Is B or H the Fundamental Magnetic Field

Is B or H the fundamental magnetic field ?

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Johannes Hendrik Cloete Department of Electrical and Electronic Engineering

University of Stellenbosch, Stellenbosch 7600 South Africa

emaii: [email protected]

So Maxwell had four field vectors - E, D, B and H - the D and H were hidden ways o f not paying attention to what was going on in the material. People tended to think that H was the magnetic field. But , as we h,ave seen, B and E are physically the fundamental fields, and H is a derived idea. R. P. Feynman, The Feynman Lectures on Physics. [l, pp. II-32-4 and 11-36-12]

B y comparing the interaction product quantities with the corresponding quantities derived from th,e Lorent:- force laws and the laws of mechanics, we shall conclude that the E-H formulation is the correct formulation of macroscopic electrodynamics.

R. M. Fano, L. J . Chu and R. B. Adler, Electromag- netic Fields, Energy and Forces, [2, p. 4861

What’s i n a name 1 That which we call a rose B y any other word would smell as sweet. William Shakespeare] Romeo and Juliet, II.2.43-44

To be or not t o be: that is the question.

William Shakespeare, Hamlet, Prince of Denmark, Act 111, Scene I

Introduction

Circuit theory based on Kirchhoff’s laws is not powerful enough for the purposes of many engineers. They must understand] model and analyze electromagnetic phenomena a t the level of Maxwell’s theory using quasistat- ic or fully dynamic formulations. Problems which require Maxtvellian models arise in the design of electric and magnetic machines; high voltage engineering; radio frequency and microwave engineering; antenna design; propagation and scattering predictions; and the design of dielectric, ferroelectric, ferrimagnetic, ferromagnetic, semiconductor and superconductor materials and devices.

Maxwell’s differential equations for the electrodynaiiiics of matter, e.g. [3, pp. 126-1271 and [4, p. 3111, may be written in the form

curl E + d B / d t = 0 (Faraday’s law)

curl H - a D / d t = Jfree (Amp&re’s law) d i v D = pfree (Gauss’s law) div B = 0 (No magnetic monopoles)

Two of these, the laws of Faraday and the divergence law for B are homogeneous, thus source free. The other two, the laws of Ampkre and Gauss are in general not source free. The free charge density, pfree (C/m3), in Gauss’s law. and the free current density, Jfree (A/m’), of Amptre‘s law, are related by the law of charge conservation divJf,,,+ dpfree/dt = 0.

The charge and current densities are free in the sense that they are associated with the movement of charge carrier- s which are not bound to stationary atoms or molecules [4, pp. 172-3, 257-8, 309-111. The free current density in linear, isotropic matter can often be modelled as a diffu- sion current which satisfies Ohm’s law according to the constitutive relation Jfree = uE, with U the conductivi- ty of the medium and E the electric field. For unipolar convection currents like an electron, proton or ion “beam” Jfree = lpfreeIpE is a good model, with ,U the mobility of the charged particles 15, p. 2521.

This paper is concerned with the meaning of the field vectors E and D, which are associated with electricity, and B and HI which are associated with magnetism, in the Maxwell equations stated above.

For linear, isotropic, electrically polarizable matter the electricity vectors E and D are often linked by the constitutive relation D = [E and it may be tempting to conclude that E and D represent the same electrical phenomenon with the only difference being the “factor” E [4, pp. 174- 1’751. However E, as defined in Section 2, is clearly the electric field, and as shown in Section 3 the D field is a secondary or auxiliary field which may be useful when dealing with dielectrics. There now seems to be unan- imous agreement amongst both engineers and physicists about the primary status of E and the secondary status of D [6, p. 2711.

N o t so in the case of the B and H fields which are used in the description of magnetic phenomena. Here confusion and disagreement still reigns, certainly among undergraduate students hut even among eminent and experienced engineers and physicists. There seem to be two camps [7]. In the first. where physicists are in the majority, it

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is taken as axiomatic from the microscopic Lorentz force law that B is the primary magnetic field; then H is derived as an auxiliary field which is useful in the description of phenomena involving magnetically polarizable matter. The second camp, of mainly electrical engineers, thinks that H is the primary magnetic field which “causes” the B field in magnetizable matter.

This s ta te of affairs, more than a century since Faraday’s field concepts were first presented in mathematical form by Maxwell, is surprising a t first. However, although some of the arguments for H as the fundamental field are superficial there is also a deep theory formulated by Chu a t hlIT in the fifties [2], which has been carefully supported by Haus and Penfield, e.g. [5, pp. xxii, 509-5101.

In t,his paper the arguments in support of the primacy of the B field are given first, a t an introductory level, by drawing in particular on the thinking of Feynman’ [l], Purcell’ [8] and Raab [9]. Then, in Section 6, Chu’s model in which the H field is considered to be the fundamental magnetic field is briefly outlined.

Actually a t issue is whether the classical electromagnetic field should be represented by the pairing E - B or E-H. (Ot,lier possibilities, such as the representation in terms of die scalar electric potential and the magnetic vector potential [l , p. 11-15-14]) will not be considered here.)

Does it really matter which point of view is held ? Cer- tainly ! For scientific and philosophical reasons students of electromagnetic theory, whether they be aspirant engineers or physicists, deserve to be taught which model is most consistent with the laws of nature as understood a t present. If the two models are equivalent, or have com- plementary strengths and weaknesses, then this must be clarified. If one of the representations is significantly infe- rior then i t should be rejected in our teaching and practice.

2 The Lorentz force law defines the E and B fields

The E and B fields are defined by the microscopic force equation

which is associated with the name of Lorentz3. This turn of the century law of physics, which has been extensively tested by experiment and practice, attributes the force f experienced in an electromagnetic field by a particle with charge q and velocity v partially to the influence of the electric field E and partially to the magnetic field B, [l, p. 11-13-1 and p. 11-15-14], [8, p. 2081.

Thus the Lorentz force law does two things [ll , pp. 71- 741. It provides the link between classical mechanics and

f = qE+ qv x B (1)

Nobel physics prize, 1965: quantum electrodynamics. 2Nobel physics prize, 1952: nuclear magnetic resonance. Rindler [lo] provides interesting historical background on the

Lorentz force law by tracing its magnetic component back to Heaviside.

electromagnetic theory by allowing us to predict the mo- tion of charged particles such as electrons, and it gives a unique definition of the electric and magnetic fields, a,nd hence the symbols E and B. in terms of the partial forces experienced by the particle. (In mat ter the macroscopic E and B fields are related to the wildly fluctuating microscopic fields by spatial averaging over an appropriate macroscopic volume. Time averaging is not necessary. [ la , pp. 226-2291.)

Lorentz’s law is consistent with Einstein’s special theory of relativity since i t can be derived from Coulomb’s law for the force between stationary electric charges, and Ein- stein’s two postulates [13, pp. 224-2311. It is classed as a “classical” law because i t fails to explain certain quantum mechanical phenomena, as demonstrated by the Bohm- Aharanov double slit experiment, and has to be replaced by a more general force law in terms of the electric scalar and the magnetic vector potentials [l, pp. 11-15-8 to 141.

Clearly, in the International System of Units (SI), the electric Lorentz force term, f, = qE, defines the unit of the electric field, E, as being newton/coulomb (N/C). By introducing the concept of electric potential and considering the potential energy of a charged particle in an electric field the unit for the electric field can also be shown to be volt/meter (V/m). The rnagnet.ic Lorentz force term, f, = qv x B, defines the unit of the magnetic field, B, as newton/(coulomb meter/second). This physically mean- ingful but cumbersome unit is by definition equal to the tesla (T), the magnetic field unit which h a replaced the weber/meter2 (Wb/m2) in modern practice. The gauss (G) an early, non-SI, unit is convenient when working with weak magnetic fields. (The earth’s magnetic field is approximately 0.5 gauss and 1 gauss =

The microscopic Lorentz force is manifest in a great variety of devices: in the ubiquitous cathode ray tube - J. J . Thomson’s “cathode ray” being a unipolar current of electrons - the beam direction is controlled by electric and magnetic fields; in magnetrons and travelling wave tubes; in the electron microscope; in the workings of diodes, tran- sistors and Hall effect magnetic field sensors; in particle accelerators such as the cyclotron; and, most notably, in moving free electrons through the vast global network of metallic conductors to make energy available to humans in the form of heat, light or mechanical work.

In the latter context electrical engineers are familiar with the macroscopic form of the magnetic Lorentz force law,

tesla.)

AF, = I x BAL,

which is used to quantify the macroscopic force on a seg- ment, of length AL, of a current carrying conductor [l, p. 11-13-31. It forms the basis for our understanding of electrical machines and generators and is derived from the fundamental microscopic law.

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3 Gauss?s law in dielectrics and For linear, isotropic matter the reiatioiisliip I>etween P and E may be written in the forin the D field

P = toxeE Most, if not all. modern textbooks on electromagnet’ism for undergraduate courses in physics. and electrica.1 and electronic engineering, agree tha t E is tslie primary electrical field while D is an induced field which is used t,o describe the interaction between the primary field, E, aiid electrically polarizable matt,er. The int,eraction resdts in electrical dipoles and a.lso the higher niultipoles (electric quadrupoles, electric octopoles etc.) being induced in the medium. In a general model t,lie contribut#ion of the higher order multipole moment,s [8. pp. 352-3551, [9], [la, pp.

by introducing the macroscopic elect,rical susceptibility, -ye. This equation also makes it explicitly clear that E is t,he cause of P. and hence D. (The elect,rical susceptibility, x e , is related to the microscopic elect,rical polarizability of the constituent atoms or niolecules t8hrough the (Ilausius- Mossoti, Lorentz-Lorenz and other niodels for the local electric field [l, p. 11-11-7’. 11-32-71, [12, pp. 152-1551. [13. 11. 3431, [14! p. Er].) Thus for linear5. isotropic dielectrics D can be written. in the electric dipole approximation, as

. . . .

136-146, 226-2351 to the polarization density must also be

Section 5. taken into account, but a discussion of this is deferred until D = EOE + P == t o ( 1 + xe)E = tE = cot,.E.

Thus the following development is based on the electric dipole approximation in which. from a ina.croscopic viewpoint, the interaction is modelled as a nett dei1sit.y of elec- t,rical dipole moments, P, which is induced in matt,er by t,he primary field.

By distinguishing explicitly between t,he free and the bound charge densities, t.he lat.ter associa.ted with polarization, Gauss’s law can be written in the form

Then the secondary field D, is defined by the linear super- posit,ion equat,ion

and it satisfies a modified form of Gauss’s law.,

cliv D = div (EOE + P) = pfree,

explicitly involving only the free charge density. In the electric dipole approximation the polarization charge density appears implicitly t,hrougli the relat.ion

A variety of names for D abound in the literature. Ex- amples are the ”electric flux density”, the “free electric flux density”, the “electric displacement”, the “electric displacement flus density”. These terms, inspired by Maxwell’s “ d i ~ p l a c e m e n t ” ~ current [G, p. 41 11, frequently cause more confusion than enlightenment , aiid the practice of Shadowitz [G, p. 2693 and Purcell [S, pp. 432-4331 in simply referring to the “D field” may be preferable. A common name for the E field is the “electric field inten- sit,y”, but liere the term “electric field” or just “E field” will be used in accordance with modern practice.

*See Purcell’s footnote regarding Maxwell being led astray by his displacement concept [8, p. 3821.

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where the electrical “I>eriiiit,t,ivity” is defined in ternis of t,he electrical susceptibilit>- as E = tOcr = C O ( 1 + ,ye) of the medium.

The constitutive relation D = c,JE,.E mliicli is coniiiioii- ly used in engineering practice unfortunat,ely olmxres t,he fundament,al phxsical role of the polarization density P. Thus nmny student,s and engineers may not fully appre- ciate that D is a composite field iiivolving the primary field, E, and the induced polarizat,ion density, P. This is a pity since. besides being responsible for increa.sed capac- itance as originally explained by Fa.raday, the polarization mechanism is exploited in a variety of important scientific and engineering applicat ious such as electrophoresis a.ncl the industrial scale electrical separators which break down oil-water emulsions and countless other mixtures [15]. The physical reality of the forces result,ing from the polarization mechanism have been clearly demonstrat,ed by Melch- er [16, pp. 127-1301, [5, pp. -294-495, 514-5161. Neverthe- less, despite hiding the role of the polarization charge density, there is no implicit physical misrepresent,at,ion of t,he macroscopic electric polarizability phenomenon or mecha- nisin in the use of D = cE provided the tlielect,ric may be taken to be linear and isotropic.

Authors like Feyiiinan [ l , pp. 11-10-7, 11-32-4 to 51 (see quote on the first page), Purcell [8, pp. 381-352, 4331, Shadowitz [6 , pp. 269-2721 and Griffiths (4, p. 2601 do not. consider D to be very useful physical concept,. For esam- ple, in practical electrocluasist,atics the electrical potential difference between the electrodes of capacitive systems is usually the independent varkble; and E is directly con- trollable by the potential difference. The resulting distri- bution of free charge, which determines div D, is usually incidental and not a design parameter.

However the result div D = 0 when pfree = 0, with the resulting integral form fA D . da = 0, does provide a help-

’The electric susceptibility and permittivity concepts are often implicitly extended into the non-linear, and possibly hysteretic regimes, of dielectrics such as the ferroelect.ric materials but then it is impossible to talk of the susceptibility because i t is dependent on the amplitude of the electric field. Similarly for the magnetic susceptibility and the permeability of magnetic materials discussed in Section 4. 356

ful intermediate step when solving problems involving dielectrics where the free charge density in the material is negligible; especially so if the probleiii is highly syniniet- ric. Of par t icuhr utility in boundary value problems is the coiitiiiuity of the normal component of D a t the interface between two different dielect,ric media i n the absence of free surface cha.rge.

Finally, it is important to note that D = EOE,E is a frequency domain representation of the dielectric properties of a material. It is implicit, that the medium is being excited time harmonically at. a certain frequency and in this representation the permittivity E, ( U ) is complex to account for dissipation, arid frequency dependent because all physical materials are dispersive. There is a t present great! interest in the behaviour of materials sub- ject, to impulsive excitation and a variety of phenomena are being studied in the tinie domain using experimental, analytical aiid numerical methods. Time domain analysis followed by Fourier transformation of the results has also become a popular method of obta.ining frequency domain data. The representation of D as the product of E with the frequency domain permitt,ivity leads to a cumbersome time domain convolution integral representation in which the “permittivit,y impulse response” of the material is the Fourier transform of cT(w). N o such coniplicat.ions arise in transforming the funda.menta1 form D = toE + P to the time domain after invoking Lorentz or Debye models. In particular this representation now seems t,o have signifi- cant adva.iitages in the numerical modelling of dispersive ma.t#erials by means of finite difference t,ime-doniaiu algo- rithms [17], [18].

4 Amp&re’s law in magnetizable matter and the H field

Whereas clarity exists rega.rding the relationship bet.ween E and D there are, as mentioned in the Introduction, two schools of thought regarding the relationship between the B a i d H fields. One6 superficial argument is based on the a.pparent analogy between the popular constitutive relations D = tE and B = pH which seem to pair E with H; hence the conclusions that H is t,he fundamental magnetic field, and that the magnetic “permeability” p is analogous to the electrical “permittivity“ E.

In this Section it is taken as a basic postulate that the Lorentz force law defines B as the fundamental ma.gnetic field. I t is also assumed7 tha t only magnetic “dipoles” contribute to the magnetization density M. The general relationship between B and H is then developed from Am- pkre’s law [l, pp. 11-36-1 to 61, [4, pp. 251-264, 304-3111, [G, pp. 306-3211, [8, pp. 423-4371 and [13, pp. 408-4121.

“Another is that the ratio E I H has the unit of impedance, QED. ’The approximation in Section 3 that P is only due to electric

dipoles, and which resulted in the definition D = coE+P, seems to be consistent with the magnetic dipole assumption of this Section. However, as discussed in Section 5 , this is not the case because electric quadrupoles contribute to the same order as magnetic dipoles [9, 191.

.-

After introducing the notion of inagnetlic susceptibility and permeabilit,y, the appropriat,e form of the magnettic con- d t u t i v e relatioil will be shown to be not B = p H but H = p L - l B .

Ampkre‘s law in niatt.er can be stated in the differential form

curl B - p o ~ o d E / B t = pO.Jtotal

xhicli involves the B a i d E fields a t a macroscopic ..point’’. the vacuum parameters E O and po, arid Jtotal,

the total current density at t,liat point. In matter which contains free charge carriers, and which has dielect,ric and magnetic properties. the total current density may be de- composed and expressed as t.he sum of the free current density. Jfree, and t.he bound current density, J b o u n d . In turrl t>he bound current densit.y can be considered to be the suni the polarization current density, Jpo19 and the magnetization, or -linpi-rian, current density, Jmag. Thus

Jtotal = Jfree + Jbound Jfree f Jpol + Jmag. ( 5 )

An engineering example of a material in which all three t.ypes of current. densities might flow is a microwave ab- sorbing comp0sit.e consisting of a fine magnetite powder embedded in a polyurethane host with die magnetrite particles packed close enough to make cont’act with each other arid thus allow ohmic currents to perco1at.e.

So Ampi-re’s law i n matter can be written in the general form

By pursuing the development in Sectioii 3 the polarization current density, in t,he electric dipole approximation, can be relat,ed t.o the electric polarizatiori density P by

It can also be shown that , in the magnetic dipole approx- ima.tion, the magnetization current density is relat,ed to t,he magnetization density, M. by

Jmag = curl M (8)

Thus Ampkre’s law can be rearranged into the form

curl ( p ; l ~ - M) - ~ ( E , , E + = J~~~~

Now the H field can be defined as

in terms of the B field and the iiiagnet,ization density, M.

Also, the term EOE + P on t,he right hand side may be replaced with the D vector which was defined in Sectmion 3. This yields Ampkre’s law in terms of H and D as given in the Introduction

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From the viewpoint developed in this Section the status of H = p ; l B - M is no different from tha t of D = EOE + P. Both are considered to be auxiliary, or induced, fields which conveniently summarize, but also obscure, the interaction between the fundamental fields, B and E, and matter .

This view does not detract from curl H - d D / d t = Jfree being a powerful statement of Amptre’s law. This is par- ticularly so when dealing with engineering applications involving ferromagnetic and ferrimagnetic materials a t frequencies which are sufficiently low to regard the system as magnetoquasistatic (MQS) in the sense that aD/at x 0 [5, pp. 70-711. The utility of the MQS approximation curl H x Jfree is evident by considering the integral form

H . ds M S, Jfree . da = Ifree.

It is now clear tha t the line integral of H around any closed path C ; whether that path passes through a =-linear magnetic material like iron or not, is dependent solely on the free current enclosed by the path. The material properties are irrelevant. This fact is of great utility in dealing with the engineering applications of ferromagnetic and ferrimagnetic materials, with their non-linear and hysteretic properties, in the design of the magnetic circuits of elec- tromagnets, transformers or machines, e.g. [l, pp. 11-36-6 to 111. Here the free currents and their paths are primary design variables. Purcell [8, p. 4331 and Griffiths [4, p. 2601 nicely discuss the utility of H in the context of the free current being under the control of the designer. They also use the opportunity, as also mentioned in Section 3, to juxtapose the D field as being of minor importance because in electrostatic or electroquasistatic systems the free charge density is rarely a design variable. Rather, it is the electric potential, and thus the electric field, which is directly controlled.

Nevertheless, despite its practical utility in solving magnetic circuit problems, H is the auxiliary magnetic field of the first camp’. Thus their preferred nomenclature is that B is the “magnetic field” instead of the “magnetic flux density”, or “magnetic induction” of historical tra- dition. Also, for H just the name “H field” instead of the traditional “magnetic field intensity” [l, pp. 11-32-4 to 5, 11-36-12], [4, pp. 260-2611, [6, pp. 312-3211, [8, pp. 431-4371,

By analogy with the electric susceptibility x e r Section 3, the magnetic susceptibility can be defined by relating the magnetization M to the B field by the relation M = pi l&B. By this definition H is expressed in terms of B as H = p i l ( l - x:,)B. However i t is convention to define the magnetic susceptibility as M = xmH, which yield- s the constitutive relation H = p;’(l + xm)-’B. The two definitions of magnetic susceptibility are related by (1 - x:,~) = (1 + x,,~)-‘ and since (1 + x,~)-‘ w (1 - x m ) if << 1 it is evident tha t for paramagnetic and dia- magnetic materials the two definitions are essentially e-

‘See the interesting remark attributed to Sommerfeld concerning Maxwell and H [4, p. 260, footnote]

quivalent. This is not the case for ferromagnetic or ferrimagnetic materials. For them it is convenient to use the conventional definition M = xmH because, as shown above, the H field under magnetoquasistatic conditions is completely determined by the free currents and is independent of the material properties, whereas the Ampkrian current density Jmag = curl M, which is dependent on the material properties, also contributes to the B field. Then, after defining the relative “permeability” of the material as p,. = 1 +xm the magnetic material constitutive relation is

Much confusion as to the status of B ,uis-a-vis H arises because this relation is commonly presented as B = pH, a form which Elliott [7] regards as being an historical ac- cident, and because the magnetization properties of ferromagnetic and other non-linear and hysteretic material- s are often presented as B-H graphs a i d not as B-Ifree

graphs. See Shadowitz [ G , pp. 312-3211 and Purcell [8, pp. 422-423, 435-4371 for additional perspectives on magnetic susceptibility and permeability.

H = p;’(l+ xm)-lB = po -1 p,. -1B- - p- ’B .

5 Maxwell’s equations and multipole moments

By using Equations (2) and (6) to explicitly show the con- bributions of the bound charge and current densities the D and H fields can be eliminated from the laws of Gauss and Amptre and Gauss in Section 1, e.g. [4, pp. 307-3111. Thus

c u r l E + d B / d t = 0 curl B - po€odE/dt = ,%(Jfree 4- Jbound)

div EoE = Pfree Pbound

div B = 0

Then from Equation (4) it would seem that the bound charge density is Pbound = -div P, while Equations (5), (7) and (8) suggest tha t the bound current density is Jbound = aP/& + curl M. These conclusions are however flawed. Raab and his colleagues have proved [9] that if the contribution of magnetic dipoles are taken into account, as they are in the expression for J b o u n d by the term curl M, then the electric quadrupoles must also be accounted for. Thus, using tensor notation, the correct form for the i- component ( i stands for z, y or z ) of the bound current density in the magnetic dipole-electric quadrupole approximation is [9, eqn. 341

Jhound, Pi + f i j k v j h l k - $vjQij -k ‘ ’ ‘

This is just J b o u n d = aP/at + curl M with the contribution of the time varying electric quadrupole density, Qij addedg. If this is not done the resulting hIaxwell’s equa-

’The dot notation is used to indicate partial derivatives with respect to time, and c l j k , is the alternating tensor which allows vector products to be written in tensor form.

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tions are not origin independent, in violation of a fundamental requirement of physics. (The laws of physics may not depend on an arbitrary choice for the origin of the reference frame.)

The underlying reason for the violation of t*ranslational symmetry is t ha t the relative magnitudes of the multipole contributions to an electromagnetic effect are ordered as [SI, [ l a , pp. 391-401 and Chapter 161

electric quadrupole magnetic dipole >> electric dipole >>

Thus electric quadrupoles contribute effects to the same order as magnetic dipoles. The correct forms for the D and H fields to this order are [9, eqns. 37 and 381

Here the expression for H corresponds exactly to Equa- tion (9) but the expression for D differs from Equation (3) because the contribution of the electric quadrupole density was neglected in the latter. Raab and Cloete [20] have shown, for non-ferromagnetic matter, that when modelling the electromagnetic behaviour of anisotropic chiral matter the electric quadrupole contribution to the predicted optical activity is approximately the same as that of the magnetic dipoles. This has been confirmed, both theoret- ically and experimentally by Theron and Cloete [21, 221. Thus care must be taken to correctly include the correct multipole contributions in J b o u n d and Pbound, with the order of the multipole expansion depending on the physical phenomena which are under investigation [9].

Complete expressions for the multipole moment densities Pi, Qlj and llfi in terms of the E and B fields and their spatial and t ime derivatives are given in [19, eqns. 9-11] to the order of electric quadrupoles and magnetic dipoles. These can then be used with the above expressions for D, and H; to obtain physically rigorous constitutive relations.

6 Chu’s E-H model

The 1960 text book by Fano, Chu and Adler [2] was written for use by third year students of electrical engineering a t MIT. T h e material is based on an E - H model of the electromagnetic field which was developed by Chu and which is presented in the book. In their 1989 textlo, also for use by undergraduate students of electrical engineering at MIT, Haus and Melcher [5] adopt Chu’s formulation of electrodynamics which uses a magnetic-charge dipole model, not the generally accepted microscopic current loop or Ampkrian model, for magnetization. This approach leads

“It. is dedicated t o Adler, Chu and Fano

naturally to the electromagnetic field being represented by pairing E with H: not B.

Chu postulates [2, Sections 5.4, 7.10-11 and Appendix 11 that the predictions of his magnetic-charge dipole model can not be distinguished (with particular emphasis on ferromagnetism. ferrimagnetism and paramagnetism) by external field or force measurements from the prediction- s of the Ampkrian current loop model. Thus: the argument goes, the magnetic-charge dipole model is preferable because of its simplicity (no moving parts in the rest frame of the material [23]) and because it allows the treatment of magnetic polarization phenomena to be develope- d by direct analogy with electric polarization. Chu also takes issue specifically with the Ampilrian current density Jmag = curl M, Equation ( 8 ) , arguing that its use in Poynting’s theorem leads to inconsistencies.

Tellegen 1241 disagreed, giving reasons, and in turn he raised an objection to Chu’s model based on an appar- ent difference between the predicted forces on a magnetic- charge dipole and a microscopic Ampbrian current loop. However, Haus and Penfield [25], [5, pp. n i i , 509-5101, [26, Section VI11 have shown that Tellegen had neglected a relativistic effect and that the two models in fact predict the same force for a magnetic dipole of constant momen- t in. And in t,lieir 1967 book [27] they have apparently shown” [23] that the same magnetic force densities are predicted by choosing either B or H as the fundamental magnetic field. However Shadowitz [6, p. 3151 still demurs.

In his 1989 paper Rindler [lo], a,pparently unaware of the work of Chu and the ensuing controversy, again raises the problem of the equivalence of the magnetic-charge and current loop dipoles. He concludes that the magnetic-charge pair dipole “gives exact answers for the torque on small current loops a.nd, under certain circumstances, also for the corresponding force.” However i t is not clear from a superficial reading whether his analysis has the depth and generality of the Haus and Penfield treatment of the problem [as]. Thus the current s ta tus of the literature on Chu’s model seems to be that nobody has formally proved it to be physically inconsistent within the espistemological ambit of classical electromagnetic theory.

7 Conclusion. B or H ?

Some of the reasons for considering B t o be the fundamental magnetic field are given by Raab [28].

1. B (and E ) as defined by the Lorentz force law, Equa- tion l , is unique and origin independent in vacuum and also in matter, the latter by spatial averaging over a macroscopic volume. (Although the posit,ion, I*, of the charge is origin dependent its velocity, v, and acceleration are not because the time derivative of an arbitrary but constant offset is zero. Acceleration is mentioned because it is proportional to the force in

“ I have been unable to obtain a copy of the book.

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non-relativistic (Newtonian) mechanics.) Origin in- dependence of its laws is a fundamental requirement of physics, as mentioned earlier.

est,ernal field measurement . . . (or) by external force niea- surement,s”. Also, according to Haus [23] the magnetic- charge dipole is a inucli simpler model because it contains “no moving part,s” . However the Einstein-de H a m esper-

2 . H enters the E-B formuhtion through Ainpkre’s law, ilnelit does nlov.ng W I ~ ~ ~ ~ magllet,izatioll of a bar of iron, suspellded and coaxially ill a solenoid, is reversed by changing the direction of curren-

0‘111s to t, flow through t,he solenoid’s windings the bar be,’ rot,ate about its axis if the experiment is very carefully

Equation 10 and it is not uniquely defined in matter because its divergence is arbitrary.

3. By using the appropriate niultipole expansions, as discussed in Sect,ion 5, Amplre’s law can be written in a origin independent form. However H (and D) is itself in general not origin independent as can be seen from Einstein interpreted this inacroscopic effect, tlie gyromag- it,s expansion in tmerlns of multipole moments - the mo- inelits depend 011 the choice of origin. (Graham and

dent form: in the 11lagllet,ic dipole-elect,ric WadruPole effect is in t,ernls of electroll spin and approximatiolll for momentum, a much more abstract quantum mechanical ditions at the interface of b e h e e n the vacuum and a concept Alllpkre.s electric [30. pp, non-nqnet ic chiral mediuiii. However this formula- 246-2471. N ~ ~ , ~ ~ ~ ~ ~ ~ I ~ ~ ~ , ~ i ~ ~ ~ ~ ~ i ~ ~ - d ~ llaas effect, does

by lvriting .‘\$re have given firlll proof of of ~ l l l p ; r e ~ s lllolecu~ar CUrrelltS’’ [30, PD. 245-

Raab ‘lave recelltlY derived an Origin indePeli- 2.16], The 1nodern illterpretat,ioli of tlie EinsteiIl-de Haas associated

to describe ‘lie boundary

tion is generally aI.‘I)licable to forms Of seelll to veto t,lle Cllu‘s lllaglletic-charge dipole Jf,it,]i llo

moving part.s, and thus presuinably 110 intrinsic a.iigular momentum. This would in turn significa.ntly undermine t8he physical founda.t,ion of Chu’s E-H formulation for t’he classical electromagnet.ic field.

4. Tile special TI^^^^^ of relativity s~lOvvs E B are illtilnately alld ~ul ldal l lenta~~y lillkec1, e,g, [ I , Sections 11-13-6 alld 7. cllapters 11-25 and 261, [13, Chapters 4 and 51.

5 . See [:32, pp. 21-27] and [20, pp. 1085-10861 for fur-

Acknowledgements ther points concerning ;\laswell’s equations, Lorentz covariance and the constitutive relations.

I wish to thank Professor R,oger Raab for ma.ny discussioii- s on tlie topic of t,liis paper. in particular for introducing me t.0 the multipole representation of the interacttion he-

Thus the case for B as the fundamental magnetic field in t,he st,andard E-B formulation is ext,remely strong.

H ~ ~ ~ ~ ~ ~ , cllu’s H field call Ilot he crit,icized oll the grounds of origin dependence or llon-uniqueness because his definition [2, p. 121, [5, pp. 364-3651 of H is in terms

t,ween the electromagnetic field and niatt.er, and for always finding time to reveal some of his deep pl:ysica,l insight to me.

of the Lorentz force law, albeit. t,he following non-standard form,

I also wish to thank Professor Hermann Haus for his illu- minating communicat.ion [23]. f = q E + q v xpoH,

and not via Ampkre’s law. Here is tlie vacuum constant and so poH in the Chu formulation is identical to B in the standard formulat,ion. In his formulation the B field is defined, via Faraday’s law, as B = po(H + M) and it, is interpreted to be a secondary field needed only for dealing with magnetizable matter. This definition is directly analogous to the definition of the D field in terms of the electric dipole density as D = toE + P. Although Chu and Haus were apparently not aware that this pairing by analogy of electric and magnetic dipoles is flawed, as discussed in Section 5, the definition of D can be recti- fied by including the electric quadrupole coutributioiis to make the definitions consistent in the magnetic dipole and electric quadrupole approximation.

Thus the question arises whether there is any experiment

Xly colleague hlr. from Hamlet !

KOOS Holtzhausen provided the pun

References

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[3] S. Ramo, J . R. \.l’hinnery and T. van Duzer, Fields and Waves an Coniniunacaiaoii Electronacs, Third Edi- tion. New York: John Wiley and Sons, 1993.

which vetos Chu’s magnetic-charge dipole model in favour of ~ ~ ~ ; ~ i ~ ~ current loop nlodel, ~l~~ ~ i ~ ~ ~ ~ ~ i ~ - d ~ [4] D. J . Griffiths, Iniroductzon t o Electrodynamzcs, Sec- Haas experiment of 1915 seeins t,o do so 130, pp. 245- ond Edition, Englewood Cliffs: Prentice Hall, 1989.

- ..

2491, [31, PP. 449-4551? [I, P. 11-37-51. As 111entioned in Section 6 , Chu selected the magnetic-charge dipole model instead of the Amphian current loop model partly because [a, p. 1751 “the two models cannot be distinguished by any

0-7803-3019-6/96/$3.00 0 1996 IEEE.

[5] H. A. Haus and J . R. hilelclier, Electronzagnefzc Fzelds a n d Energy, Englewood Cliffs: Prentice-Hall, 1989.

3 60

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[21] I.P. Tlieron and J.H. Cloete. “The optical act,ivity of an artificial non-magnetic uniaxial chiral crystal a t microwave frequencies,” Journal of Electromagnetic Waves and Xpplica.t,ions, vol. 10, no. 4 , pp. 539-561, ~ ~ ~ i 1 1996 [7] R. S. Elliott, “Some useful analogies in the t,eacliiiig

of electroniagiiet,ic theory,” IEEE Trans. Educat.ion, vol. E-22. no. 1. pp. 7-10, February 19i9. (Sce also R. S. Elliott, “Electromagnetic theory: a siiiiplified representation,“ IEEE Trans. Education. vol. E-24. no. 1, pp . 294-298, November 1981.)

[22] I.P. Tlieron and J . H . Cloete. “The electric quadrupole contribution to the circular birefringence of nonmag- netic a.nisot,ropic cliiral media.: a circular waveguide esperimeiit,” IEEE Transactions on Microwave The- ory and Techniques. Scheduled for publication in vol.

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[23] H . A. Ha.us. “H versus B”, Private communication by [9] E. R. Crraliarri! J . Pierrus, arid R. E. Raab, “hlulti-

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force on a magnetic monopole.” Am. .J. Pliys, vol. :j7. no. 11, pp. 993-994, November 1989. [25] H . A . Haus and P. Penfield, “Force on a conducting

current loop,” Physics., vol. 42, pp. 447-4511, 1969. [ l l ] C:. I T ’ . Misner, I<. S. Thorne and J . A. Wheeler. Grnv-

i tntzon. New York: \V. H. Freemaii and Company, [26] H. A . Haus and J . R. hlelclier, “Fields that are always 1973. dyna.mic,” IEEE Tra.ns. Education, vol. 3 3 , no. 1 pp.

35-46. Februa.ry lC)90. See also H. A. Haus and J . R. Melcher: “Electric ancl iiiagnetic fielcls,” Proc. IEEE, vol. 59, no. 6 , pp. 887-

[12] J . D. Jackson, Classical Electrodyriamics, Second Edi- tion, New York: John IViley arid Sons, 1975.

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ing d l e d i n . Cambridge: MIT Press, 1967 [14] M. Born and E. TVolf. Principles of Optics. Perganion Press. London. 1959. [ 2 S ] R. E. Ra.ab, “Digression on the 13-13 cont,roversy” ,

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proach”, .J Opticti1 Soczely of Ainericu A , accepted for publication October 1995. (Additional det,ails are available in .‘New insights into the Ma.xwel1 bounclary conditions”, private communication, 10 March 1995.)

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hne-tluriiaiii modeling of iioiiperfectly conducting metallic thin-film gratings,” Journal Optical Society of America A, vol. 12, no. 9, pp. 1974-1983, Septeni- ber 1995.

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[19] E. B. Graliam and R. E. Raab, “Magnetic effects in antiferromagnetic crysta.ls in the electric quadrupole- magnetic dipole approxiiiia.tioii,” Philosophical Mag- azine B, vol. 66, no. 2. pp. 269-284, 1992.

[20] R.E. Raab and J .H. Cloete, “An eigenvalue theory of circular birefringence and dichroism in a non- magnetic chiral medium,” Journal of Electromagnetic Waves and Applications, vol. 8, no. 8, pp. 1073-1089, August 1994

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Documents

Is B or H the Fundamental Magnetic Field