Static Electromagnetic Fields in Matter
7.3 Microscopic Properties of Matter
paramagnetic and diamagnetic,|χm| 1. Whenχm1, the material is ferromag- netic. In this latter case the relationB =µ0(H +M) still holds, butM is usually a very complicated, nonlinear, multivalued function ofH.
It is worth pointing out that to this pointH andD appear to be nothing but mathematical constructs derivable from the fields B and E. The Lorentz force on a charged particle is F = q(E +v×B). This means the fields E and B are detectable through forces they exercise on charges. The magnetic flux through a loop remains Φ = B ·dS. However, as we are gradually converting to the view that fields, possessing momentum and energy have an existence independent of interacting particles, there is no reason to suggest that D and H should have less reality thanE andB. As we recast Maxwell’s equations into their eventual form, H and D will assume an equal footing with E and B. It is worthwhile to shift our perspective on the fields, regardingE andB as the fields responsible for forces on charged particles, whereas H and D are the fields generated by the sources.
In gravitational theory, by contrast, the gravitational mass and inertial mass are identical meaning that the source and force fields are the same.
In the following sections we will briefly discuss the microscopic behavior of materials responsible for polarization and magnetization. We also touch briefly on conduction in 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 have the dipoles oriented randomly (to maximize entropy) in the absence of an electric field. In an exceedingly strong field all the dipoles will align with the electric field, giving a maximum polarization P =np(n is the number density of molecules and
pis the dipole moment of each). At field strengths normally encountered, thermal randomizing will oppose the alignment to some extent. The average polarization may be found from thermodynamics.
According to Boltzmann statistics, the probability of finding a molecule in a state of energy W is proportional to e−W/kT. We consider only the energy of the dipole in the electric field, W = −p·E =−pEcosθ. The mean value of pmust just be the component along E, the perpendicular components averaging to zero.
Hence the mean polarization is
pcosθ=
pcosθe+pEcosθ/kTdΩ
e+pEcosθ/kTdΩ
=p
cothpE kT −kT
pE
(7–34)
a result known as the Langevin formula.
The low field limit of the polarization is readily found. We abbreviate x = pE/kT to write
Figure 7.6: The mean dipole moment of a molecule in thermal equilibrium in an electric field. The low field susceptibility is 13p2E/kT.
pcosθ=p
cothx−1 x
=p
ex+e−x ex−e−x −1
x
(7–35) This expression may be expanded for smallx as
p
2 +x2 2(x+x3/3!)− 1
x
p
(2 +x2)(1−x2/3!)
2x − 1
x
p 1
x+(x2−x2/3)
2x −1
x
=px 3 = p2E
3kT (7–36)
The Langevin-Debye results as well as their smallx limit are plotted in Figure 7.6.
We obtain the polarization by multiplying the average dipole moment by n, the number of dipoles per unit volume:
P = np2E
3kT =χε0E (7–37)
The susceptibility is read directly from (7–37).
7.3.2 Nonpolar Molecules
Taking a simple classical harmonic oscillator model for an atom or molecule with
‘spring’ constantmω20, we find that the displacement of charge at frequencies well below the resonant (angular) frequencyω0 is given by
∆r= q E
mω20 (7–38)
where q and m are, respectively, the charge and the reduced mass of the electron.
The induced molecular dipole moment is then
p=q∆r= q2E
mω20 (7–39)
Figure 7.7: The electric field in the cavity is equal to the field in dielectric augmented by the field resulting from the exposed ends of the molecular dipoles.
We deduce that the polarizability isα=q2/ε0mω20 and the susceptibility becomes χ=nq2/ε0mω02. Thus for molecular hydrogen with its lowest electronic resonance near ω0 1.8×1016 sec−1(λ100nm) at STP (standard temperature and pres- sure), n = 2.69×1025m−3) we obtain χ 2.64×10−4. The experimental value is also (somewhat fortuitously) 2.64×10−4. Such good agreement should not be expected for substances other than hydrogen and helium; generally a sum over all resonant frequencies is required to obtain reasonable agreement. It is worth noting that this value should be fairly good up to and above optical frequencies. By con- trast, the orientation of polar molecules fails for frequencies approaching rotational frequencies of the molecule, typically a few GHz. Thus water has χ 80 (it has a strong dependence on temperature, varying from 87 at 0◦ C to 55 at 100◦ C) at low frequencies, decreasing to χ .8 at optical frequencies. It will be recognized that an exact calculation of the molecular dipole moment will require the quantum mechanical evaluation of the expectation value of the dipole momentψ|#
eiri|ψ, where|ψis the ground state of the atom or molecule involved.
7.3.3 Dense Media—The Clausius-Mosotti Equation
In 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. In gases, where the molecular distances are large, there is little difference between the macroscopic field and the field acting on any molecule. In dense media, however, the closely spaced neighboring dipoles give rise to an internal fieldEi at any given molecule that must be added to the externally applied fieldE. A useful dodge is to exclude the field arising from molecules within some small sphere of radiusRabout the chosen molecule (small on the scale of inhomogeneities ofP but still containing many molecules) and then to add the near fields of the molecules contained in the sphere. As we will show in example 7.8, the electric field in the spherical cavity formed by the removal of all the near neighbors is given by
Ecav=E0+ P
3ε0 (7–40)
The physical origin of the polarization contribution to the field is evident from
Figure 7.7. The calculation of the field from the nearby molecules is more difficult, depending on the structure of the medium. In a simple cubic lattice of dipoles this field vanishes at any lattice point, and it seems reasonable that the field will also vanish for amorphous materials including liquids. Under this condition the polarizing field for the molecule of interest is just the electric field in the cavity (7–40). Therefore, we find
p=ε0α
E0+ P 3ε0
(7–41) The polarization due ton such induced dipoles per unit volume is
P =nε0α
E0+ P 3ε0
(7–42) which, solved forP, gives
P = nα
1−nα/3ε0E (7–43)
The electric susceptibility may now be read from (7–43) χ= nα
1−nα/3 (7–44)
The relationship (7–44) is known as the Clausius-Mosotti equation. When nα is small, as is the case for a dilute gas, thenα/3 in the denominator is inconsequential.
For denser liquids, nαis of order unity and is not negligible.
7.3.4 Crystalline Solids
The 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 this discussion let us assume that the we can replace nα/3 in (7–43) by nα/η with η not very different from 3 to account for the field from nearby molecules.
A number of materials, when cooled in an electric field, freeze in an electric polarization. A piece of such a material is called an electret. Electrets are much less noticeable than magnets because the surfaces very quickly attract neutralizing charges. When the polarization of the electret is changed, however, a net charge will appear on the surface. This change of polarization may be brought about by exceedingly small changes in the physical parameters when nα is near η. Thus, heating a crystal decreases the density, n, giving rise to the pyroelectric effect.
Compressing the crystal increasesn sometimes producing very large voltages. This and the inverse effect are known as the piezoelectric effect.
At first sight it would appear that there is nothing to preventnαfrom exceeding η, resulting in a negative susceptibility, χ. Physically, however, as nαis increased from less than η, the polarization becomes greater, in turn giving an increased local field. If, in small field, nα is larger than η, then the extra field produced by the polarization is larger than the original field producing it. The polarization grows spontaneously until nonlinearities prevent further growth. A material with
Figure 7.8:As the flux through the loop is increased, the electron is tangen- tially accelerated. The change in field resulting from this acceleration must oppose the externally imposedd B/dt.
this property is ferroelectric. On heating the material it is possible to decrease the density until at the Curie point nαno longer exceedsη and the material ceases to have a spontaneous polarization. For BaTiO3 the Curie point is 118◦ C. Slightly above this temperatureχmay be as large as 50,000.
7.3.5 Simple Model of Paramagnetics and Diamagnetics
All materials exhibit diamagnetism. To better understand its origin, let us consider the atoms and molecules of matter as Bohr atoms with electrons in plane orbit about the nucleus. The orbiting electrons have magnetic moments, but because the moments are randomly oriented, no net magnetization results. When a magnetic field is introduced, Lenz’ law predicts that the electron orbits ought to change in such a manner that field from their change in magnetic moment opposes the applied field, yielding a negative magnetic susceptibility. Let us make this observation somewhat more quantitative by considering an electron in a circular orbit (Figure 7.8).
The electromotive force around the loop of the electron’s orbit is given by E= 1
e F ·d=− d B
dt ·dS (7–45)
Replacing 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 obtain
d|p|
dt 2πr=eπr2dB
dt (7–46)
Integrating this expression over time, we have
∆p= er∆B
2 (7–47)
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 electron with massmeis given by
m= ev×r
2 = ep×r 2me
(7–48)
leading to change in magnetic moment in response to the introduction of the mag- netic field
∆|m|=e|∆p×r|
2me = e2r2∆B
4me (7–49)
For several electrons inside an atom, the planes of the orbit will clearly not all be perpendicular to the field and r2 should be replaced by r2cos2θ, where θ is the inclination of the orbit (the component ofH perpendicular to the orbit and the component of ∆malongH are each decreased by a factor cosθ). For an isotropic distribution of orbits,
r2cos2θ=r2
cos2θdΩ 4π
=r2 2
π
0
cos2θsinθdθ=13r2 (7–50) Since the currents inside atoms flow without resistance, the dipoles created by the imposition of the field will persist until the magnetic induction field is turned off again. The resulting magnetic susceptibility isχm=−121ne2r2µ0/me.
Paramagnetism arises when the molecules’ nuclei have a nonzero magnetic mo- ment that attempts to align with the localB in much the same fashion that polar molecules align with E. This interaction tends to be very similar in size to the diamagnetic 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 dipole–field interaction is so much smaller than that for the electric field, large alignments are attainable only at very low temperatures. A few molecules with un- paired electrons such as O2, NO, and GdCl3 have a paramagnetic susceptibility several hundred times larger due to the much larger (spin) magnetic moment of the electron (compared to that of the nucleus).
Although it is tempting to ascribe ferromagnetism to a mechanism similar to that of ferroelectricity, the magnetic dipole–field interaction is so much weaker than the electric dipole–field interaction that thermal agitation would easily overwhelm the aligning tendencies. A much stronger quantum mechanical spin–spin exchange interaction must be invoked to obtain sufficiently large aligning forces. With the exchange force responsible for the microscopic spin–spin interaction, the treatment of 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 are usually very nonlinear, and the magnetization depends on the history of the mate- rial.
7.3.6 Conduction
Qualitatively, a conductor is a material that contains (sub-) microscopic charged particles or charge carriers that are free to move macroscopic distances through the medium. In the absence of an electric field these charges move erratically through the conductor in a random fashion. When an electric field is present, the charges
accelerate briefly in the direction of (or opposite to) the field before being scattered by other relatively immobile components. The random component of the carriers’
velocity yields no net current, but the short, directed segments yield a drift velocity along the field (for isotropic conductors). This leads us to postulate that
J=g(E)E (7–51)
For many materials,g(E) is almost independent ofE, in which case the material is labelled ohmic with constitutive relation J =g E. The constant g is called the conductivity of the material and is generally a function of temperature as well as dislocations in the material. (Many authors useσto denote the conductivity.) The resistivityη ≡1/g is also frequently employed.
A rough microscopic description can be given in terms of the carriers’ mean time between collisions, τ, since v 12aτ = 12q Eτ /m. The quantity qτ /m is commonly called the carrier mobility. Computing the net current density as J = nqv= 12(nq2τ /m)E wheren is the carrier number density, leads us to write the conductivity as
g=nq2τ
2m (7–52)
When a magnetic field is present, we expect the current to be influenced by the magnetic force on the charge carriers. The modified law of conduction should read
J=g(E +v ×B) (7–53)
This form of the conduction law governs the decay of magnetic fields in conduc- tors. Substituting (7-53) into Amp`ere’s law and assuming, that inside the conductor
∂ E/∂tis sufficiently small to ignore, we have ∇ × B =µ J. Taking the curl once more we obtain
∇2B =gµ
∂ B
∂t −∇ × (v ×B)
(7–54) If the conductor is stationary, the equation above reduces to a diffusion equation.
The time-independent simplified Maxwell equations (3–27) are modified in the pres- ence of matter to read
∇ · D =ρ ∇ × E = 0
∇ · B = 0 ∇ × H =J
(7–55)
whereD =ε0E+P andB =µ0(H +M). The first and the last of these equations may be integrated to give, respectively, Gauss’ law
S
D ·dS=
τ
ρd3r (7–56)