Chapter 7 - Static Electromagnetic Fields in Matter 175paramagnetic and diamagnetic,||1. When1, the material is ferromag-mmnetic. In this latter case the relationB=(H+M) still holds, butMis usually0a very complicated, nonlinear, multivalued function ofH.It is worth pointing out that to this pointHandDappear to be nothing butmathematical constructs derivable from the fieldsBandE. The Lorentz forceon a charged particle isF=q(E+v B). This means the fieldsEandBaredetectable through forces they exercise on charges. The magnetic ux through aloop remains F =B dS. However, as we are gradually converting to the viewthat fields, possessing momentum and energy have an existence independent ofinteracting particles, there is no reason to suggest thatDandHshould have lessreality thanEandB. As we recast Maxwells equations into their eventual form,HandDwill assume an equal footing withEandB. It is worthwhile to shiftour perspective on the fields, regardingEandBas the fields responsible for forceson charged particles, whereasHandDare the fields generated by the sources.In gravitational theory, by contrast, the gravitational mass and inertial mass areidentical meaning that the source and force fields are the same.7.3 Microscopic Properties of MatterIn the following sections we will brie y discuss the microscopic behavior of materialsresponsible for polarization and magnetization. We also touch brie y on conductionin metals for the sake of completeness.7.3.1 Polar Molecules (Langevin-Debye Formula)A polar molecule has a permanent dipole moment. If the nearest neighbor inter-action energies are small, a material made of such molecules will normally havethe dipoles oriented randomly (to maximize entropy) in the absence of an electricfield. In an exceedingly strong field all the dipoles will align with the electric field,giving a maximum polarizationP=np(nis the number density of molecules andpis the dipole moment of each). At field strengths normally encountered, thermalrandomizing will oppose the alignment to some extent. The average polarizationmay be found from thermodynamics.According to Boltzmann statistics, the probability of finding a molecule in astate of energyWis proportional toe. We consider only the energy of the- W/k Tdipole in the electric field,W=-p E=-pEcos. The mean value ofpmustjust be the component alongE, the perpendicular components averaging to zero.Hence the mean polarization ispcosed+pEco s/kTpEpcos==pcoth(734)kT - kTpEed+pEc o s/ kTa result known as the Langevin formula.The low field limit of the polarization is readily found. We abbreviatex=pE/kTto write 176Classical Electromagnetic TheoryFigure 7.6:The mean dipole moment of a molecule in thermal equilibriumin an electric field. The low field susceptibility is1p2E/kT.31+e1x- xpcos=pcothx -=p e-(735)xe- exx- xThis expression may be expanded for smallxas2+x1(2 +x)(1- x/3!)1222p-2(x+x/3!)x p2x -x31x- x/3)122p+ (x2x -xpxpE2=(736)3 =3kTThe Langevin-Debye results as well as their smallxlimit are plotted in Figure 7.6.We obtain the polarization by multiplying the average dipole moment byn,thenumber of dipoles per unit volume:npE2P==eE(737)03kTThe susceptibility is read directly from (737).7.3.2 Nonpolar MoleculesTaking a simple classical harmonic oscillator model for an atom or molecule withspring constantm, we find that the displacement of charge at frequencies well20below the resonant (angular) frequencyis given by0qEr=(738)m20whereqandmare, respectively, the charge and the reduced mass of the electron.The induced molecular dipole moment is thenqE2p=qr=(739)m20 Chapter 7 - Static Electromagnetic Fields in Matter 177Figure 7.7:The electric field in the cavity is equal to the field in dielectricaugmented by the field resulting from the exposed ends of the moleculardipoles.We deduce that the polarizability isa=q/emand the susceptibility becomes2200=nq/em. Thus for molecular hydrogen with its lowest electronic resonance2200near1.810sec(100nm) at STP (standard temperature and pres-1 6-10sure),n=2.6910m) we obtain2.6410. The experimental value2 5-3-4is also (somewhat fortuitously) 2.6410. Such good agreement should not be-4expected for substances other than hydrogen and helium; generally a sum over allresonant frequencies is required to obtain reasonable agreement. It is worth notingthat this value should be fairly good up to and above optical frequencies. By con-trast, the orientation of polar molecules fails for frequencies approaching rotationalfrequencies of the molecule, typically a few GHz. Thus water has80 (it hasa strong dependence on temperature, varying from 87 at 0Cto55at100C) atlow frequencies, decreasing to.8 at optical frequencies. It will be recognizedthat an exact calculation of the molecular dipole moment will require the quantummechanical evaluation of the expectation value of the dipole moment| er|,iiwhere|is the ground state of the atom or molecule involved.7.3.3 Dense MediaThe Clausius-Mosotti EquationIn the foregoing treatment, we have tacitly assumed that the electric field experi-enced by a molecule is in fact the average macroscopic field in the dielectric. Ingases, where the molecular distances are large, there is little di erence between themacroscopic field and the field acting on any molecule. In dense media, however,the closely spaced neighboring dipoles give rise to an internal fieldEat any givenimolecule that must be added to the externally applied fieldE. A useful do dge is toexclude the field arising from molecules within some small sphere of radiusRaboutthe chosen molecule (small on the scale of inhomogeneities ofPbut still containingmany molecules) and then to add the near fields of the molecules contained in thesphere. As we will show in example 7.8, the electric field in the spherical cavityformed by the removal of all the near neighbors is given byPE=E+(740)ca v03e0The physical origin of the polarization contribution to the field is evident from 178Classical Electromagnetic TheoryFigure 7.7. The calculation of the field from the nearby molecules is more di cult,depending on the structure of the medium. In a simple cubic lattice of dipolesthis field vanishes at any lattice point, and it seems reasonable that the field willalso vanish for amorphous materials including liquids. Under this condition thepolarizing field for the molecule of interest is just the electric field in the cavity(740). Therefore, we findPp=ea E+(741)003e0The polarization due tonsuch induced dipoles per unit volume isPP=nea E+(742)003e0which, solved forP, givesnaP=eE(743)01- na/3The electric susceptibility may now be read from (743)na=1- na/3 (744)The relationship (744) is known as the Clausius-Mosotti equation. Whennaissmall, as is the case for a dilute gas, thena/3 in the denominator is inconsequential.For denser liquids,nais of order unity and is not negligible.7.3.4 Crystalline SolidsThe near fields on a molecule within a crystal will not vanish for all crystal struc-tures; nonetheless, the net result is generally not large. For the purpose of thisdiscussion let us assume that the we can replacena/3 in (743) byna/withnot very di erent from 3 to account for the field from nearby molecules.A number of materials, when cooled in an electric field, freeze in an electricpolarization. A piece of such a material is called an electret. Electrets are muchless noticeable than magnets because the surfaces very quickly attract neutralizingcharges. When the polarization of the electret is changed, however, a net chargewill appear on the surface. This change of polarization may be brought about byexceedingly small changes in the physical parameters whennais near. Thus,heating a crystal decreases the density,n, giving rise to the pyro electric e ect.Compressing the crystal increasesnsometimes producing very large voltages. Thisand the inverse e ect are known as the piezoelectric e ect.At first sight it would appear that there is nothing to preventnafrom exceeding, resulting in a negative susceptibility,. Physically, however, asnais increasedfrom less than, the polarization becomes greater, in turn giving an increasedlocal field. If, in small field,nais larger than, then the extra field pro ducedby the polarization is larger than the original field producing it. The polarizationgrows spontaneously until nonlinearities prevent further growth. A material with Chapter 7 - Static Electromagnetic Fields in Matter 179Figure 7.8:As the ux through the loop is increased, the electron is tangen-tially accelerated. The change in field resulting from this acceleration mustoppose the externally imposeddB/dt.this property is ferroelectric. On heating the material it is possible to decrease thedensity until at the Curie pointnano longer exceedsand the material ceases tohave a spontaneous polarization. For BaTiOthe Curie point is 118C. Slightly3above this temperaturemay be as large as 50,000.7.3.5 Simple Model of Paramagnetics and DiamagneticsAll materials exhibit diamagnetism. To better understand its origin, let us considerthe atoms and molecules of matter as Bohr atoms with electrons in plane orbitabout the nucleus. The orbiting electrons have magnetic moments, but because themoments are randomly oriented, no net magnetization results. When a magneticfield is intro duced, Lenz law predicts that the electron orbits ought to change in sucha manner that field from their change in magnetic moment opposes the applied field,yielding a negative magnetic susceptibility. Let us make this observation somewhatmore quantitative by considering an electron in a circular orbit (Figure 7.8).The electromotive force around the loop of the electrons orbit is given byE= 1=- dB(745)e F ddt dSReplacing the force by the rate of change of momentum, we evaluate the two inte-grals when the loop is perpendicular to the magnetic field to obtaind|p |dB2pr=epr(746)2dtdtIntegrating this expression over time, we haveerBp=2 (747)We would like to relate the change of momentum to the change of the (orbital)magnetic moment of the electron. The magnetic moment of an orbiting electronwith massmis given byeev rep rm=(748)2 =2me 180Classical Electromagnetic Theoryleading to change in magnetic moment in response to the introduction of the mag-netic fielde |p r |erB22|m|==(749)2m4meeFor several electrons inside an atom, the planes of the orbit will clearly not allbe perpendicular to the field andrshould be replaced byrcos, whereis222the inclination of the orbit (the component ofHperpendicular to the orbit and thecomponent ofmalongHare each decreased by a factor cos). For an isotropicdistribution of orbits,cosd2rcos=r2224prp2=cossind=r(750)212230Since the currents inside atoms ow without resistance, the dipoles created bythe imposition of the field will persist until the magnetic induction field is turnedo again. The resulting magnetic susceptibility is=-ner/m.122m0e1 2Paramagnetism arises when the molecules nuclei have a nonzero magnetic mo-ment that attempts to align with the localBin much the same fashion that polarmolecules align withE. This interaction tends to be very similar in size to thediamagnetic interaction so that it is hard to predict whether any particular sub-stance will have net positive or negative susceptibility. Because the nuclear mag-netic dipolefield interaction is so much smaller than that for the electric field, largealignments are attainable only at very low temperatures. A few molecules with un-paired electrons such as O, NO, and GdClhave a paramagnetic susceptibility23several hundred times larger due to the much larger (spin) magnetic moment of theelectron (compared to that of the nucleus).Although it is tempting to ascribe ferromagnetism to a mechanism similar tothat of ferro electricity, the magnetic dipolefield interaction is so much weaker thanthe electric dipolefield interaction that thermal agitation would easily overwhelmthe aligning tendencies. A much stronger quantum mechanical spinspin exchangeinteraction must be invoked to obtain su ciently large aligning forces. With theexchange force responsible for the microscopic spinspin interaction, the treatmentof ferromagnetism parallels that of ferroelectricity.In metallic ferromagnets, magnetic moments over large distances (magnetic do-mains) spontaneously align. An applied field will reorient or expand entire domains,resulting in a very large magnetization. Materials exhibiting ferromagnetism areusually very nonlinear, and the magnetization depends on the history of the mate-rial.7.3.6 ConductionQualitatively, a conductor is a material that contains (sub-) microscopic chargedparticles or charge carriers that are free to move macroscopic distances through themedium. In the absence of an electric field these charges move erratically throughthe conductor in a random fashion. When an electric field is present, the charges Chapter 7 - Static Electromagnetic Fields in Matter 181accelerate brie y in the direction of (or opposite to) the field before being scatteredby other relatively immobile components. The random component of the carriersvelo city yields no net current, but the short, directed segments yield a drift velocityalong the field (for isotropic conductors). This leads us to postulate thatJ=g(E)E(751)For many materials,g(E) is almost independent ofE, in which case the materialis labelledohmicwith constitutive relationJ=gE. The constantgis called theconductivity of the material and is generally a function of temperature as well asdislocations in the material. (Many authors usesto denote the conductivity.) Theresistivity=1/gis also frequently employed.A rough microscopic description can be given in terms of the carriers meantime between collisions,t, sincevat=qEt/m. The quantityqt/mis1122commonly called the carrier mobility. Computing the net current density asJ=nq v=(nqt/m)Ewherenis the carrier number density, leads us to write the122conductivity asnqt2g=(752)2mWhen a magnetic field is present, we expect the current to be in uenced by themagnetic force on the charge carriers. The modified law of conduction should readJ=g(E+v B) (753)This form of the conduction law governs the decay of magnetic fields in conduc-tors. Substituting (7-53) into Amp`eres law and assuming, that inside the conductorE/ tis su ciently small to ignore, we haveB=J. Taking the curl oncemore we obtainB=g B(v B) (754)2t - If the conductor is stationary, the equation above reduces to a di usion equation.7.4 Boundary Conditions for the Static FieldsThe time-independent simplified Maxwell equations (327) are modified in the pres-ence of matter to readD=E=0(755)B=0H=JwhereD=eE+PandB=(H+M). The first and the last of these equations00may be integrated to give, respectively, Gauss lawD dS=dr(756)3St