(2008-03-20) The densities of magnetic dipoles and electric dipoles, respectively.
AlthoughMaxwell's equationsdo describe electromagnetism both in vacuum and in the midst of matter, it's useful to make a distinction betweenelectromagnetic sources which are either free or bound to matter at the atomic level.
Ultimately, this allows a different presentation of Maxwell's equations (where bound sources are suitably hidden) which can be better suited to a description of electromagnetismwithin the bulk of dense matter. First things first:
In one nice continuous model of matter, the microscopic electromagnetic sources bound to matter are simply approximated by a distribution ofdipoles. (This dipolar approximation usually capturesdirectly the main aspects of things, but it may be awkwardin some cases, including antiferromagnetic materials.) The fields created by those so-called molecular sources is simply superimposed to the fieldcreated by the free charges and currents (henceforth subscripted with a nought).
The total current density j (which appears inthe ordinaryMaxwell equations) is the sum of three terms: free current, magnetization current and polarizationcurrent. Likewise, the total electric charge density is the sum of thefree charge density and the density (-div P) implied by the conservation of bound charges:
j = jo+ rot M + P/t = o divP
Of the three components of the total current density,the magnetization current or bound current (rot M) is the most difficult to fathom. Let's explain...
(2008-03-23)
Mathematically, the fields produced by any smooth distribution ofelectromagnetic sources can be equated to what's produced by somesmooth distribution of electromagnetic dipoles. However, this may involve "unphysical"dipolar densities (P andM) which grow without bounds over time, or over space. Thus, there's a finiteness requirement which helps definemathematically whatportion of the electromagnetic sources ought to be considered free in the above sense.
The same distribution of charges and currentsis obtained if P and M are respectivelyreplaced by the following electric polarization and magnetization densities (for arbitrary fields Z and k of respective units C/m and A) :
P + rot ZMZ/t grad k
(2008-02-24) Maxwell's equations "in matter" feature free charges and currents only.
Defining magnetizationand polarization densities (M &P) asabove, the twoequations of Maxwellinvolving electromagnetic sources become:
div (0E) = o divP rot (B /0) (0E)/t = jo + rot M + P/t
This strongly suggests bundling P with E and M with B as follows...
The electric displacementD is defined asa function of the electric fieldE and electric polarization densityP (in C/m2) namely:
D = 0E + P
Likewise, the magnetic field strengthH (alsocalled magnetizing field ormagnetizing force, in the magnet trade) depends on the magnetic inductionB and magnetizationM (magnetic momentper unit of volume, in A/m) :
H = B /0M
Those definitions give Maxwell's equations the following simple form:
Maxwell's Equations in Matter (1864)
rot E +
B
= 0
div D =
o
t
rot H
D
= jo
div B = 0
t
If and when there's no risk of confusion, the nought subscripts (denoting free charges and currents) may be dropped.
The misleading term "displacement current" for D/t was coined by Maxwell himself, in 1861, when he had to introduce it to make Ampère's Lawcome out right! In a dense medium, some of itcan be interpreted as an actual current (the polarization current P/t ). In vacuum, however, none of this "current" is real;it's simply a mathematical artefact whichmakes Maxwell's equations consistent.
The above equations form a framework which must be supplemented by specificrelations giving D and H in terms of E and B for a particular medium. Such relations are known as electromagnetic constitutive relations.
This may be applied to the D and H fieldswhich result from non-dipolar expressions of bound sources, although the constitutive relations for multipolar expressions of D and H are rarely considered.
(2008-02-24) e and m ) A medium responds to a field with polarization densities (P andM).
An external electromagnetic field can disturb the equilibrium of charges and spins inordinary matter. Some of the ensuing disturbances may be described classically insuch terms that the macroscopic electromagnetic fields appear to obey a modifiedversion ofMaxwell's equations.
A simple way matter can react to a driving electromagnetic field is by creatingelectric and magneticdipoles in its midst with densities P and M, respectively.
The simplest response of matter to a driving electromagnetic field at a givenfrequency is the creation of varying dipoles proportional to the fields. The coefficients of proportionality are scalars in an isotropic medium,but they are generally tensors. The coordinates of those tensors arecomplex numbers whose imaginary part vanishes at low frequency (because the lag time in the response of matter to electromagnetic excitationscan then be neglected).
(2008-02-25) Functions of electric susceptibility and magnetic susceptibility.
In an isotropic nondispersive medium...
(2008-03-03) A susceptibility inversely proportional to temperature.
Permanent magnetic dipoles in thermal equilibrium tend to align themselves with theapplied magnetic field. Such a model of matter yields a magnetic susceptibilitywhich is inversely proportional to the temperature T:
m = C / T
The constant of proportionality C is called the Curie constant. Such a relation was first recorded (in the case of oxygen) by Pierre Curie in 1895.
To account for this, Langevin proposed (in 1905) that molecules havepermanent magnetic moments of magnitude , oriented according to Boltzmann statistics.
(2008-03-02) Materials with negative susceptibilities repel both poles of a magnet.
In 1778, S.J. Brugmans (of Leyden University) noted that bismuth weakly repels both poles of a magnet. In 1827, Le Baillif described the same effect for antimony (see p. 144 ofLight on Electricity by John Tyndall, 1871). Also in 1827, [Antoine César]Becquerel noticed the effect for wood. In 1828,Seebeckreported it for several other substances...
In 1845,Michael Faraday (1791-1867)started to investigatethe phenomenon systematically and called it diamagnetism, because a small rod of a diamagnetic substance (like bismuth) tends to align itself across the magnetic field lines (as each part of the rod tries to get asfar away from the nearest magnetic pole as possible).
It turns out that all substances have diamagnetic propertiesbut the diamagnetic repulsionis usually masked by attractive paramagnetic or ferromagneticproperties, which are much stronger if at all present (especially the latter).
If measured for a given number of atoms ormoles, diamagnetism (unlike paramagnetism and ferromagnetism) does not depend on temperature. Thus, for a given volume of a certain substance, diamagnetism simplyvaries with temperature as thedensityof the substance (this amounts to very little dependence on temperaturefor solids or liquids).
Here is how diamagnetism could be explained in semi-classical terms: The Lorentz force applied to an orbiting electron changes its centripetal accelerationand modifies its orbital magnetic moment in a direction opposing the applied external magnetic field. The size of the orbitswould have to be obtained from quantum considerations.
Classical Diamagnetism (Paul Langevin, 1905)
= N
0 q 2
iri2
6 m
Paul Langevinobtained that result in 1905 by a classical argument which takes into accountthe Larmor precession of each electron about the applied magnetic field. In the above formula, N is the number of atoms per unit of volume, q and m are the charge and the mass of the electron. The summation extends over all the electrons in each atom to yield thesum of the mean squares of their orbital radii.
(2008-03-05) Levitation without active devices defies Earnshaw's Theorem (1842).
In 1842,Samuel Earnshaw(1805-1888) proved that permanent magnets are unable to produce stable levitation. This theorem can be extended to include ferromagnetic or paramagnetic materials.
In 1845, Faraday rediscovered diamagnetism. In 1847,Lord Kelvin recognizedthat Earnshaw's theorem would not apply to diamagnetic materials. Static magnetic levitation is indeed possible if diamagnets areinvolved.
Because of their negative susceptibility,diamagnetic bodies seek equilibrium at a minimum of the magneticfield... Although diamagnetic effects are small, they can be large enoughto oppose Earth gravity (for thin shim of pyrographite) or, at least,combine with stronger magnetic fields to obtain stablelevitation in midair, at room temperature.
Martin D. Simon designed a diamagnetic levitation stand (seevideo) at UCLA which included two disks of pyrolytic graphite (PG). According toMeredith Lamb,it was on January 17, 2000 that those PG disks were first used to make thin floatersthat could levitate over a pattern of alternating poles formed by four neodymium block magnets.
We are not discussing here the use of electromagnets (which consume some power) to achieve the illusion of stabilityby a dynamic control of the magnetic field, using sensors which monitor the position ofa permanent magnet floating over another one. This does have greatentertainment value,though.
(2008-06-23) At room temperature, this is the most diamagnetic substance known.
Pyrolytic graphite (PG or PyC) is a layered form of pure carbon with a density between 1.7 and 2.0.
It's obtained from short hydrocarbon gases (mostly methane or propane) by chemical vapor deposition (CVD) at high temperature (up to 2000°C) under a low partial pressure (10 mmHg or less) which prevents the formation of carbon black (this can be achieved by dilution in an inert gas, like helium, argon, nitrogen or hydrogen). The process is fairly slow: A thickness of just 1 mm requires typically 48 hours (but as little as 1 hour for low-grade stuff).
In medical applications (replacement joints and heart valves) material coated with pyrolytic carbon is marketed under the name of PyroCarbon by companies like AscensionOrthopedics (Austin, Texas) andNexa (the Tornier group acquired the relevant implant technology from the Frenchfirm BioProfile). PyroCarbon was first used to manufacture heart valves in 1968.
For experimental purposes, pyrolytic graphite is available from SciToys.
(2008-03-18) Classical diamagnetism and paramagnetism cancel each other...
As part of his doctoral dissertation (Copenhagen 1911) Niels Bohr (1885-1962) introduced aclassicalargument which would later be developed by Hendrika JohannaVan Leeuwen (1887-1974) in her own doctoral dissertation (Leiden 1919, Journal de Physique 1921) under the guidance of H.A. Lorentz and Paul Ehrenfest.
The remark, known as the Bohr-Van Leeuwen Theorem, is that the ordinary laws of classical and statistical physics (outside of quantum theory) imply that an external magnetic field will notinduce any net magnetization in a set of moving electric chargesat thermal equilibrium. Thus, classically, the diamagnetic and paramagnetic effects cancel each other exactly !
Of course, this flies in the face of experimental results and merely goes to showthat classical physics by itself cannot produce an adequate theory of magnetism. Some form of quantization is needed to resolve this and other issues and reconciletheory with experiment (the magnetic dipoles postulated byLangevin in his theory of paramagnetism can be construed asa good substitute for such a quantization).
JohnHasbrouck Van Vleck (1899-1980) discusses the theorem in Theory of Electric and Magnetic Susceptibilities (1934). In hisNobellecture (1970) he argues that this particular pointmay have been one of the main motivations which led Niels Bohr himself to propose quantum conditions for the structure of the atom, in 1913 (thereby founding the so-called Old Quantum Theory).
: (The following argument is based on whatRichard Feynman saysin section 34-6 (vol. 2and vol. 3) of The Feynman Lectures on Physics.)
A system of moving charges has a probability proportional to e-U/kT to have a state of motion of energy U at thermal equilibrium (temperature T). This energy U includes only the kinetic energy of the particlesand their electric potential energy. It's unaffectedby the existence of any additional magnetic field.
Thus, the exact same statistical distribution of charge velocities is achievedat thermal equilibrium whether an external magnetic field is applied or not.
If we assume, as we do within a strict classical framework, that magneticmoments are entirely due to the circulating currents formed by movingcharges, then we come to the conclusion that no magnetic moments at all are induced. In other words, the net magnetic susceptibility is zero!
(2008-03-18) Electromagnetic interactions of moving charges and magnetic dipoles.
To avoid the blatant contradiction of experimental evidence embodiedby the aboveTheorem of Bohr and Van Leeuwen,a semiclassical discussion of magnetism should at least allowthe existence of fundamental magnetic dipoles (elementary particles endowed with a magnetic moment notdue to a rotation of electric charges). The energy of such a beast does depend on the magnetic field it is subjected to.
(2008-03-05) Magnets, hysteresis, Weiss domains and Bloch Walls.
The Weiss magneton (empirical molecular magneton) is roughlyequal to 1.853 10-24 J/T (or about 20% of aBohr magneton).
In ferromagnetic materials,the magnetization of the medium itself can create a magneticfield which greatly exceeds a typical external field. Furthermore, a remanent magnetization may exist in theabsence of any external field. In 1906-1907,Pierre Weissdiscovered that such materials are always subdividedinto variously oriented domains where the magnetization has its full saturation value. Those domains are now known as Weiss domains. To explain this, Weiss proposed the so-called molecular field hypothesis whereby molecules could be endowedwith tinymagnetic dipole moments whichtend to align with their neighbors within each Weiss domain. The boundaries between Weiss domains are called Bloch walls, in honor of the Swiss physicistFelix Bloch (1905-1983;Nobel 1952) who investigated them.
Saturation Magnetization :
Ferromagnetism is such that the magnetic moments created at the atomic leveltend to be aligned in each Weiss domain. It's useful to estimate what themaximum magnetization can be under such conditions. The contribution of each atom in the material is mostly due to its electrons,either from their orbital motion or their intrinsicspinwhich are respectively quantized (nonrelativistically) to a wholeor half-integer multiple of theBohr magneton. In the main, we neglect theinterestingmagnetic effects due to the nucleons, which are 3 orders of magnitude smaller.
The magnetic energy density of a ferromagnet (in joules per cubic meter or, equivalently, in pascals) is the product of the remanent fluxdensity (i.e., the magnetic inductionB,in teslas) by the density of magnetization M. A non-SI unit commonly used in the trade for this is the megagauss-oersteds (MG.Oe) :
1 T = 104 G 1 A/m = 4 10-3 Oe 1 T.A/m = 1 J/m3 = 1 Pa = 40 G.Oe = 125.6637... G.Oe Conversely, 1 MG.Oe = 106/40 Pa = 7957.747... J/m3
For example, the theoretical maximum for a neodymium-iron-boron magnet ("NIB" or "neo") is quotedto be 64 MGOe while the best available grade is currently 54 MGOe (that's what the designation "N54" means). In more readable SI units,those numbers correspond respectively to 0.51 MPa and 0.43 MPa. In other words, an N54 neodymium magnetpacks ideally a magnetic energy of 0.43 J per cubic centimeter.
(2008-03-09) Magnetic multipoles dominate when adjacent dipoles cancel.
Antiferromagnetism occurs below a certain transition temperature, called the Néel temperature TN , which varies fromone antiferromagnetic material to the next:
(2008-03-09) Several types of dipoles may partially cancel each other in a crystal.
The most famous example of a ferrimagnetic substance is lodestone (whichGilbertspelled loadstone ) which is the traditional name formagnetite, the most magnetic substance amongnaturally occurring minerals. In fact, magnetism derives its namefrom magnetite, not the other way around...
Magnetite (Fe3O4) is also called ferrous-ferric oxide. An expanded chemical formula (FeO, Fe2O3) better reflects the structure of its crystal...
In the lattice, ferrous ions (Fe++) and ferric ions (Fe+++)tend to have antiparallel dipole moments. However, the ferrous and the ferric magnetic moments arenot equal in magnitude, so there's a net localmagnetization.
(2008-04-04) In an active crystal, light polarization is rotated by a magnetic field.
Faraday effect (transmitted beam) and Kerr effect (reflected beam).The magnetization may be polar (perpendicular to the diopter) longitudinal(parallel to both the diopter and the plane of incidence) or transverse(parallel to the diopter, perpendicular to the plane of incidence).
The first nonlinear-optical effect was the quadraticKerr effect (quadratic electro-optic effect, QEO effect)described in 1875 by the Reverend John C.Kerr (1824-1907).
In 1893,Pockels(1865-1913) discovered that a birefringence proportional to the applied field exists in some crystals(Pockels Effect).
(2008-02-24) . Ohm's law. Current density (j) is proportional to the electric field (E) : j = E
In an ideal conductor (a superconductor) the conductivity is infinite and, therefore, E = 0. There's no electric field and the magnetic field doesn't change.
Ordinary substanceshave a finite conductivity which varies with temperature.
Conductivity , in S/m = (.m)-1
Substance
0°C
20°C
Silver (Ag)
6.82 107
6.301 107
Copper (Cu)
6.48 107
5.96 107
Gold (Au)
4.88 107
4.52 107
Aluminum (Al)
4.14 107
3.78 107
Tungsten (W)
2.07 107
1.89 107
Zinc (Zn)
1.67 107
Brass
1.5 107
Nickel (Ni)
1.44 107
Iron (Fe)
1.167 107
1.013 107
Lead (Pb)
5.21 106
4.81 106
Mercury (Hg)
1.044 106
Graphite (C)
6.10 104
Pencil Lead
1.869 102
Glass
3.0 10-9
Diamond (C)
1.0 10-12
Polyurethane
1.0 10-15
Sulfur (S)
5.0 10-16
Fused Quartz
2.0 10-16
(2012-08-05) Longitudinal and tranverse relaxation times for magnetization.
A phenomenological description...
(2015-04-18) Interaction of light with chiral molecules. The Cotton effect.
In 1953, a prestigious yearly prize for promising French researchers wascreated to honor the memory of Aimé Cotton. That prize was awarded in 1971 to Serge Haroche (b. 1944) for his doctoral work. Haroche went on to earn the NobelPrize in Physics, in 2012.
