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.