative sign is a way of showing the observed left-handed relationship, d. This is similar to the structure of of Ampère's law:

which also relates the curliness of a field to something that is going on nearby (a current, in this case).

It's important to note that even though the emf, Γ_E, has units of volts, it isn't a voltage. A voltage is a measure of the electrical energy a charge has when it is at a certain point in space. The curly nature of nonstatic fields means that this whole concept becomes nonsense. In a curly field, suppose one electron stays at home while its friend goes for a drive around the block. When they are reunited, the one that went around the block has picked up some kinetic energy, while the one who stayed at home hasn't. We simply can't define an electrical energy U_e=qV so that U_e+K stays the same for each electron. No voltage pattern, V, can do this, because then it would predict the same kinetic energies for the two electrons, which is incorrect. When we're dealing with nonstatic fields, we need to think of the electrical energy in terms of the energy density of the fields themselves.

It might sound as though an electron could get a free lunch by circling around and around in a curly electric field, resulting in a violation of conservation of energy. The following examples, in addition to their practical interest, both show that energy is in fact conserved.

Discussion Questions

Why induction?

Faraday's results leave us in the dark about several things:

• They don't explain why induction effects occur.
• The relationship Γ_E ∝ -∂ B/∂ t tells us that a changing magnetic field creates an electric field in the surrounding region of space, but the phrase “surrounding region of space” is vague, and needs to be made mathematical.
• Suppose that we can make the “surrounding region of space” idea more well defined. We would then want to know the proportionality constant that has been hidden by the ∝ symbol. Although experiments like Faraday's could be used to find a numerical value for this constant, we would like to know why it should have that particular value.

We can get some guidance from the example of a car's alternator (which just means generator), referred to in the self-check on page 637. To keep things conceptually simple, I carefully avoided mentioning that in a real car's alternator, it isn't actually the permanent magnet that spins. The coil is what spins. The choice of design h/1 or h/2 is merely a matter of engineering convenience, not physics. All that matters is the relative motion of the two objects.

This is highly suggestive. As discussed at the beginning of this chapter, magnetism is a relativistic effect. From arguments about relative motion, we concluded that moving electric charges create magnetic fields. Now perhaps we can use reasoning with the same flavor to show that changing magnetic fields produce curly electric fields. Note that figure h/2 doesn't even require induction. The protons and electrons in the coil are moving through a magnetic field, so they experience forces. The protons can't flow, because the coil is a solid substance, but the electrons can, so a current is induced.⁷

Now if we're convinced that figure h/2 produces a current in the coil, then it seems very plausible that the same will happen in figure h/1, which implies the existence of induction effects. But this example involves circular motion, so it doesn't quite work as a way of proving that induction exists. When we say that motion is relative, we only mean straight-line motion, not circular motion.

A more ironclad relativistic argument comes from the arrangement shown in figure i. This is also a generator --- one that is impractical, but much easier to understand.

Flea 1 doesn't believe in this modern foolishness about induction. She's sitting on the bar magnet, which to her is obviously at rest. As the square wire loop is dragged away from her and the magnet, its protons experience a force out of the page, because the cross product F=qv×B is out of the page. The electrons, which are negatively charged, feel a force into the page. The conduction electrons are free to move, but the protons aren't. In the front and back sides of the loop, this force is perpendicular to the wire. In the right and left sides, however, the electrons are free to respond to the force. Note that the magnetic field is weaker on the right side. It's as though we had two pumps in a loop of pipe, with the weaker pump trying to push in the opposite direction; the weaker pump loses the argument.⁸ We get a current that circulates around the loop.⁹ There is no induction going on in this frame of reference; the forces that cause the current are just the ordinary magnetic forces experienced by any charged particle moving through a magnetic field.

Flea 2 is sitting on the loop, which she considers to be at rest. In her frame of reference, it's the bar magnet that is moving. Like flea 1, she observes a current circulating around the loop, but unlike flea 1, she cannot use magnetic forces to explain this current. As far as she is concerned, the electrons were initially at rest. Magnetic forces are forces between moving charges and other moving charges, so a magnetic field can never accelerate a charged particle starting from rest. A force that accelerates a charge from rest can only be an electric force, so she is forced to conclude that there is an electric field in her region of space. This field drives electrons around and around in circles, so it is apparently violating the loop rule --- it is a curly field. What reason can flea 2 offer for the existence of this electric field pattern? Well, she's been noticing that the magnetic field in her region of space has been changing, possibly because that bar magnet over there has been getting farther away. She observes that a changing magnetic field creates a curly electric field.

We therefore conclude that induction effects must exist based on the fact that motion is relative. If we didn't want to admit induction effects, we would have to outlaw flea 2's frame of reference, but the whole idea of relative motion is that all frames of reference are created equal, and there is no way to determine which one is really at rest.

This whole line of reasoning was not available to Faraday and his contemporaries, since they thought the relative nature of motion only applied to matter, not to electric and magnetic fields.¹⁰ But with the advantage of modern hindsight, we can understand in fundamental terms the facts that Faraday had to take simply as mysterious experimental observations. For example, the geometric relationship shown in figure d follows directly from the direction of the current we deduced in the story of the two fleas.

Faraday's law

We can also answer the other questions posed on page 638. The divide-and-conquer approach should be familiar by now. We first determine the circulation Γ_E in the case where the wire loop is very tiny, j. Then we can break down any big loop into a grid of small ones; we've already seen that when we make this kind of grid, the circulations add together. Although we'll continue to talk about a physical loop of wire, as in figure i, the tiny loop can really be just like the edges of an Ampèrian surface: a mathematical construct that doesn't necessarily correspond to a real object.

In the close-up view shown in figure j, the field looks simpler. Just as a tiny part of a curve looks straight, a tiny part of this magnetic field looks like the field vectors are just getting shorter by the same amount with each step to the right. Writing d x for the width of the loop, we therefore have

for the difference in the strength of the field between the left and right sides. In the frame of reference where the loop is moving, a charge q moving along with the loop at velocity v will experience a magnetic force . In the frame moving along with the loop, this is interpreted as an electrical force, . Observers in the two frames agree on how much force there is, so in the loop's frame, we have an electric field . This field is perpendicular to the front and back sides of the loop, BC and DA, so there is no contribution to the circulation along these sides, but there is a counterclockwise contribution to the circulation on CD, and smaller clockwise one on AB. The result is a circulation that is counterclockwise, and has an absolute value

Using a right-hand rule, the counterclockwise circulation is represented by pointing one's thumb up, but the vector ∂ B/∂ t is down. This is just a rephrasing of the geometric relationship shown in figure d on page 636. We can represent the opposing directions using a minus sign,

Although this derivation was carried out with everything aligned in a specific way along the coordinate axes, it turns out that this relationship can be generalized as a vector dot product,

Finally, we can take a finite-sized loop and break down the circulation around its edges into a grid of tiny loops. The circulations add, so we have

This is known as Faraday's law. (I don't recommend memorizing all these names.) Mathematically, Faraday's law is very similar to the structure of Ampère's law: the circulation of a field around the edges of a surface is equal to the sum of something that points through the

If the loop itself isn't moving, twisting, or changing shape, then the area vectors don't change over time, and we can move the derivative outside the sum, and rewrite Faraday's law in a slightly more transparent form:

A changing magnetic flux makes a curly electric field. You might think based on Gauss' law for magnetic fields that Φ_B would be identically zero. However, Gauss' law only applies to surfaces that are closed, i.e., have no edges.

11.6 Maxwell's Equations

Induced magnetic fields

We have almost, but not quite, done figuring out the complete set of physical laws, called Maxwell's equations, governing electricity and magnetism. We are only missing one more term. For clarity, I'll state Maxwell's equations with the missing part included, and then discuss the physical motivation and experimental evidence for sticking it in:

The Φ_E equation is Gauss' law: charges make diverging electric fields. The corresponding equation for Φ_B tells us that magnetic “charges” (monopoles) don't exist, so magnetic fields never diverge. The third equation says that changing magnetic fields induce curly electric fields, whose curliness we can measure using the emf, Γ_E, around a closed loop. The final equation, for Γ_B, is the only one where anything new has been added. Without the new time derivative term, this equation would simply be Ampère's law. (I've chosen to move the c² over to the left because it simplifies the writing, and also because it more clearly demonstrates the analogous roles played by charges and currents in the Φ_E and Γ_B equations.)

This new ∂Φ_E/∂ t term says that just as a changing magnetic field can induce a curly electric field, a changing electric field can induce a curly magnetic field. Why should this be so? The following examples show that Maxwell's equations would not make sense in general without it.

Figure b shows a mysterious curly magnetic field. Magnetic fields are supposed to be made by moving charges, but there don't seem to be any moving charges in this landscape. Where are they? One reasonable guess would be that they're behind your head, where you can't see them. Suppose there's a positively charged particle about to hit you in the back of the head. This particle is like a current going into the page. We're used to dealing with currents made by many charged particles, but logically we can't have some minimum number that would qualify as a current. This is not a static current, however, because the current at a given point in space is not staying the same over time. If the particle is pointlike, then it takes zero time to pass any particular location, and the current is then infinite at that point in space. A moment later, when the particle is passing by some other location, there will be an infinite current there, and zero current in the previous location. If this single particle qualifies as a current, then it should be surrounded by a curly magnetic field, just like any other current.¹¹

This explanation is simple and reasonable, but how do we know it's correct? Well, it makes another prediction, which is that the positively charged particle should be making an electric field as well. Not only that, but if it's headed for the back of your head, then it's getting closer and closer, so the electric field should be getting stronger over time. But this is exactly what Maxwell's equations require. There is no current I_through piercing the Ampèrian surface shown in figure c, so Maxwell's equation for Γ_B becomes c²Γ_B = ∂Φ_E/∂ t. The only reason for an electric field to change is if there are charged particles making it, and those charged particles are moving. When charged particles are moving, they make magnetic fields as well.

Note that the above example is also sufficient to prove the positive sign of the ∂Φ_E/∂ t term in Maxwell's equations, which is different from the negative sign of Faraday's -∂Φ_B/∂ t term.

The addition of the ∂Φ_E/∂ t term has an even deeper and more important physical meaning. With the inclusion of this term, Maxwell's equations can describe correctly the way in which disturbances in the electric and magnetic fields ripple outwards at the speed of light. Indeed, Maxwell was the first human to understand that light was in fact an electromagnetic wave. Legend has it that it was on a starry night that he first realized this implication of his equations. 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.

To see how the ∂Φ_E/∂ t term relates to electromagnetic waves, let's look at an example where we would get nonsense without it. Figure d shows an electron that sits just on one side of an imaginary Ampèrian surface, and then hops through it at some randomly chosen moment. Unadorned with the ∂Φ_E/∂ t term, Maxwell's equation for Γ_B reads as c²Γ_B = 4π k I_through, which is Ampère's law. If the electron is a pointlike particle, then we have an infinite current I_through at the moment when it pierces the imaginary surface, and zero current at all other times. An infinite magnetic circulation Γ_B can only be produced by an infinite magnetic field, so without the ∂Φ_E/∂ t term, Maxwell's equations predict nonsense: the edge of the surface would experience an infinite magnetic field at one instant, and zero magnetic field at all other times. Even if the infinity didn't upset us, it doesn't make sense that anything special would happen at the moment the electron passed through the surface, because the surface is an imaginary mathematical construct. We could just as well have chosen the curved surface shown in figure e, which the electron never crosses at all. We are already clearly getting nonsensical results by omitting the ∂Φ_E/∂ t term, and this shouldn't surprise us because Ampère's law only applies to statics. More to the point, Ampère's law doesn't have time in it, so it predicts that this effect is instantaneous. According to Ampère's law, we could send Morse code signals by wiggling the electron back and forth, and these signals would be received at distant locations instantly, without any time delay at all. This contradicts the theory of relativity, one of whose predictions is that information cannot be transmitted at speeds greater than the speed of light.

Discussion Questions

Light waves

We could indeed send signals using this scheme, and the signals would be a form of light. A radio transmitting antenna, for instance, is simply a device for whipping electrons back and forth at megahertz frequencies. Radio waves are just like visible light, but with a lower frequency. With the addition of the ∂Φ_E/∂ t term, Maxwell's equations are capable of describing electromagnetic waves. It would be possible to use Maxwell's equations to calculate the pattern of the electric and magnetic fields rippling outward from a single electron that fidgets at irregular intervals, but let's pick a simpler example to analyze.

The simplest wave pattern is a sine wave like the one shown in figure f. Let's assume a magnetic field of this form, and see what Maxwell's equations tell us about it. If the wave is traveling through empty space, then there are no charges or currents present, and Maxwell's equations become

The equation Φ=0 has already been verified for this type of wave pattern in example 33 on page 580. Even if you haven't learned the techniques from that section, it should be visually plausible that this field pattern doesn't diverge or converge on any particular point.

Geometry of the electric and magnetic fields

The equation c²Γ_B=∂Φ_E/∂ t tells us that there can be no such thing as a purely magnetic wave. The wave pattern clearly does have a nonvanishing circulation around the edge of the surface suggested in figure g, so there must be an electric flux through the surface. This magnetic field pattern must be intertwined with an electric field pattern that fills the same space. There is also no way that the two sides of the equation could stay synchronized with each other unless the electric field pattern is also a sine wave, and one that has the same wavelength, frequency, and velocity. Since the electric field is making a flux through the indicated surface, it's plausible that the electric field vectors lie in a plane perpendicular to that of the magnetic field vectors. The resulting geometry is shown in figure h. Further justification for this geometry is given later in this subsection.

h / The geometry of an electromagnetic wave.

One feature of figure h that is easily justified is that the electric and magnetic fields are perpendicular not only to each other, but also to the direction of propagation of the wave. In other words, the vibration is sideways, like people in a stadium “doing the wave,” not lengthwise, like the accordion pattern in figure i. (In standard wave terminology, we say that the wave is transverse, not longitudinal.) The wave pattern in figure i is impossible, because it diverges from the middle. For virtually any choice of Gaussian surface, the magnetic and electric fluxes would be nonzero, contradicting the equations Φ_B=0 and Φ_E=0.¹²

Polarization

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.

The speed of light

What is the velocity of the waves described by Maxwell's equations? Maxwell convinced himself that light was an electromagnetic wave partly because his equations predicted waves moving at the velocity of light, c. The only velocity that appears in the equations is c, so this is fairly plausible, although a real calculation is required in order to prove that the velocity of the waves isn't something like 2c or c/π --- or zero, which is also c multiplied by a constant! The following discussion, leading up to a proof that electromagnetic waves travel at c, is meant to be understandable even if you're reading this book out of order, and haven't yet learned much about waves. As always with proofs in this book, the reason to read it isn't to convince yourself that it's true, but rather to build your intuition. The style will be visual. In all the following figures, the wave patterns are moving across the page (let's say to the right), and it usually doesn't matter whether you imagine them as representing the wave's magnetic field or its electric field, because Maxwell's equations in a vacuum have the same form for both fields. Whichever field we imagine the figures as representing, the other field is coming in and out of the page.

The velocity of the waves is not zero. If the wave pattern was standing still in space, then the right sides of the Γ equations would be zero, because there would be no change in the field over time at a particular point. But the left sides are not zero, so this is impossible.¹³

The velocity of the waves is a fixed number for a given wave pattern. Consider a typical sinusoidal wave of visible light, with a distance of half a micrometer from one peak to the next peak. Suppose this wave pattern provides a valid solution to Maxwell's equations when it is moving with a certain velocity. We then know, for instance, that there cannot be a valid solution to Maxwell's equations in which the same wave pattern moves with double that velocity. The time derivatives on the right sides of Maxwell's equations for Γ_E and Γ_B would be twice as big, since an observer at a certain point in space would see the wave pattern sweeping past at twice the rate. But the left sides would be the same, so the equations wouldn't equate.

The velocity is the same for all wave patterns. In other words, it isn't 0.878c for one wave pattern, and 1.067c for some other pattern. This is surprising, since, for example, water waves with different shapes do travel at different speeds. Similarly, even though we speak of “the speed of sound,” sound waves do travel at slightly different speeds depending on their pitch and loudness, although the differences are small unless you're talking about cannon blasts or extremely high frequency ultrasound. To see how Maxwell's equations give a consistent velocity, consider figure j. Along the right and left edges of the same Ampèrian surface, the more compressed wave pattern of blue light has twice as strong a field, so the circulations on the left sides of Maxwell's equations are twice as large.¹⁴ To satisfy Maxwell's equations, the time derivatives of the fields must also be twice as large for the blue light. But this is true only if the blue light's wave pattern is moving to the right at the same speed as the red light's: if the blue light pattern is sweeping over an observer with a given velocity, then the time between peaks is half as much, like the clicking of the wheels on a train whose cars are half the length.¹⁵

We can also check that bright and dim light, as shown in figure k, have the same velocity. If you haven't yet learned much about waves, then this might be surprising. A material object with more energy goes faster, but that's not the case for waves. The circulation around the edge of the Ampèrian surface shown in the figure is twice as strong for the light whose fields are doubled in strength, so the left sides of Maxwell's Γ equations are doubled. The right sides are also doubled, because the derivative of twice a function is twice the derivative of the original function. Thus if dim light moving with a particular velocity is a solution, then so is bright light, provided that it has the same velocity.

We can now see that all sinusoidal waves have the same velocity. What about nonsinusoidal waves like the one in figure l? There is a mathematical theorem, due to Fourier, that says any function can be made by adding together sinusoidal functions. For instance, 3sin x-7cos 3x can be made by adding together the functions 3sin x and -7cos 3x, but Fourier proved that this can be done even for functions, like figure l, that aren't obviously built out of sines and cosines in the first place. Therefore our proof that sinusoidal waves all have the same velocity is sufficient to demonstrate that other waves also have this same velocity.

We're now ready to prove that this universal speed for all electromagnetic waves is indeed c. Since we've already convinced ourselves that all such waves travel at the same speed, it's sufficient to find the velocity of one wave in particular. Let's pick the wave whose fields have magnitudes

which is about as simple as we can get. The peak electric field of this wave has a strength , and the peak magnetic field is . The sine functions go through one complete cycle as x increases by 2π=6.28…, so the distance from one peak of this wave to the next --- its wavelength --- is 6.28... meters. This means that it is not a wave of visible light but rather a radio wave (its wavelength is on the same order of magnitude as the size of a radio antenna). That's OK. What was glorious about Maxwell's work was that it unified the whole electromagnetic spectrum. Light is simple. Radio waves aren't fundamentally any different than light waves, x-rays, or gamma rays. ¹⁶

The justification for putting x+vt inside the sine functions is as follows. As the wave travels through space, the whole pattern just shifts over. The fields are zero at x=0, t=0, since the sine of zero is zero. This zero-point of the wave pattern shifts over as time goes by; at any time t its location is given by x+vt=0. After one second, the zero-point is located at x=-(1 s)v. The distance it travels in one second is therefore numerically equal to v, and this is exactly the concept of velocity: how far something goes per unit time.

The wave has to satisfy Maxwell's equations for Γ_E and Γ_B regardless of what Ampèrian surfaces we pick, and by applying them to any surface, we could determine the speed of the wave. The surface shown in figure m turns out to result in an easy calculation: a narrow strip of width and height h, coinciding with the position of the zero-point of the field at t=0.

Now let's apply the equation c²Γ_B=∂Φ_E/∂ t at t=0. Since the strip is narrow, we can approximate the magnetic field using sin x≈ x, which is valid for small x. The magnetic field on the right edge of the strip, at , is then , so the right edge of the strip contributes to the circulation. The left edge contributes the same amount, so the left side of Maxwell's equation is

The other side of the equation is

Maxwell's equation for Γ_B therefore results in

An application of Γ_E=-∂Φ_B/∂ t gives a similar result, except that there is no factor of c²

(The minus sign simply represents the right-handed relationship of the fields relative to their direction of propagation.)

Multiplying these last two equations by each other, we get

This is the desired result. (The plus or minus sign shows that the wave can travel in either direction.)

As a byproduct of this calculation, we can find the relationship between the strengths of the electric and magnetic fields in an electromagnetic wave. If, instead of multiplying the equations and , we divide them, we can easily show that .

n / The electromagnetic spectrum.

Figure n shows the complete spectrum of light waves. The wavelength λ (number of meters per cycle) and frequency f (number of cycles per second) are related by the equation c=fλ. 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 surface. Likewise, light waves can be focused with a curved, dish-shaped mirror, and Hertz demonstrated the same thing with a dish-shaped radio antenna.

Momentum of light waves

A light wave consists of electric and magnetic fields, and fields contain energy. Thus a light wave carries energy with it when it travels from one place to another. If a material object has kinetic energy and moves from one place to another, it must also have momentum, so it is logical to ask whether light waves have momentum as well. It can be proved based on relativity that it does, and that the momentum and energy are related by the equation U=p/c, where p is the magnitude of the momentum vector, and U=U_e+U_m is the sum of the energy of the electric and magnetic fields (see homework problem 16 on page 405). We can now demonstrate this without explicitly referring to relativity, and connect it to the specific structure of a light wave.

The energy density of a light wave is related to the magnitudes of the fields in a specific way --- it depends on the squares of their magnitudes, E² and B², which are the same as the dot products E⋅E and B⋅B. We argued on page 534 that since energy is a scalar, the only possible expressions for the energy densities of the fields are dot products like these, multiplied by some constants. This is because the dot product is the only mathematically sensible way of multiplying two vectors to get a scalar result. (Any other way violates the symmetry of space itself.)

How does this relate to momentum? Well, we know that if we double the strengths of the fields in a light beam, it will have four times the energy, because the energy depends on the square of the fields. But we then know that this quadruple-energy light beam must have quadruple the momentum as well. If there wasn't this kind of consistency between the momentum and the energy, then we could violate conservation of momentum by combining light beams or splitting them up. We therefore know that the momentum density of a light beam must depend on a field multiplied by a field. Momentum, however, is a vector, and there is only one physically meaningful way of multiplying two vectors to get a vector result, which is the cross product (see page 848). The momentum density can therefore only depend on the cross products E×E, B×B, and E×B. But the first two of these are zero, since the cross product vanishes when there is a zero angle between the vectors. Thus the momentum per unit volume must equal E×B multiplied by some constant,

This predicts something specific about the direction of propagation of a light wave: it must be along the line perpendicular to the electric and magnetic fields. We've already seen that this is correct, and also that the electric and magnetic fields are perpendicular to each other. Therefore this cross product has a magnitude

where in the last step the relation |E|=c|B| has been used.

We now only need to find one physical example in order to fix the constant of proportionality. Indeed, if we didn't know relativity, it would be possible to believe that the constant of proportionality was zero! The simplest example of which I know is as follows. Suppose a piece of wire of length is bathed in electromagnetic waves coming in sideways, and let's say for convenience that this is a radio wave, with a wavelength that is large compared to , so that the fields don't change significantly across the length of the wire. Let's say the electric field of the wave happens to be aligned with the wire. Then there is an emf between the ends of the wire which equals , and since the wire is small compared to the wavelength, we can pretend that the field is uniform, not curly, in which case voltage is a well-defined concept, and this is equivalent to a voltage difference between the ends of the wire. The wire obeys Ohm's law, and a current flows in response to the wave.¹⁷ Equating the expressions d U/d t and IΔ V for the power dissipated by ohmic heating, we have

for the energy the wave transfers to the wire in a time interval d t.

Note that although some electrons have been set in motion in the wire, we haven't yet seen any momentum transfer, since the protons are experiencing the same amount of electric force in the opposite direction. However, the electromagnetic wave also has a magnetic field, and a magnetic field transfers momentum to (exerts a force on) a current. This is only a force on the electrons, because they're what make the current. The magnitude of this force equals (homework problem 6), and using the definition of force, dp/d t, we find for the magnitude of the momentum transferred:

We now know both the amount of energy and the amount of momentum that the wave has lost by interacting with the wire. Dividing these two equations, we find

which is what we expected based on relativity. This can now be restated in the form dp = (constant)E×B d v (homework problem 40).

Note that although the equations p=U/c and dp = (constant)E×B d v are consistent with each other for a sine wave, they are not consistent with each other in general. The relativistic argument leading up to p=U/c assumed that we were only talking about a single thing traveling in a single direction, whereas no such assumption was made in arguing for the E×B form. For instance, if two light beams of equal strength are traveling through one another, going in opposite directions, their total momentum is zero, which is consistent with the E×B form, but not with U/c.

Some examples were given in chapter 3 of situations where it actually matters that light has momentum.

Angular momentum of light waves

For completeness, we note that since light carries momentum, it must also be possible for it to have angular momentum. If you've studied chemistry, here's an example of why this can be important. You know that electrons in atoms can exist in states labeled s, p, d, f, and so on. What you might not have realized is that these are angular momentum labels. The s state, for example, has zero angular momentum. If light didn't have angular momentum, then, for example, it wouldn't be possible for a hydrogen atom in a p state to change to the lower-energy s state by emitting light. Conservation of angular momentum requires that the light wave carry away all the angular momentum originally possessed by the electron in the p state, since in the s state it has none.

11.7 Electromagnetic Properties of Materials

Different types of matter have a variety of useful electrical and magnetic properties. Some are conductors, and some are insulators. Some, like iron and nickel, can be magnetized, while others have useful electrical properties, e.g., dielectrics, discussed qualitatively in the discussion question on page 544, which allow us to make capacitors with much higher values of capacitance than would otherwise be possible. We need to organize our knowledge about the properties that materials can possess, and see whether this knowledge allows us to calculate anything useful with Maxwell's equations.

Conductors

A perfect conductor, such as a superconductor, has no DC electrical resistance. It is not possible to have a static electric field inside it, because then charges would move in response to that field, and the motion of the charges would tend to reduce the field, contrary to the assumption that the field was static. Things are a little different at the surface of a perfect conductor than on the interior. We expect that any net charges that exist on the conductor will spread out under the influence of their mutual repulsion, and settle on the surface. As we saw in chapter 10, Gauss's law requires that the fields on the two sides of a sheet of charge have |E_⊥,1-E_⊥,2| proportional to the surface charge density, and since the field inside the conductor is zero, we infer that there can be a field on or immediately outside the conductor, with a nonvanishing component perpendicular to the surface. The component of the field parallel to the surface must vanish, however, since otherwise it would cause the charges to move along the surface.

On a hot summer day, the reason the sun feels warm on your skin is that the oscillating fields of the light waves excite currents in your skin, and these currents dissipate energy by ohmic heating. In a perfect conductor, however, this could never happen, because there is no such thing as ohmic heating. Since electric fields can't penetrate a perfect conductor, we also know that an electromagnetic wave can never pass into one. By conservation of energy, we know that the wave can't just vanish, and if the energy can't be dissipated as heat, then the only remaining possibility is that all of the wave's energy is reflected. This is why metals, which are good electrical conductors, are also highly reflective. They are not perfect electrical conductors, however, so they are not perfectly reflective. The wave enters the conductor, but immediately excites oscillating currents, and these oscillating currents dissipate the energy both by ohmic heating and by reradiating the reflected wave. Since the parts of Maxwell's equations describing radiation have time derivatives in them, the efficiency of this reradiation process depends strongly on frequency. When the frequency is high and the material is a good conductor, reflection predominates, and is so efficient that the wave only penetrates to a very small depth, called the skin depth. In the limit of poor conduction and low frequencies, absorption predominates, and the skin depth becomes much greater. In a high-frequency AC circuit, the skin depth in a copper wire is very small, and therefore the signals in such a circuit are propagated entirely at the surfaces of the wires. In the limit of low frequencies, i.e., DC, the skin depth approaches infinity, so currents are carried uniformly over the wires' cross-sections.

We can quantify how well a particular material conducts electricity. We know that the resistance of a wire is proportional to its length, and inversely proportional to its cross-sectional area. The constant of proportionality is 1/σ, where σ (not the same σ as the surface charge density) is called the electrical conductivity. Exposed to an electric field E, a conductor responds with a current per unit cross-sectional area J=σE. The skin depth is proportional to , where f is the frequency of the wave.

Dielectrics

A material with a very low conductivity is an insulator. Such materials are usually composed of atoms or molecules whose electrons are strongly bound to them; since the atoms or molecules have zero total charge, their motion cannot create an electric current. But even though they have zero charge, they may not have zero dipole moment. Imagine such a substance filling in the space between the plates of a capacitor, as in figure a. For simplicity, we assume that the molecules are oriented randomly at first, a/1, and then become completely aligned when a field is applied, a/2. The effect has been to take all of the negatively charged black ands of the molecules and shift them upward, and the opposite for the positively charged white ends. Where the black and white charges overlap, there is still zero net charge, but we have a strip of negative charge at the top, and a strip of positive charge at the bottom, a/3. The effect has been to cancel out part of the charge that was deposited on the plates of the capacitor. Now this is very subtle, because Maxwell's equations treat these charges on an equal basis, but in terms of practical measurements, they are completely different. The charge on the plates can be measured be inserting an ammeter in the circuit, and integrating the current over time. But the charges in the layers at the top and bottom of the dielectric never flowed through any wires, and cannot be detected by an ammeter. In other words, the total charge, q, appearing in Maxwell's equartions is actualy q=q_free-q_bound, where q_free is the charge that moves freely through wires, and can be detected in an ammeter, while q_bound is the charge bound onto the individual molecules, which can't. We will, however, detect the presence of the bound charges via their electric fields. Since their electric fields partially cancel the fields of the free charges, a voltmeter will register a smaller than expected voltage difference between the plates. If we measure q_free/V, we have a result that is larger than the capacitance we would have expected.

Although the relationship E≤ftrightarrow q between electric fields and their sources is unalterably locked in by Gauss's law, that's not what we see in practical measurements. In this example, we can measure the voltage difference between the plates of the capacitor and divide by the distance between them to find E, and then integrate an ammeter reading to find q_free, and we will find that Gauss's law appears not to hold. We have E≤ftrightarrow q_free/(constant), where the constant fudge factor is greater than one. This constant is a property of the dielectric material, and tells us how many dipoles there are, how strong they are, and how easily they can be reoriented. The conventional notation is to incorporate this fudge factor into Gauss's law by defining an altered version of the electric field,

and to rewrite Gauss's law as

The constant ε is a property of the material, known as its permittivity. In a vacuum, ε takes on a value known as ε_o, defined as 1/(4π k). In a dielectric, ε is greater than ε_o. When a dielectric is present between the plates of a capacitor, its capacitance is proportional to ε (problem 38). The following table gives some sample values of the permittivities of a few substances.

A capacitor with a very high capacitance is potentially a superior replacement for a battery, but until the 1990's this was impractical because capacitors with high enough values couldn't be made, even with dielectrics having the largest known permittivities. Such supercapactors, some with values in the kilofarad range, are now available. Most of them do not use dielectric at all; the very high capacitance values are instead obtained by using electrodes that are not parallel metal plates at all, but exotic materials such as aerogels, which allows the spacing between the “electrodes” to be very small.

Although figure a/2 shows the dipoles in the dielectric being completely aligned, this is not a situation commonly encountered in practice. In such a situation, the material would be as polarized as it could possibly be, and if the field was increased further, it would not respond. In reality, a capacitor, for example, would normally be operated with fields that produced quite a small amount of alignment, and it would be under these conditions that the linear relationship D=εE would actually be a good approximation. Before a material's maximum polarization is reached, it may actually spark or burn up.

Magnetic materials

Atoms and molecules may have magnetic dipole moments as well as electric dipole moments. Just as an electric dipole contains bound charges, a magnetic dipole has bound currents, which come from the motion of the electrons as they orbit the nucleus, c/1. Such a substance, subjected to a magnetic field, tends to align itself, c/2, so that a sheet of current circulates around the externally applied field. Figure c/3 is closely analogous to figure a/3; in the central gray area, the atomic currents cancel out, but the atoms at the outer surface form a sheet of bound current. However, whereas like charges repel and opposite charges attract, it works the other way around for currents: currents in the same direction attract, and currents in opposite directions repel. Therefore the bound currents in a material inserted inside a solenoid tend to reinforce the free currents, and the result is to strengthen the field. The total current is I=I_free+I_bound, and we define an altered version of the magnetic field,

and rewrite Ampère's law as

The constant μ is the permeability, with a vacuum value of μ_o=4π k/c². Here are the magnetic permeabilities of some substances:

The electromagnet of example 23 could also be used as an inductor, and its inductance would be proportional to the permittivity of the core. This makes it possible to construct high-value inductors that are relatively compact. Permeable cores are also used in transformers.

A transformer or inductor with a permeable core does have some disadvantages, however, in certain applications. The oscillating magnetic field induces an electric field, and because the core is typically a metal, these currents dissipate energy strongly as heat. This behaves like a fairly large resistance in series with the coil. Figure f shows a method for reducing this effect. The iron core of this transformer has been constructed out of laminated layers, which has the effect of blocking the conduction of the eddy currents.

Surprisingly, some materials have magnetic permeabilities less than μ_o. This cannot be accounted for in the model above, and although there are semiclassical arguments that can explain it to some extent, it is fundamentally a quantum mechanical effect. Materials with μ>μ_o are called paramagnetic, while those with μ<μ_o are referred to as diamagnetic. Diamagnetism is generally a much weaker effect than paramagnetism, and is easily masked if there is any trace of contamination from a paramagnetic material. Diamagnetic materials have the interesting property that they are repelled from regions of strong magnetic field, and it is therefore possible to levitate a diamagnetic object above a magnet, as in figure h.

A complete statement of Maxwell's equations in the presence of electric and magnetic materials is as follows:

Comparison with the vacuum case shows that the speed of an electromagnetic wave moving through a substance described by permittivity and permeability ε and μ is . For most substances, μ≈μ_o, and ε is highly frequency-dependent.

Suppose we have a boundary between two substances. By constructing a Gaussian or Ampèrian surface that extends across the boundary, we can arrive at various constraints on how the fields must behave as me move from one substance into the other, when there are no free currents or charges present, and the fields are static. An interesting example is the application of Faraday's law, Γ_H=0, to the case where one medium --- let's say it's air --- has a low permeability, while the other one has a very high one. We will violate Faraday's law unless the component of the H field parallel to the boundary is a continuous function, H_||,1=H_||,2. This means that if μ/μ_o is very high, the component of B=μH parallel to the surface will have an abrupt discontinuity, being much stronger inside the high-permeability material. The result is that when a magnetic field enters a high-permeability material, it tends to twist abruptly to one side, and the pattern of the field tends to be channeled through the material like water through a funnel. In a transformer, a permeable core functions to channel more of the magnetic flux from the input coil to the output coil. Figure j shows another example, in which the effect is to shield the interior of the sphere from the externally imposed field. Special high-permeability alloys, with trade names like Mu-Metal, are sold for this purpose.

j / A hollow sphere with μ/μ_o=10, is immersed in a uniform, externally imposed magnetic field. The interior of the sphere is shielded from the field. The arrows map the magnetic field B. (See homework problem 47 on page 677.

The very last magnetic phenomenon we'll discuss is probably the very first experience you ever had of magnetism. Ferromagnetism is a phenomenon in which a material tends to organize itself so that it has a nonvanishing magnetic field. It is exhibited strongly by iron and nickel, which explains the origin of the name.

k / A model of ferromagnetism.

Figure k/1 is a simple one-dimensional model of ferromagnetism. Each magnetic compass needle represents an atom. The compasses in the chain are stable when aligned with one another, because each one's north end is attracted to its neighbor's south end. The chain can be turned around, k/2, without disrupting its organization, and the compasses do not realign themselves with the Earth's field, because their torques on one another are stronger than the Earth's torques on them. The system has a memory. For example, if I want to remind myself that my friend's address is 137 Coupling Ct., I can align the chain at an angle of 137 degrees. The model fails, however, as an explanation of real ferromagnetism, because in two or more dimensions, the most stable arrangement of a set of interacting magnetic dipoles is something more like k/3, in which alternating rows point in opposite directions. In this two-dimensional pattern, every compass is aligned in the most stable way with all four of its neighbors. This shows that ferromagnetism, like diamagnetism, has no purely classical explanation; a full explanation requires quantum mechanics.

Because ferromagnetic substances “remember” the history of how they were prepared, they are commonly used to store information in computers. Figure l shows 16 bits from an ancient (ca. 1970) 4-kilobyte random-access memory, in which each doughnut-shaped iron “core” can be magnetized in one of two possible directions, so that it stores one bit of information. Today, RAM is made of transistors rather than magnetic cores, but a remnant of the old technology remains in the term “core dump,” meaning “memory dump,” as in “my girlfriend gave me a total core dump about her mom's divorce.” Most computer hard drives today do store their information on rotating magnetic platters, but the platter technology may be obsoleted by flash memory in the near future.

The memory property of ferromagnets can be depicted on the type of graph shown in figure m, known as a hysteresis curve. The y axis is the magnetization of a sample of the material --- a measure of the extent to which its atomic dipoles are aligned with one another. If the sample is initially unmagnetized, 1, and a field H is externally applied, the magnetization increases, 2, but eventually becomes saturated, 3, so that higher fields do not result in any further magnetization, 4. The external field can then be reduced, 5, and even eliminated completely, but the material will retain its magnetization. It is a permanent magnet. To eliminate its magnetization completely, a substantial field must be applied in the opposite direction. If this reversed field is made stronger, then the substance will eventually become magnetized just as strongly in the opposite direction. Since the hysteresis curve is nonlinear, and is not a function (it has more than one value of M for a particular value of B), a ferromagnetic material does not have a single, well-defined value of the permeability μ; a value like 4,000 for transformer iron represents some kind of a rough average.

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Homework Problems

a / Problem 13.

d / Problem 20.