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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 592, 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.

11.7.1 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 \(|\mathbf{E}_{\perp,1}-\mathbf{E}_{\perp,2}|\) 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 \(1/\sigma\), where \(\sigma\) (not the same \(\sigma\) as the surface charge density) is called the electrical conductivity. Exposed to an electric field \(\mathbf{E}\), a conductor responds with a current per unit cross-sectional area \(\mathbf{J}=\sigma\mathbf{E}\). The skin depth is proportional to \(1/\sqrt{f\sigma}\), where \(f\) is the frequency of the wave.

11.7.2 Dielectrics

a / A capacitor with a dielectric between the plates.

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 \(q=q_{\text{free}}-q_{\text{bound}}\), where \(q_{\text{free}}\) 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 \(q_{\text{free}}/V\), we have a result that is larger than the capacitance we would have expected.

Although the relationship \(\mathbf{E}\leftrightarrow q\) 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 \(\mathbf{E}\), and then integrate an ammeter reading to find \(q_{\text{free}}\), and we will find that Gauss's law appears not to hold. We have \(\mathbf{E}\leftrightarrow q_{\text{free}}/(\text{constant})\), 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,

\[\begin{equation*} \mathbf{D}=\epsilon \mathbf{E} , \end{equation*}\]

and to rewrite Gauss's law as

\[\begin{equation*} \Phi_D = q_{\text{in, free}} . \end{equation*}\]

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

substance

εε0 at zero frequency

vacuum 1
air 1.00054
water 80
barium titanate 1250

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 \(\mathbf{D}=\epsilon\mathbf{E}\) would actually be a good approximation. Before a material's maximum polarization is reached, it may actually spark or burn up.

 

Figure b: A stud finder is used to locate the wooden beams, or studs, that form the frame behind the wallboard. It is a capacitor whose capacitance changes when it is brought close to a substance with a particular permittivity. Although the wall is external to the capacitor, a change in capacitance is still observed, because the capacitor has “fringing fields” that extend outside the region between its plates.

 

self-check:

Suppose a parallel-plate capacitor is built so that a slab of dielectric material can be slid in or out. (This is similar to the way the stud finder in figure b works.) We insert the dielectric, hook the capacitor up to a battery to charge it, and then use an ammeter and a voltmeter to observe what happens when the dielectric is withdrawn. Predict the changes observed on the meters, and correlate them with the expected change in capacitance. Discuss the energy transformations involved, and determine whether positive or negative work is done in removing the dielectric.

(answer in the back of the PDF version of the book)

11.7.3 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: The magnetic version of figure a. A magnetically permeable material is placed at the center of a solenoid.

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 \(I=I_{\text{free}}+I_{\text{bound}}\), and we define an altered version of the magnetic field,

\[\begin{equation*} \mathbf{H}= \frac{\mathbf{B}}{\mu} , \end{equation*}\]

and rewrite Ampère's law as

\[\begin{equation*} \Gamma_H = I_{\text{through, free}} . \end{equation*}\]

The constant \(\mu\) is the permeability, with a vacuum value of \(\mu_\text{o}=4\pi k/c^2\). Here are the magnetic permeabilities of some substances:

substance

µµ0

vacuum 1
aluminum 1.00002
steel 700
transformer iron 4,000

µ-metal

20,000
Example 24: An iron-core electromagnet

d / Example 24: a cutaway view of a solenoid.

\(\triangleright\) A solenoid has 1000 turns of wire wound along a cylindrical core with a length of 10 cm. If a current of 1.0 A is used, find the magnetic field inside the solenoid if the core is air, and if the core is made of iron with \(\mu/\mu_\text{o}=4,000\).

e / Example 24: without the iron core, the field is so weak that it barely deflects the compass. With it, the deflection is nearly 90\(°\).

\(\triangleright\) Air has essentially the same permability as vacuum, so using the result of example 13 on page 674, we find that the field is 0.013 T.

We now consider the case where the core is filled with iron. The original derivation in example 13 started from Ampère's law, which we now rewrite as \(\Gamma_H = I_{\text{through, free}}\). As argued previously, the only significant contributions to the circulation come from line segment AB. This segment lies inside the iron, where \(\mathbf{H}=\mathbf{B}/\mu\). The \(\mathbf{H}\) field is the same as in the air-core case, since the new form of Ampère's law only relates \(\mathbf{H}\) to the current in the wires (the free current). This means that \(\mathbf{B}=\mu\mathbf{H}\) is greater by a factor of 4,000 than in the air-core case, or 52 T. This is an extremely intense field --- so intense, in fact, that the iron's magnetic polarization would probably become saturated before we could actually get the field that high.

The electromagnet of example 24 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.

f / A transformer with a laminated iron core. The input and output coils are inside the paper wrapper. The iron core is the black part that passes through the coils at the center, and also wraps around them on the outside.

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.

Example 25: A ferrite bead

Cables designed to carry audio signals are typically made with two adjacent conductors, such that the current flowing out through one conductor comes back through the other one. Computer cables are similar, but usually have several such pairs bundled inside the insulator. This paired arrangement is known as differential mode, and has the advantage of cutting down on the reception and transmission of interference. In terms of transmission, the magnetic field created by the outgoing current is almost exactly canceled by the field from the return current, so electromagnetic waves are only weakly induced. In reception, both conductors are bathed in the same electric and magnetic fields, so an emf that adds current on one side subtracts current from the other side, resulting in cancelation.

The opposite of differential mode is called common mode. In common mode, all conductors have currents flowing in the same direction. Even when a circuit is designed to operate in differential mode, it may not have exactly equal currents in the two conductors with \(I_1+I_2=0\), meaning that current is leaking off to ground at one end of the circuit or the other. Although paired cables are relatively immune to differential-mode interference, they do not have any automatic protection from common-mode interference.

 

Figure g: Example 25: ferrite beads. The top panel shows a clip-on type, while the bottom shows one built into a cable.

Figure g shows a device for reducing common-mode interference called a ferrite bead, which surrounds the cable like a bead on a string. Ferrite is a magnetically permeable alloy. In this application, the ohmic properties of the ferrite actually turn put to be advantageous.

Let's consider common-mode transmission of interference. The bare cable has some DC resistance, but is also surrounded by a magnetic field, so it has inductance as well. This means that it behaves like a series L-R circuit, with an impedance that varies as \(R+i\omega L\), where both \(R\) and \(L\) are very small. When we add the ferrite bead, the inductance is increased by orders of magnitude, but so is the resistance. Neither \(R\) nor \(L\) is actually constant with respect to frequency, but both are much greater than for the bare cable.

Suppose, for example, that a signal is being transmitted from a digital camera to a computer via a USB cable. The camera has an internal impedance that is on the order of 10 \(\Omega\), the computer's input also has a \(\sim10\) \(\Omega\) impedance, and in differential mode the ferrite bead has no effect, so the cable's impedance has its low, designed value (probably also about 10 \(\Omega\), for good impedance matching). The signal is transmitted unattenuated from the camera to the computer, and there is almost no radiation from the cable.

But in reality there will be a certain amount of common-mode current as well. With respect to common mode, the ferrite bead has a large impedance, with the exact value depending on frequency, but typically on the order of 100 \(\Omega\) for frequencies in the MHz range. We now have a series circuit consisting of three impedances: 10, 100, and 10 \(\Omega\). For a given emf applied by an external radio wave, the current induced in the circuit has been attenuated by an order of magnitude, relative to its value without the ferrite bead.

Why is the ferrite is necessary at all? Why not just insert ordinary air-core inductors in the circuit? We could, for example, have two solenoidal coils, one in the outgoing line and one in the return line, interwound with one another with their windings oriented so that their differential-mode fields would cancel. There are two good reasons to prefer the ferrite bead design. One is that it allows a clip-on device like the one in the top panel of figure g, which can be added without breaking the circuit. The other is that our circuit will inevitably have some stray capacitance, and will therefore act like an LRC circuit, with a resonance at some frequency. At frequencies close to the resonant frequency, the circuit would absorb and transmit common-mode interference very strongly, which is exactly the opposite of the effect we were hoping to produce. The resonance peak could be made low and broad by adding resistance in series, but this extra resistance would attenuate the differential-mode signals as well as the common-mode ones. The ferrite's resistance, however, is actually a purely magnetic effect, so it vanishes in differential mode.

Surprisingly, some materials have magnetic permeabilities less than \(\mu_\text{o}\). 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 \(\mu>\mu_\text{o}\) are called paramagnetic, while those with \(\mu\lt\mu_\text{o}\) 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.

 

h / A frog is levitated diamagnetically by the nonuniform field inside a powerful magnet. Evidently frog has \(\mu\lt\mu_\text{o}\).

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

\[\begin{align*} \Phi_D &= q_\text{free} \\ \Phi_B &= 0 \\ \Gamma_E &= -\frac{d \Phi_B}{dt} \\ \Gamma_H &= \frac{d \Phi_D}{dt} + I_\text{free} \end{align*}\]

Comparison with the vacuum case shows that the speed of an electromagnetic wave moving through a substance described by permittivity and permeability \(\epsilon\) and \(\mu\) is \(1/\sqrt{\epsilon\mu}\). For most substances, \(\mu\approx\mu_\text{o}\), and \(\epsilon\) is highly frequency-dependent.

i / At a boundary between two substances with \(\mu_2>\mu_1\), the \(\mathbf{H}\) field has a continuous component parallel to the surface, which implies a discontinuity in the parallel component of the magnetic field \(\mathbf{B}\).

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, \(\Gamma_H=0\), 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 \(\mathbf{H}\) field parallel to the boundary is a continuous function, \(\mathbf{H}_{\parallel,1}=\mathbf{H}_{\parallel,2}\). This means that if \(\mu/\mu_\text{o}\) is very high, the component of \(\mathbf{B}=\mu\mathbf{H}\) 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.

 

Figure j: A hollow sphere with \(\mu/\mu_\text{o}=10\), 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 \(\mathbf{B}\). (See homework problem 47, page 727.)

 

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.

 

Figure 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.

 

Figure l: Magnetic core memory.

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.

 

Figure m: A hysteresis curve.

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 \(\mu\); a value like 4,000 for transformer iron represents some kind of a rough average.

Example 26: The fluxgate compass

The fluxgate compass is a type of magnetic compass without moving parts, commonly used on ships and aircraft. An AC current is applied in a coil wound around a ferromagnetic core, driving the core repeatedly around a hysteresis loop.

 

n / A fluxgate compass.

Because the hysteresis curve is highly nonlinear, the addition of an external field such as the Earth's alters the core's behavior. Suppose, for example, that the axis of the coil is aligned with the magnetic north-south. The core will reach saturation more quickly when the coil's field is in the same direction as the Earth's, but will not saturate as early in the next half-cycle, when the two fields are in opposite directions. With the use of multiple coils, the components of the Earth's field can be measured along two or three axes, permitting the compass's orientation to be determined in two or (for aircraft) three dimensions.

11.7.4 Maxwell's equations in differential form

 

This chapter is summarized on page 960. Notation and terminology are tabulated on pages 941-942.

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