1. POLARIZATION AND MAGNETIZATION

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1 1. POLARIZATION AND AGNTIZATION 1.1. The acroscopic Form of the axwell quations On the microscopic level, the electric field and magnetic induction B are described by the axwell equations C Q d A =, (1) ε B d A =, () d ds= d B A, (3) d B ds = μi + με d A, (4) C where Q is the charge inside a closed surface and I is the current flowing through a surface bounded by the closed contour C. Within the realm of classical electrodynamics, these equations provide a correct description of the universe and all that is in it. However, this description is useful only for problems that involve just a small number of particles. On the other hand, even the tiniest macroscopic object consists of prodigious numbers of particles, and the large variety of interesting phenomena that arise from macroscopic collections of particles must be described by a theory that smoothes out the microscopic complexity of matter. The key step in the development of the macroscopic form of the axwell equations is the division of the charges and currents into what are generally called free and bound charges and currents. Free charges are those that can move around the system over macroscopic distances. For example, the conduction electrons in a metal and the positive and negative ions in an electrolyte are free charges. Bound charges are those that are confined near a particular atom in the solid, liquid or gas. The atom itself can move according to the macroscopic motions of the material, but the bound charge moves with the atom. We therefore write Q= Q + Q, (5) f f b b I = I + I (6) where Q f and Q b represent the microscopic average free and bound charges, and I f and I b the free and bound current, respectively. It might be thought that if the bound charges are tied to the atoms in the material, which are intrinsically neutral, then they could not contribute to a net charge inside the surface. However, this is not true. Consider the surface imbedded in a polarized dielectric as shown in 1

2 Figure 1 Figure Figure 1. At the surface, the dipoles with a dipole moment in the direction of the surface normal leave behind a negative charge inside the surface. The net bound charge may therefore be represented by an integral over the surface of the form Qb = P d A (7) where P is a vector proportional to the polarization per unit volume. ubstituting this into Gauss s law, we get We can write this in the simple form where the so-called displacement is defined ε da = Qf P da, (8) D d = Qf (9) D= ε + P (1) In its new form, Gauss s law (9) looks just like the old law except that now we don t have to worry about the bound charge. Thus, we can carry over much of what we ve already learned about electrostatics in vacuum. At the surface of a dielectric, the normal component of the displacement D is continuous. That is, Dn ˆ is the same on both sides of the surface. To see this we consider the pillboxshaped Gaussian surface shown in Figure in the limit when the top and bottom faces approach the surface of the dielectric. ince there is no free charge in the pillbox, the net flux D d A out of the pillbox must vanish, so ignoring the flux though the sides of the pillbox we get ( ) = Doutside Dinside A, where A is the vector area of the top of the pillbox. If we substitute this into the vacuum form (1) of Gauss s law, we find that ( ) ε ε A P A σ, (11) = = A outside inside

3 Figure 3 where σ ˆ = Pn is an effective surface charge density and ˆn is a unit vector normal to the surface. The physical origin of the net surface charge is illustrated in Figure 3. With this we can now show that the polarization P is just the dipole moment per unit volume. Consider a cylinder of area A and length L uniformly polarized along its length. At each end there is a net charge q = PA positive at one end and negative at the other. Thus, the dipole moment of the cylinder is μ total = ql = PV, where V = AL is the volume of the cylinder. But the total dipole moment is the sum of the dipole moments of all the molecules in the cylinder, so we see that P = nμ (1) where μ is the average molecular dipole moment and n is the number of molecules per unit volume. In many substances, the molecular dipole moment is proportional to the electric field. In this case, the polarization is P= ε χ (13) where the dielectric susceptibility χ is a constant, and the displacement is D= ε (14) 1 where the permittivity ( ) ε = + χ ε is likewise a constant. Again it might be thought that currents localized within neutral molecules could not contribute to a significant macroscopic current density, but again this is wrong. In fact, there are two sources of bound currents, I b = I + I, the first due to the polarization and the second due to the magnetization of the medium. To find the current density J P due to the polarization, we differentiate Gauss s law (1) with respect to time and get dqb dp = d = P d A J A (15) where the last integral follows from conservation of charge, which says that the rate of change of the charge inside is the negative rate of flow of charge out through the surface. The polarization current flowing through an area bounded by the contour C is then 3

4 Figure 4 Figure 5 I P d = d P A (16) A physical picture of macroscopic currents arising from microscopic bound currents is illustrated in Figure 4. As shown there, a magnetic dipole aligned along the contour C has a net current upward on the inside of the contour. The net current through the contour C due to the magnetization may therefore be represented by the integral around the contour I = d s (17) C where the magnetization is a vector proportional to the magnetic moment per unit volume. ubstituting (16) and (17) into the axwell-ampere law (4), we get d d B ds= μ d + d + I f + με d P A s A, (18) C C Rearranging this in an obvious way, we get the simpler equation d H ds= d + I f D A, (19) C where we have introduced the magnetic field 1 H = B () μ In its new form, (19) looks just like the original axwell-ampere law except that it depends only on the current due to free charges and hides the current due to bound charges. trictly speaking, the field B is called the magnetic induction and H is called the magnetic field. ore commonly, however, physicists refer to B as the magnetic field and H as the H field, or more simply, just H. Note that the textbook defines the quantity Β = μ to describe the magnetization, and the quantity B = B B = μh in place of the conventional magnetic field H At the surface of a magnetized material, the tangential component of the magnetic field H is continuous. To see this we consider the Amperian loop shown in Figure 5 in the limit as the top and bottom approach the surface of the material. ince there is no free current and the 4

5 Figure 6 area of the loop vanishes in the limit, (19) tells us that the integral H d s around the loop must H H L =, where vanish. Ignoring the integrals along the ends of the loop we get ( ) L is the vector length of the top of the loop. ubstituting this into the vacuum form (4) of the axwell-ampere law, we find that ( ) = = I = outside inside outside B B L μ L μ μ σ L, (1) where the vector surface current density is σ = n ˆ and ˆn is a unit vector normal to the surface. A physical picture of the origin of the bound surface currents is shown in Figure 6. To show that the magnetization is just the magnetic dipole moment per unit volume, we consider again a cylinder of area A and length L uniformly magnetized along its length. Around the curved surface there is a surface current I = σ L, so the magnetic moment of the cylinder is μ total = IA= σ V, where V = AL is the volume of the cylinder. But the total magnetic dipole moment is the sum of the magnetic dipole moments of all the molecules in the cylinder, so inside = nμ, () where μ is the average molecular magnetic dipole moment and n is the number of molecules per unit volume. In many substances, the magnetic dipole moment of a molecule is proportional to the magnetic induction B. In this case, the magnetization is given by χ = B= χ H (3) 1 χ for some constant χ called the magnetic susceptibility. The magnetic field H is H = B (4) μ where the permeability is μ = μ /1 ( χ ). For ferromagnetic materials, the permeability can 4 5 be very large, as large as 1 1 in so-called mu metals, which are nickel-iron alloys that are used for magnetic shielding. In summary, in the presence of dielectric and magnetic materials, the axwell equations become 5

6 Figure 7 D d = Qf, (5) where C B d A =, (6) d ds= d B A, (7) d H ds= d + I f D A, (8) C D= ε + P (9) 1 H = B μ (3) The homogeneous equations (6) and (7) are unchanged from their vacuum form. 1.. xamples As an illustration of polarization phenomena, we consider the simple dielectric-filled capacitor illustrated in Figure 7. In the absence of the dielectric, we find from the vacuum form (1) of Gauss s law that the field between the plates of the capacitor is = σ / ε, where σ is the charge density on the surface of the capacitor plates. The voltage across the capacitor is then V = d = σ d / ε, where d is the separation of the plates, and the capacitance is C = Q/ V = ε A/ d, where A is the area of the plates. When the dielectric is placed between the plates, a surface charge appears due to the displaced bound charge in the dielectric. This partially cancels the electric field in the dielectric. ince the fields are normal to the surface of the dielectric, the induced surface charge density is just the polarization P, so the field inside the dielectric is inside = ( σ P) / ε. For a linear dielectric with susceptibility χ, the polarization is P= ε χ, so inside inside σ P 1 σ outside = = =. (31) ε ε 1+ χ ε 1+ χ 6

7 Figure 8 Figure 9 The electric field in the dielectric is reduced by the factor ( χ ) 1+, and the capacitance is increased. For example, if the space between the plates is completely filled with mineral oil, a 1+ =.1. common dielectric material, the capacitance is increased by the factor ( ) Alternately, we can use the macroscopic form (5) of Gauss s law. Using a Gaussian surface around the upper plate of the capacitor, we find that the displacement is D = σ, (3) everywhere between the plates, both inside the dielectric and outside. The electric field is then D= ε outside = σ outside the dielectric and D= ε ( 1 inside + P= ε + χ ) inside = σ inside the dielectric. The concept of magnetization is illustrated by the Rowland ring shown in Figure 8. This consists of N turns of wire wound on a toroidal iron core with a major radius R and minor radius r. The field is confined to the interior of the toroid, and is aligned along the centerline of the toroid. In the absence of the iron core, we can use the vacuum form (4) of the axwell- Ampere law to find the magnetic field inside the toroid. In the steady-state case ( d / = ), we find for r << Rthat the field is B = μ NI /πr, where I is the current in the windings. When the iron core is included, we can use the media form (8) of the law in the same way. Integrating around a contour on the centerline of the core, we get NI H = (33) π R To find the magnetic field B we need to know the properties of the iron. These are summarized B H on the graph. by the curve B( H ) shown in Figure 9. Given H, we can simply look up ( ) The magnetic field with the iron core is generally much larger than the field without the iron core. We can understand this in the following way. Within the iron toroid, the magnetization is aligned along the centerline of the toroid, in the direction of the magnetic field. As described above, this creates a bound surface current perpendicular to the direction of the magnetization, which in this case corresponds to a surface current wrapped around the toroid parallel to and in the same direction as the current in thee windings. This surface current amplifies the effect of the current in the windings, and is generally much larger than the current in the windings themselves. For example, for I = 1 A and N / π R= 1 turns/m, we have 3 H = 1 A/m. This is enough to saturate mild steel, and the corresponding magnetization is χ 7

8 Figure 1 Figure = 1 A/m. The surface current is σ = 1 A/m, which is 1 times larger than the current in the windings! Another example is provided by the magnetic circuit shown in Figure 1, which consists of an iron core wrapped with wire, and a small air gap. The magnetic field is largely confined to the iron core and the region of the gap. Inside the iron, the surface currents increase the magnetic induction by adding their current to that of the windings. ince the magnetic induction B forms continuous loops around the magnetic circuit, including he gap, the magnetic induction in the gap is the same as that in the core. On the other hand, the magnetic field H is quite different in the air gap and in the iron core. To find H, we apply the macroscopic form (8) of the axwell-ampere law, and use the contour indicated in Figure 1. Then H d l = NI, (34) where N is the number of turns and I is the current in the winding. To simplify matters we assume that in the gap the magnetic induction B is uniform and inside the iron core the magnetic field H is uniform. In the gap B = μh, so the line integral is B HL+ g = NI, (35) μ where L is the length of the core and g is the length of the gap. The magnetic induction B in the gap is the same as that in the core, since the lines of force form continuous loops. But in the core, the induction B( H ) is a function of the field H, which can be represented by a curve like that in Figure 11. Therefore, to find the magnetic induction in the gap we rearrange (35) and solve the equation μ B ( H) = ( NI HL), (36) g in which each side is a function of the magnetic field H. This points out the usefulness of the historical convention of representing the magnetization and magnetic induction B= μ ( +H ) as functions of H, rather than making B the independent variable. For a nonlinear magnetic material such as iron, the solution to (36) must be obtained graphically, as indicated in Figure 11, even for the simple geometry we consider here. The righthand side of (36) is a straight line, as shown in Figure 11, and where it intersects the curve B( H ) representing the left-hand side we find the solution. To deal with a more complicated 8

9 geometry, or to learn anything about the details of the field, such as the uniformity, it is necessary to resort to numerical computation. Fortunately, powerful computer codes now exist for this purpose, and they are widely available. 9

10 . PROPRTI OF DILCTRIC AND AGNTIC ATRIAL.1. Dielectric aterials We take up first the phenomenon of polarization. ince the materials in which we are interested are composed of atoms and molecules, the polarization of the medium is related to the polarization of the individual atoms or molecules. In the case of crystalline solids, the unit cell takes on the role of a molecule in this discussion. The degree to which the atoms and molecules are polarized depends first of all on whether or not the atom or molecule possesses an intrinsic dipole moment, and secondarily on the degree to which it is influenced by its neighbors. Polar dielectrics tend to have a larger dielectric susceptibility than nonpolar dielectrics, and a different temperature dependence. In the absence of an intrinsic dipole moment, the only electric dipole moment an atom or molecule can have is that due to distortion of the charge distribution by the electric field in which the atom finds itself. As a simple model we may imagine an atom as consisting of an electron bound to a nucleus. If the nucleus is regarded as fixed and the electron is bound to the nucleus like a harmonic oscillator, then the spring constant of the electron is k = mω, where m is the mass and ω the oscillation frequency of the electron. The dipole moment of the atom is then where the molecular polarizability is μ = ε α, (37) α in which q is the electron charge, take part in the polarization, and polarization of the medium is Z mol q = eff mol εmω eff, (38) Z eff is the number of electrons in the molecule that effectively ω eff is an average oscillation frequency for these electrons. The P = Np= ε χ, (39) e where N is the number density of atoms and is the electric field in the material, and the dielectric susceptibility χ is e χ NZ q = =. (4) ε ε ω P eff m eff The frequency ω eff of electron oscillations is the same as the frequency of the light they emit. This is typically in the visible or ultraviolet region, and (4) gives the correct order of magnitude for the polarizability of nonpolar materials. 1

11 Figure 1 Unshaded elements are paramagnetic; shaded elements are diamagnetic; gradient shading indicates no data; Fe, Co, and Ni are ferromagnetic. Next we turn to macroscopic materials composed of polar molecules. When the molecules have an intrinsic dipole moment and the dipole moment is free to rotate, as in a liquid, an externally applied electric field causes the dipoles to line up parallel to the field and create a net polarization of the medium. The molecular polarizability of such a medium is generally larger than that of a nonpolar medium. In a liquid the rotational motions are damped by the surrounding molecules, so it is appropriate to consider the state of the system at thermal equilibrium. We recall that the energy of a permanent dipole p in an electric field is / B W = p. At thermal equilibrium the number of dipoles with the energy W is dn e W k T, where k B is Boltzmann s constant and T the temperature. Under ordinary conditions the energy of interaction is small compared with the thermal energy ( p / kbt << 1), and it takes only a bit of algebra to show that the susceptibility is.. agnetic aterials Np χ = e 3ε kt. (41) B The magnetic susceptibilities of the elements span nearly five orders of magnitude, and unlike the dielectric susceptibility, the magnetic susceptibility can be either positive or negative. aterials for which the susceptibility is negative are called diamagnetic, and those for which the susceptibility is positive are called paramagnetic or ferromagnetic. As shown in Figure 1, the susceptibility varies from element to element in a regular way, depending on the electron configuration. lements with valence electrons in p-orbitals tend to be diamagnetic. This corresponds to elements on the right-hand side of the periodic table. lements with valence electrons in s-orbitals tend to be weakly paramagnetic, while those with valence electrons in d- or f-orbitals tend to be strongly paramagnetic (we ll discuss s-, p-, and d-orbitals later in this course). Thus, the elements in columns I and II of the periodic table tend to be weakly paramagnetic, while elements toward the center of the periodic table are strongly paramagnetic. 11

12 At room temperature just three elements, iron, cobalt, and nickel, are ferromagnetic. There is, of course, an explanation for this regularity, and for the exceptions. Although we don t have time to discuss all the details, it s not hard to understand the general principles. Diamagnetism is observed in materials whose molecules have no intrinsic magnetic dipole moment. The diamagnetic effect arises because as the magnetic field at the position of the molecule increases, the orbits adjust themselves in accordance with Lenz s law. If we imagine that each electron in its orbit is equivalent to a submicroscopic circuit carrying a tiny current, then Lenz s law tells us that the current in the circuit changes in a way that opposes the change in the magnetic flux through the circuit. Diamagnetism, then, corresponds to a reduction of the magnetic induction due to the response of the orbital magnetic moment. Diamagnetism is a 4 small effect. In bismuth, for example, the magnetic susceptibility is χ m = Actually, diamagnetism is present in all materials, but in most materials it is masked by paramagnetic or ferromagnetic effects, which are much larger. We can describe diamagnetism classically by means of Larmor s theorem. We consider an atom with Z electrons of charge q and mass m in a uniform magnetic field B. The electrons are assumed to interact with each other and with the central potential of the nuclear 1 db attraction. As the magnetic field is applied, a tangential electric field = ρ is induced at the radius ρ from the axis of the molecule in the field, according to Faraday s law. This dω q q db tangential field causes a rotational acceleration of the electrons = =. mρ m Integrating this we find that when the field reaches the value B, the rotation rate is qb Ω L = (4) m This is Larmor's theorem. Note that the Larmor frequency Ω L is half the cyclotron frequency of the particles in the same magnetic field, and the direction of the rotation vector Ω L is opposite the direction of the magnetic field for positive charges, in accordance with Lenz s law. The atomic magnetic moment created by this rotation is πρ i for each electron, where the current is i =Ω q/ π, so the magnetic moment of the atom is L Zq μ = ρ B, (43) 4m where ρ the average radius of the electrons in the atom relative to the axis of the magnetic field. For a spherical atom, or for an average over all orientations of a nonspherical atom, ρ = 3 r, where r is the radial distance from the nucleus. The diamagnetic susceptibility is then χ μnzq r 1 χ = (44) 6m 1

13 This is called the Langevin formula. Paramagnetism is observed in materials whose atomic or molecular constituents possess a permanent magnetic dipole moment. Whereas diamagnetic materials have a negative susceptibility because the induced magnetization opposes the applied field, paramagnetic materials have a positive susceptibility. Under the influence of the magnetic field the dipole moments of the atoms and molecules line up with the magnetic field and create a relatively large magnetization. The situation is analogous to the dielectric susceptibility of polar substances, and we can make use of the results we obtained there. The energy of a magnetic dipole μ in the magnetic inductance B is W = μ B, which replaces the energy W = μ in dielectrics. Because we define the magnetic susceptibility in terms of H rather than B we replace χ by χ /1+ χ. The paramagnetic susceptibility is therefore ( ) χ μnμ 1+ χ =, (45) 3kT B provided that the interaction energy is small compared with the thermal energy ( μ B/ kbt << 1). The inverse temperature dependence of the paramagnetic susceptibility was actually discovered experimentally, and is known as Curie s law (for Pierre Curie). In condensed matter the magnetization of an atom or molecule is influenced by the magnetization of its neighbors. In most materials the effect is very small. The exception occurs in ferromagnets. A quantum mechanical effect links the spins of certain electrons and causes them to align parallel to one another. Although the individual interactions are small, the overall effect corresponds to a large magnetization. A completely magnetized crystal of iron has a 6 magnetization = A/m, which is equivalent to a magnetic induction B =. T. This 6 corresponds to a surface current σ = A/m! Due to the coupling, all the spins in a crystal of iron line up spontaneously. However, the crystal subdivides itself into magnetic domains that are arranged in a way that minimizes the energy of the field. oreover, ordinary polycrystalline material consists of many small regions that are separated by grain boundaries and oriented randomly. When a magnetic field is applied to a sample of ferromagnetic material that is initially unmagnetized, the first thing that happens is that the boundaries between the domains move to allow the more favorably oriented domains to grow at the expense of the others. This process is reversible, but at some point the domain boundaries bump up against grain boundaries and other crystal imperfections, and further growth in the magnetization occurs only by displacements of the domain boundaries across the imperfections. These displacements are irreversible in the sense that as the magnetic field is Figure 13 13

14 reduced the domain boundaries are pinned by the crystal imperfections, so the domains do not return to the original configuration. Domain growth saturates at very high fields, and further increase in the magnetization occurs only by rotation of the magnetization to crystal orientations that are otherwise energetically less favorable. This overall behavior is illustrated in Figure 13, where we have plotted the total magnetic induction B= μ ( H + ) as a function of the applied field H. 14