arXiv:cond-mat/0010261v1 [cond-mat.soft] 18 Oct 2000 I.Srs,FreadPressure and Force Stress, III. I yrdnmcTheories Hydrodynamic II. .Introduction I. .EulbimSraeFre15 Force Surface Equilibrium D. .ToCvas5 Caveats Two D. .Budr odtos11 Conditions Boundary D. .TeBl oc est 12 Density Force Bulk The A. .TeCas-rie oc 1 Force Coarse-Grained The A. .Dlt lsa6 Plasma Dilute A. .TeMxelSrs nEulbim13 Equilibrium in Stress Maxwell The C. .Cneto hsAtce4 Article this of Content C. .Hdoyai awl hoy10 Theory Maxwell Hydrodynamic C. .TeIcmrsil ii 13 Limit Incompressible The B. .TeMxelSrs esr2 Tensor Stress Maxwell The B. .Wal iscae a 7 Gas Dissociated Weakly B. .TemdnmcDrvto fteSrs 17 Stress the of Derivation Thermodynamic E. .Ntso nt 5 Units on Notes E. lcrmgei oc n h awl tesTno nCond in Tensor Stress Maxwell the and Force Electromagnetic .QaiEulbim14 13 Quasi-Equilibrium 2. Equilibrium Total 1. .ATcnclFa 3 3 3 3 Problems Conceptual 4. Flaw Technical A Validity 3. of Range The 2. Objections Obvious Three 1. .“-ed hoy8 7 6 Theory 7 “2-field” 2. Contributions Material The 6 1. Conservation Momentum and Energy 3. Contributions Field equation The Newtonian 2. of Version Euler 1. osdrd nadto oehnigteapeito fth i of and formulas, appreciation derived the and concepts enhancing introduced to the of addition grasp In considered. awl testno a ehnld niigterssan theorists inviting handled, quantity. be useful can tensor stress Maxwell in assign ambiguities validity hidden of and range scrutinized, a are with misconceptions Simplifica equilibrium, un the equations. from Maxwell especially starts the It and thermodynamics media. laws, condensed in force electromagnetic oekyqatte nmcocpcelectrodynamics. redu macroscopic unnecessarily in conf Max circumstances quantities even These key the times some as secret. at kept opaque, such well fairly expressions a is by derivation given Their and force. complicated, more is ubro hnmn yia fsrnl oaial syste polarizable strongly of typical phenomena of number A hsatcepeet hruhytpdggclderivatio pedagogical yet thorough a presents article This hl h lcrmgei oc smcocpclysimply microscopically is force electromagnetic the While Contents ntttfu hoeicePyi,Universit¨at Hannover Physik, f¨ur Theoretische Institut eto hsk Universit¨at M¨unchen, Physik, Sektion 06 anvr emn,EC Germany, Hannover, 30167 03 ¨nhn emn,EC Germany, M¨unchen, 80539 Dtd eray1 2008) 1, February (Dated: lu Stierstadt Klaus ai i [*] Liu Mario 12 1 6 and rnil,oemytk h oet oc sa exact an the as obtain [1], to force force it Lorentz macroscopic In coarse-grain the and density. total expression take force the microscopic varying may to spatially fer- only one the not as principle, to respond (such also these liquids but as for force, 4]), true 3, on especially [2, force rofluids is electromagnetic This the matter. concerns question central V Experiments IV. .Appendix V. tdigplrzbeadmgeial aeil,one materials, magnetizable and polarizable Studying .Mgei -ig n ca eaain24 Separation Scrap and O-Rings Magnetic D. References .FedIdcdVraini este 22 Densities in Variation Induced Field A. .Hdottc ntePeec fFed 23 Fields of Presence the in Hydrostatics C. .CretCryn etclWr 23 Wire Vertical Carrying Current B. .Elpia eomto fDolt 25 Droplets of Deformation Elliptical E. .AbgosNttos19 Notations Ambiguous F. dt ahepeso.Sm widespread Some expression. each to ed oua oain revealed. notations popular sn,adterrneo aiiyalbut all validity of range their and using, xeietlssaiet mrc this embrace to alike experimentalists d eteueuns n rswrhns of trustworthiness and usefulness the ce h oet oc,ismcocpcform macroscopic its force, Lorentz the eosrtstees ihwihthe which with ease the demonstrates t vral cetdipt:conservation inputs: accepted iversally .Kli n emot oc 21 18 17 20 19 Force Helmholtz and Kelvin 3. Equation Bernoulli Magnetic Pressures 2. Field Zero Different 1. Part Magnetic The 2. Part Electric The 1. in r osdrdfrvroslimits, various for considered are tions s ytm,i ep osldf the solidify to helps it systems, ese fteMxelsrs esrand tensor stress Maxwell the of n .TeCas-rie Force Coarse-Grained The A. s seilyfroud r then are ferrofluid, especially ms, elsrs esradteKelvin the and tensor stress well .INTRODUCTION I. , ne Systems ensed 22 26 26 2

or equally justified, as Mi∇ Bi . Consequently and re- markably, the Kelvin force ish noti usually a valid formula fL = ρ E + j B . (1) m h i h e i h e × i in ferrofluids, where frequently M H and χ 1 — though it is widely employed there.≈ ≈ Unfortunately, this is of little practical value, as we nei- The nowadays generally accepted method to derive the ther have, nor indeed are interested in, the detailed mi- electromagnetic, or “ponderomotive” force f P, cf 15 of croscopic information of the fields ρ , j , E, B. On the i e e Electrodynamics in Continuous Media by Landau§ and other hand, although we do have the knowledge of the Lifshitz [5], is a definit improvement over the microscopic coarse-grained fields ρ , j , E and B , or may ob- e e one given above. It is completely macroscopic and starts tain them via the macroscopich i h i h Maxwelli h equations,i it is from identifying f P as clear that the difference between the above true expres- i sion and the “fake” Lorentz force, f P = [Πtot P (T,ρ)δ ], (5) i −∇j ij − ij ρe E + j B , (2) tot h ih i h ei × h i where Πij denotes the macroscopic Maxwell stress ten- sor, and−P (T,ρ) the zero-field pressure – the pressure that may become quite important: Think of an iron nail in is there for given temperature and density, but without a magnetic field, which is obviously subject to an elec- the external field. The ponderomotive force f P reduces tromagnetic force, although the “fake” force vanishes, as i to the Kelvin force, if the polarization and magnetiza- ρ , j 0. e e tion are (assuming the dilute limit) proportional to the h Thei h differencei → between the true and fake Lorentz force density ρ. is frequently taken to be the Kelvin force, being P ∇ E i i The idea behind the identification seems attractively in the electric case [5], where P denotes the polarization,h i i simple: Since Πtot is the total force on a volume ele- with summation over repeated indices implied. Orig- j ij ment, and −∇P the force at zero field, their difference inally, the formula was microscopically derived: First i f P must be−∇ the force due to the external field. Unfortu- write down the force exerted by an external field on a i nately, as we shall show in this review, the resultant force dipole, eg by taking the derivative of the electrostatic en- is rather ambiguous. The reason lies with the zero-field ergy with respect to the position of the dipole for given pressure P (T,ρ), which is not a unique quantity, because field. Then, assuming that the dipoles are too far apart its value depends on the chosen variables, eg P (T,ρ) = to interact, and to feed back to the given field, take the P (s,ρ), where s is the entropy density. The convention-6 total force density as simply the sum of all forces ex- ally defined Kelvin force is valid only under isothermal erted on the dipoles in a unit volume. Clearly, this result conditions with the density kept constant. Under adi- neglects nonlinear terms in P (which account for inter- i abatic conditions, with s given, the temperature T is action) and is valid only in the dilute limit. This implies known to change with the field, so will P (T,ρ), the al- small polarization and therefore weak force, and one is leged zero-field pressure. Any field dependence implies free to add a second order term, writing the Kelvin force that P (T,ρ) contributes to the electromagnetic force. as Employing the force of Eq (5) alone then leads to wrong predictions. Pi∇( Ei + Pi/ǫ0) Pi∇Di/ǫ0. (3) h i ≡ In physics, we tend to start from a robust base of broad There is a more sophisticated macroscopic derivation of applicability when studying a given situation, narrowing the Kelvin force that we shall discuss next. Stating that our focus along the way only when necessary. Clearly, the same constraint applies here, and one may still write while the universally valid Lorentz force is one such base, the Kelvin force either as Pi∇ Ei or as Pi∇Di/ǫ0 is the fragile ponderomotive force, and even more specific, bound to raise a few eyebrows.h Buti as will be shown the Kelvin force are not. This makes it desirable to look in this review, it is really a rather straight-forward im- for better alternatives, one of which is employing the plication of the assumptions made during its derivation. Maxwell stress tensor directly. By the same token, for linear constitutive relation, the Kelvin force is only valid for small susceptibility, χ 1, to linear order in χ. ≪ B. The Maxwell Stress Tensor The same holds for the magnetic case, where the corre- sponding force is much larger in praxis (as it is not sub- Given in terms of thermodynamic variables, the Max- ject to ionization or residual conductivity). The Kelvin well tensor is an unequivocally macroscopic, coarse- force expression is again valid only when weak, to linear grained quantity. It holds for all conceivable systems and order in the magnetization M B , or the magnetic is subject only to the validity of local equilibrium – a con- susceptibility χm 1. And one≪ may h i take its expression straint in frequency but not in the type and strength of ≪ either in the usual form [2, 5], as interaction. Besides, more frequently than not, in under- standing the many facets of strongly polarizable systems, it is more convenient to employ the Maxwell tensor di- M ∇( B µ0M ) µ0M ∇H , (4) i h ii− i ≡ i i rectly, rather than to follow the detour via pressure and 3

P the ponderomotive force fi — in spite of having to deal Nevertheless, it is customary to take for granted that with quantities such as the chemical potential or the en- the form of the Maxwell tensor and the electromagnetic tropy. force thus derived also hold in dynamic situations, at fi- Given the Maxwell tensor’s pivotal role for a solid un- nite frequencies and nonvanishing liquid velocity. This derstanding of electromagnetism in condensed systems, may well be approximately true for some circumstances, (not least as the starting point for forces), it is espe- but which and why? Should we not rather derive dynam- cially unfortunate that one is hard pressed to find a fully ically valid expressions for the Maxwell stress tensor and convincing derivation of its form. The classic reference the electromagnetic force, and justify the employed ap- is again to the otherwise excellent book by Landau and proximations by estimating the magnitude of the terms Lifshitz [5], the relevant 15 of which, however, leaves neglected? many readers unconvinced,§ even confused. The reasons for this are presented in the four sections below. 3. A Technical Flaw

1. Three Obvious Objections Then there is a technical flaw that actually renders the derivation mathematically invalid: There is no question tot The three more obvious reasons for the frustration of that one can always obtain a tensorial quantity Πij by tot the readers are (i) the gradient terms, (ii) the magnetic evaluating the scalar Πij ξinj while varying the unit vec- field dependence, and (iii) the concentration: First, the tors ξi and nj . Taking each of the two unit vectors to Maxwell tensor is derived considering a capacitor. Its successively point in all three directions, we obtain nine tot tot geometry is so simple that terms proportional to the gra- different results for Πij ξinj , each a component of Πij . dients of temperature and field may have been missed. In [5], nj is the surface normal, ξi the direction of the Although these are asserted to be negligible in a foot- virtual displacement, and the Maxwell tensor Πtot is − ij note, one feels the need for convincing arguments. denoted as σij . Second, the magnetic terms are obtained, in 35 of [5], With the electric field being the only preferred direc- § by the replacement E H, D B. This would be jus- tion, ξi and nj need to be both along and perpendicu- tified if the system under→ consideration→ does not contain lar to the field. In 15 of [5], only the displacement of sources, and the respective Maxwell equations are the the capacitor plate is§ considered, implying that the sur- same, ∇ E = 0, ∇ D = 0 versus ∇ H = 0, ∇ B = 0. face normal nj is kept parallel to the E-field throughout. In the geometry× used· to derive the× electric part· of the Therefore, if the direction of the electric field is taken as stress tensor, however, the charge on the capacitor plates ˆex, only the components Πxx, Πyx and Πzx could have is essential for producing the electric field. Therefore, an been evaluated. A footnote states: It is not important independent consideration for the magnetic part is nec- that in this derivation E is parallel to n, since Πij can essary, with a current carrying coil replacing the charge obviously depend only on the direction of E, not on that laden capacitor. In 4 of his widely read book on fer- of n. This is incorrect, because although Π does not § ij rofluids [2], Rosensweig points to this gap, and aims to depend on n, the scalar Πij ξinj does: Take for instance fill it. Unfortunately, the calculation starts from a faulty the term DiEj , which is part of the Maxwell stress, and assumption, discrediting the derivation, cf section III E 2. which indeed does not depend on n. Yet with D, E ex, k Finally, neither Landau and Lifshitz nor Rosensweig we have DiEj nj = 0 for n ex, and DiEj nj = 0 if 6 k study the concentration dependence of the stress tensor, n ex. ⊥ although it is an important thermodynamic and hydro- dynamic variable in ferrofluids, liable to undergo much greater variations than the density, and hence of experi- 4. Conceptual Problems mental importance. Finally, there are conceptual problems of more funda- mental nature. The Maxwell stress tensor is introduced 2. The Range of Validity in [5] by the terse sentences: It is well known that the forces acting on any finite volume in a body can be reduced Next, the lack of any discussion on the range of va- to forces applied to the surface of that volume. This is lidity for the obtained expression further undermines the a consequence of the law of conservation of momentum. trust of the readers. The derivation is thermodynamic The surface force density is subsequently taken as the in nature and takes place in a stationary capacitor, so stress tensor. the result is apparently valid only in equilibrium and for Momentum conservation is without further elaboration vanishing liquid velocity. Consistent to this, the static a confusing argument, because the very idea of force is Maxwell equations for the rest frame are repeatedly em- something that alters the momentum of particles. Being ployed when deriving the electromagnetic force from the accelerated by external fields, the momentum of a lump of Maxwell tensor. material is not conserved, and there is no reason why the 4 momentum density ρv should satisfy a continuity equa- tion II A) is a dilute gas of charged particles, which is tion. On the other hand, what the electromagnetic field well accounted for by the Newtonian equation of motion does is to exchange momentum with the material, so the and the microscopic Maxwell equations. The next sys- total momentum of both remains conserved, which is, as tem (section II B) is a slightly dissociated liquid of par- we shall see, gtot = ρv +E H. This momentum density ticles possessing negligible electric and magnetic dipole × tot tot does satisfy a continuity equation,g ˙i + kΠik = 0. moments. So there are many particles, but at most one (Even in the presence of fields that accelerate∇ particles, charge carrier, per grain. The appropriate theory here the total system – including particles and the field pro- is of a mixed type, being a combination of the hydrody- ducing charge carriers – is still translational invariant, namic theory and the microscopic Maxwell theory. The with the associated total momentum a locally conserved former accounts for all particles, the latter accounts for quantity, see 75 of [5], Ch 14 of [6], and [7].) The neg- the spatially slowly varying field, either externally im- § tot ative of the associated momentum flux, Πik , is the posed or produced by the few charge carriers. Finally, properly defined, dynamically valid Maxwell− tensor. It in section II C, we consider a dense system, including holds its ground for finite frequencies, nonvanishing ve- dipole moments and hidden sources. It is to be accounted locities of the medium, and including dissipations. Un- for by the genuinely low-resolution, hydrodynamic-type derstanding the conservation of gtot = ρv +E H renders Maxwell theory that is our goal to derive and consider in the derivation of the Maxwell tensor cogent× and coher- details. ent, even though the second term is, under usual circum- In all these systems, there are a conserved energy utot tot stances in condensed matter, much smaller than the first. and a conserved momentum density, gi = ρvi, both con- sist of material and field contributions.6 These we shall choose as two of our dynamic variables, with the equa- tot tot tot tot C. Content of this Article tions of motionu ˙ + kQk = 0 andg ˙i + kΠik = 0, tot ∇ ∇ where Πik is the Maxwell stress tensor. We shall ob- − tot tot Our aim in this paper is a careful and rigorous deriva- tain explicit expressions for gi and Πik in all three sys- tion of the Maxwell tensor and the dynamically valid elec- tems — whereby finding the expressions in the genuinely tromagnetic force, in a presentation that is detailed and dense, third medium is rendered very much simpler, and pedagogical. We also consider applications, especially for more transparent, by the considerations in the previous experimentally relevant situations in ferrofluids, to give two dilute systems. hands-on illustrations on how to employ these quantities. In Chapter III, starting from the Maxwell tensor, we The paper may be divided into two parts, The first con- shall first obtain the general expression for the electro- sists of Chapter II and concentrates on the fundamental magnetic force density, valid for arbitrarily strong fields, concepts. The hydrodynamic equations for a dense sys- moving medium and finite frequencies, given essentially by (s∇T + ρ∇ξ), where ξ is the chemical potential. tem exposed to electromagnetic fields — including espe- − cially the form of the Maxwell tensor — are derived here. Although this expression is already valid without field, The second part (Chapter III and IV) is more practically it is the proper formula for the quantity usually written concerned. It starts from the Maxwell tensor, derives as the sum of the zero-field pressure gradient and the the electromagnetic force, considers different simplifica- ponderomotive force. In contrast to the latter, however, (s∇T + ρ∇ξ) is valid for any sets of variables, and for tions, and studies a number of experimentally relevant − situations. Although we believe that Chapter II provides arbitrarily large polarization and magnetization. useful and necessary insights, the second part could if In equilibrium, since both the temperature T and necessary be read alone. the chemical potential ξ are uniform, this force density Chapter II starts by differentiating two types of the- is identically zero. Nevertheless, electric and magnetic ories: First, the high-resolution theory for a low den- fields remain operative, may lift a polarizable system off sity system, with at most one particle per infinitesimal the ground, deform its shape and generally alter its hy- volume element, or per grain (as in photographs); sec- drostatics — all signs that forces of electromagnetic ori- ond, the low resolution theory for a high density sys- gin are at work. In section IIID, the responsible equi- tem, with many particles per grain. The microscopic librium force is identified as a surface force. Its form Maxwell equations and the Newtonian equation of mo- follows directly from the simplifications that occur for tion (including the Lorentz force) belong to the first type. the Maxwell stress if the system is in equilibrium (section There are no hidden charges, polarization or magnetiza- III C). As most of the experiments considered in Chapter tion here, and we know the whereabouts of every single IV reflect varying appearances of the equilibrium force, particle. The second type is represented by the macro- this is a useful and concise result. scopic Maxwell equations, or any thermodynamic and A thermodynamic rederivation of the Maxwell stress hydrodynamic theories. The question about the coarse- tensor is given in section III E, an approach which is sim- grained form of the electromagnetic force arises here. ilar in spirit to that of Landau and Lifshitz [2, 5], yet We shall consider three systems with increasing densi- carefully freed of the flaws listed in section I B above. ties, to be accounted for by theories of decreasing resolu- In comparison to the derivation given in Chapter II, this tion and increasing complexities: The first system (sec- here is narrower in focus and validity, but simpler. Chap- 5 ter III ends with section III F, which points out the in- vertical, current-carrying wire that goes through a dish herent ambiguity of the zero-field pressure P (T,ρ), as containing ferrofluid will drag up the ferrofluid along the it is independent of external fields only if T and ρ are wire. Our calculation in section IV B includes the surface kept constant. We stress this point by repeating the tension and the variation of concentration. The following separation of the zero-field pressure from the Maxwell section IVC considers how far an external field will raise stress for various combinations of variables, obtaining the level of a fluid column, and what a pressure gauge many forms of the ponderomotive force, all rather dif- actually measures within the ferrofluid that is subject to ferent from each other. Furthermore, a warning is issued external fields. Section IV D deals with rather similar on the unsuspected, narrow range of validity for various physics and considers magnetic O-rings and scrap sepa- expressions in the notation containing the zero-field pres- ration. Section IV E contains a simple calculation on the sure, especially the Maxwell stress in the approximation elliptical deformations a droplet of ferrofluid undergoes 1 2 (P + 2 H )δij HiBj , and the Kelvin force. Neither is along the external magnetic field. valid for large magnetization− and magnetic susceptibility, M H and χm 1. ≈ ≈ The very useful magnetic Bernoulli equation by Ro- D. Two Caveats sensweig [2] is unfortunately also given in this notation. An formulation free of ambiguity is found, along with the There are two points we shall not address in this arti- generalization to include the variation of concentration. cle. First, we shall not consider any dissipative terms, In Chapter IV, we study a number of phenomena typ- although the framework presented here is well suited ical of strongly polarizable systems. Most of these ex- for including them. This was a difficult decision, as periments have already been given detailed theoretical the dissipative terms account for irreversibility, without analyses, especially by Rosensweig in his comprehensive which no macroscopic, coarse-grained theory is complete, book on ferrofluids [2]. The reason to repeat them here and because they give rise to additional electromagnetic is threefold: First, to demonstrate that it is not nec- forces of comparable magnitude and experimental rele- essary to introduce the zero field pressure, that experi- vance. Yet, including these terms would have consid- mental phenomena may be well and easily accounted for erably lengthened an already lengthy article, and fur- employing the Maxwell stress directly, using a generally ther complicated its formulas and arguments. We pledge valid, unambiguous notation. Second, we believe that our amendments for the future, and for now direct the impa- derivations are well streamlined and easy to follow. This tient readers to the literature [3, 4, 9]. is achieved by stressing the simplification of equilibrium, Second, given the long and tortuous history of the and by consistently using only the Maxwell stress and the struggle to come to terms with the macroscopic electro- associated boundary conditions. In contrast, confusingly magnetic force, time and again forcing us to back up from many pressures are deemed necessary in [2]. Finally, we blind alleys, any attempt by us on a comprehensive cita- aim to amend some of the physics and especially include tion would bear historic rather than scientific interests. the concentration of magnetic particles as an indepen- As we see it, the consistent thermodynamic treatment dent variable. Being inhomogeneous where the field is, given first by Landau and Lifshitz [5], and later general- the concentration alters experimental outcomes and is ized to include nonlinearity by Rosensweig and others [2], something theoretical considerations need to include. cf the excellent summary by Byrne [10], is so much more Since mesoscopic particles diffuse slowly, the equilib- superior than the many treatises preceding them, that rium between concentration and field gradient needs time this represents the only worthwhile starting point for any (up to days, even weeks) to establish. So for times much further considerations — and consequently these two are briefer, the concentration will remain constant and may also the only ones subject to scrutiny and criticisms here. indeed be taken as dependent. On the other hand, strong gradients, small geometry, large particles will all consid- erably shorten the equilibrating time (down to fractions E. Notes on Units of a minute), then the spatially varying concentration will be an important feature in any system. Besides, long con- This paper will be in the SI units throughout, though tacts with field gradients are also of interest (eg in mag- with a little twist to render the display and manipulation netic O-rings). Although the non-uniform equilibrium of the formulas simple. We define, and from now on shall distribution of magnetic particles is not a well studied extensively use, the fields and sources subject, their diffusion under enforced non-equilibrium H = Hˆ µ , B = Bˆ/ µ , ̺ =̺ ˆ / ǫ , situation is, see [4, 8]. √ 0 √ 0 e e √ o ˆ ˆ Now a rundown of the considered experiments. In sec- E = E√ǫo, D = D/√ǫo, q =q/ ˆ √ǫo, (6) ˆ ˆ tion IV A, we study the variation of density and concen- P D E = P/√ǫo, M B H = M√µ0, tration in the presence of inhomogeneous fields: While ≡ − ≡ − the variation of density is a measured phenomenon, that where (again starting from here) the quantities with hats of the concentration is not – although its variation is are the usual ones, denominated in MKSA, or the SI- much more pronounced, by many orders of magnitude. A units. The advantage is, all new fields have the same 6 dimension, J/m3, and sensibly, H = B and D = E in Euler formulation lends itself to a scaling-up of the grains vacuum. (Electricp charge q has the dimension √Jm, and and a reduction of the resolution, we shall first find the 5 Euler formulation for the Newtonian equation, before us- its density [̺e] = J/m .) Written in the new fields, all formulas are ridp of the ubiquitous (and content-free) ing it to draw a number of conclusions useful for dense systems. factors ǫ0 and µ0. To return to the more familiar MKSA- fields at any instance, simply employ Eqs (6). (These fields are sometimes referred to as being in the Heaviside- Lorentz, or the rational, units. But saying so immedi- 1. Euler Version of Newtonian equation ately leads to heated and pointless discussions. We stick to the SI units throughout and use the above fields only Since there is at most one particle per grain, of vol- as a convenient shorthand that may be abandoned when- ume VG, we may identify the velocity, mass and charge ever one chooses.) of a volume element with that of the particle occupying Some literature, notably [5], employs cgs, or Gauss- it at a given instance, and take all three to be zero if units. To ease comparison with them, simply note that there is no particle. As a result, we obtain three (highly α the Gauss-fields, with tilde, may be obtained any time discontinuous) fields: v v(r,t), m/VG ρ(r,t), → → via q/VG ρe(r,t). The many Newtonian equations of mo- tion then→ reduce to one field equation, ˜ ˜ H = H/√4π, B = B/√4π, ̺e = √4π ̺˜e, ρ d v ρ[v˙ + (v ∇)v]= ρ (e + v b/c). (11) ˜ ˜ dt e E = E/√4π, D = D/√4π, je = √4π j˜e, (7) ≡ · × P = √4πP˜, M = √4π M˜ , (The quantity v is to be taken from the velocity of the same particle∇ at two successive moments.) It is now essential to explicitly include the continuity equation, II. HYDRODYNAMIC THEORIES ρ˙ + ∇ (ρv)=0, (12) · A. Dilute Plasma that in the Lagrange version is implicit, nearly inciden- tally contained in the fact that one does not loose any We first consider a rarefied system of charged particles, of the many equations (10). [The continuity equation for and choose a resolution that is high enough that each the charge density ρe is implied by the Maxwell equations grain (or infinitesimal volume element) contains one or Eq (8,9).] less particles. (This theory is meant as a starting point, The material contributions to the energy and momen- to clarify a few concepts important for the more complex tum density are, respectively, theories of the following chapters. So we shall simply uM = ρ(c2 + v2/2), gM = ρv. (13) discard the possibility that even in a rarefied gas, two particles will occasionally come close to each other.) The where the first comprises of the rest energy and the (non- microscopic Maxwell equations account for the time evo- relativistic) kinetic energy. Employing Eqs (11,12), we lution of the electromagnetic field for given sources, find M M ∇ e = ρ , ∇ b =0, (8) u˙ + ∇ Q = ρe(v e), (14) · e · · · ˙ M M e v b ˙e = c∇ b ρev, b = c∇ e, (9) g˙i + k Πik = ρe( + /c)i, (15) × − − × ∇ M M M × M Q = u v, Πik = gi vk. (16) while the feedback of the field on the motion of the sources is given by the Newtonian equation of motion, 2. The Field Contributions mv˙ α = q(e + vα b/c), (10) × one for each particle α. (To emphasize the fact that we The field contribution to the energy and momentum are here dealing with the high-resolution, microscopically density are accurate fields, these are denoted by the lower case letters F 1 2 2 F ˆ u = 2 (e + b ), g = e b/c. (17) e ˆe√ǫo, b b/√µ0, while the coarse-grained fields × ≡ ≡ below are denoted by the usual D, E, H, B.) Eqs (8,9,10) From the Maxwell equations (8,9) we deduce represent a conceptually simple and complete theory, F ∇ F F but it contains a notational inconsistency: The Maxwell u˙ + Q = ρe(v e), Q = c e b, (18) · F − F · × equations are an Euler type theory, accounting for the g˙i + kΠik = ρe(e + v b/c)i, (19) time evolution of fields at a given point in space, while F 2∇ 2 −F × Π = (e + b u )δik eiek bibk. (20) the Newtonian equation is of the Lagrange type, which ik − − − F F 2 concentrates on a given particle. (So the term ρev in Note the relationship g = Q /c . This is far from Eq (9) denotes the electric current at a space point, while accidental and derives from the symmetry of the rela- α F F v in Eq (10) is the velocity of a particle.) As only the tivistic energy-momentum 4-tensor, Παβ = Πβα, because 7

F F F F F F 2 cgk = Πk4, Qk /c = Π4k. Less formally, g = Q /c resolving, vacuum Maxwell equations (8,9) remain valid may also be seen as the conservation of the field an- – and with them every single of the equations of sec- gular momentum for ρe 0. The angular momentum tion II A 2, about the field contributions to energy and density ℓF r gF is a→ locally conserved quantity in momentum. The equations in section II A 1, about the ≡ × neutral systems. Rewriting Eq (19) (with ρe 0) as respective material contributions, must be modified, as F F F → ∂ℓi /∂t + m(ǫijk rj Πkm)= ǫijkΠkj , we observe that the these will now be accounted for by three hydrodynamic ∇ F variables: the entropy density s, and the coarse-grained angular momentum ℓi satisfies a continuity equation only mass density ρ and velocity v – all smooth and slowly if the stress tensor is symmetric. Although this argument h i h i seems to require merely the symmetry of the momen- varying fields. (The coarse-graining brackets are dropped F F below to retain the simplicity of display.) tum 3-tensor, Πik = Πki, we know that a nonvanishing F F Πk4 Π4k in one inertial system will foul up the symme- try of− the 3-tensor in other systems, as the antisymmetric parts of any 4-tensors mix in a Lorentz transformation. 1. The Material Contributions Yet angular momentum is conserved in every inertial sys- tem. The hydrodynamic theory of the material part of our Since this reasoning is so general, it also holds for the system is given by two continuity and two balance equa- material part, gM = QM/c2. Hence the expression for the tions, momentum density is actually gM = ρv[1 + v2/(2c2)], cf Eqs (13,16), though we are quite justified to neglect the ρ˙ + ∇ (ρv)=0, s˙ + ∇ (sv)=0, (24) · M M · second term in the nonrelativistic limit. Later, when we u˙ + ∇ Q = ρe(v e), (25) have no prior knowledge of the form of the momentum M M · · g˙ + k Π = ρe(e + v b/c)i, (26) density, we shall deduce it from that of the energy flux, i ∇ ik × QM M v M M as angular momentum is also conserved in dense systems. = (u + P ) , Πik = gi vk + Pδik. (27) We register the fact that while the expression for the For ρ 0, these are the well known hydrodynamic energy density uF = 1 (e2 + b2) is a genuine input – in- e 2 equations→ of a neutral, isotropic liquid [11]. For finite dependent of, and in addition to, the Maxwell equations, ρ , the right sides of Eqs (25, 26) have the given form the formula gF = e b/c is not, since QF is given once e because the field source terms of Eqs (18, 19) remain un- uF is. × changed, and because summing up the respective right sides must yield nil, such that total energy and momen- 3. Energy and Momentum Conservation tum are conserved, as given in Eqs(21). (Since v actu- ally denotes v now, the above formulas presume the h i validity of ˙e = c∇ b ρe v , rather than the origi- The preceding two sections allow the simple and note- nal Eqs (9). This is× justified,− h however,i because charge worthy conclusion that our starting equations imply local carriers do move with v in the absence of dissipation.) conservation of total energy, momentum and angular mo- M h iM Comparing Q and Πij here to those of Eqs (16), the mentum even in the presence of charges, ρe = 0. Taking only apparent difference is the appearance of the pressure utot uF + uM and gtot gF + gM, we find6 ≡ ≡ P , clearly an expression of the present system being dense tot ∇ tot tot tot and interacting. However, there is more than meets the u˙ + Q =0, g˙i + kΠik =0, (21) tot tot · F M tot ∇ F M eyes, and the most important one is our ignorance about Πik = Πki = Πik + Πik, Q = Q + Q , (22) the explicit expression for the energy uM. (Also, being Qtot/c2 = gtot ρv + e b/c. (23) a hydrodynamic, coarse-grained theory, the equations of ≈ × motion now contain dissipative terms to account for ir- These results have been collected here because local con- reversibility and entropy production. As mentioned in servation of these quantities is always true, independent the introduction, however, we shall neglect these in this of the above derivation tailored to a dilute and finely article.) resolved system. So we may start from them as input In the co-moving, local rest frame of the liquid, (de- next. noted by the subscript 0,) the material energy density in local equilibrium is given by the thermodynamic expres- sion, B. Weakly Dissociated Gas M du0 = T ds + ξ0dρ, (28) Now we consider a dense macroscopic system in its hy- M drodynamic regime: To the above gas of dilute charge which states the simple fact that u0 is a function of carriers we add a dense system of neutral particles with s and ρ, with the temperature T and the chemical M vanishing electric and magnetic dipole moments. One potential ξ0 defined respectively as T ∂u0 /∂s and M ≡ example of this composite system is a slightly ionized ξ0 ∂u0 /∂ρ. (The dimension of the chemical potential ≡ M gas with negligible electric and magnetic susceptibility. ξ0 is energy/mass.) The explicit form of u0 is usually This is still a comparatively simple system, as the highly unknown, it depends on microscopic specifics, and is in 8 general rather complicated. Yet, as we know from ther- an equation that will prove useful later. Note an inter- modynamics, and as we shall see below, Eq (28) is as esting feature of the hydrodynamic equations of neutral such already very useful. systems, Eqs (24,25,32) with ρe 0: They are given in The second of Eqs (13), gM = ρv, remains an excellent terms only of the quantities appearing→ in Eq (29): the M M M approximation, because the energy flux Q as given by energy u , the thermodynamic variables s,ρα, g , and 2 Eq (27) is still dominated by the term ρc v (contained the conjugate variables T, ξα, v. Without an explicit ex- in uMv), from the transport of rest energy. All the other pression for uM, the hydrodynamic theory is clearly nec- terms are (in comparison) relativistically small and can essarily written in these general and abstract quantities, be neglected. and we may take this observation as an indication that In nonrelativistic physics, it is customary to subtract the hydrodynamic theory contains no more input than d(c2ρ)= c2dρ from both sides of Eq (28), eliminating the conservation laws and thermodynamics, which is the ba- M rest energy from du0 on the left, redefining the chemi- sic reason of its general validity. We shall return to, and 2 cal potential as ξ0 c on the right, and at the same build upon, this point in the next section. 2 − M time subtracting c ρv from the energy flux Q , Eq (27). For a charged system, ρe = 0, we need to include the This represents a shift to a different but equivalent set Maxwell equations (8,9) for6 a complete description. The of independently conserved quantities: from uM and ρ to independent variables now include e and b, each with M 2 (u ρc ) and ρ. However, this operation does alter the an equation of motion. All equations also depend on ρe, link− between the momentum density and the energy flux, which however only stands for ∇ e and is not indepen- rendering it gM = QM/c2 + ρv ρv. We shall follow dent. · M ≈ this convention, so u0 and ξ0, starting from this section, no longer include the rest energy and c2, respectively. To account for a liquid with a varying local veloc- 2. “2-field” Theory ity v, we need to employ a globally valid “lab”-frame. M M 2 The material energy is then given as u = u0 + ρv /2, For our purpose of deriving the genuine low-resolution where d( 1 ρv2) = d(gM/2ρ) = v dgM 1 v2dρ. So the Maxwell theory, it is instructive to again change the no- 2 · − 2 thermodynamic expression of Eq (28) becomes duM = tation to utot and gtot by adding up the material and M 1 2 T ds + ξdρ + v dg , with ξ = ξ0 2 v the chemical po- field contributions, as we did in section II A 3. The new tential of the lab· frame. Clearly,− this expression signals theory remains closed and complete, and is given by the M M M M that u = u (s,ρα, g ) is also a function of g . Maxwell equations (8,9), the continuity equations (24) If we are dealing with a solution, a suspension, or any for mass and entropy, and Eqs (21) for utot and gtot, system with more than one conserved densities, we need where the fluxes to replace the term ξdρ with ξαdρα, introducing all con- tot M Q = (Ts + ξαρα + v g )v + c e b, (33) served densities ρα as independent thermodynamic vari- tot tot · M× 2 2 ables, Πik = ( u + Ts + ξαρα + v g + e + b )δik − M · duM = T ds + ξ dρ + v dgM, (29) +gi vk eiek bibk (34) α α · − − where summation over α = 1, 2, is implied. Being are obtained by adding the fluxes of Eq (18) and (20) to · · · conserved, all ρα obey continuity equations, so the first of that of (27). The total momentum density is still as in ∇ v Eq (23), gtot = ρv + e b/c, while the energy dutot is Eqs (24) is to be replaced byρ ˙α+ (ρα ) = 0. Usually, × it is more convenient to retain the· total density ρ as a obtained by adding Eq (29) to the differential of the first variable. For a binary mixture with the solute density of Eqs (17), yielding ρ1 and the solvent density ρ2, we therefore take ρ1 and tot M du = T ds + ξαdρα + v dg + e de + b db, (35) ρ ρ1 + ρ2 as the variables, implying that ξαdρα in · tot · · ≡ = T ds + ξ dρ + v dg + e0 de + b0 db. (36) Eq (29) is to be read as ξdρ +ξ1dρ1. α α · · · The pressure, defined as the energy change if the vol- Eq (36) is algebraically identical to Eq (35), because v ume changes at constant entropy and mass, is related d(gM gtot)c = v d(e b)= (v e) db+(v b) de·, to the variables and conjugate variables of uM via the − − · × − × · × · where e0 e + (v/c) b, b0 b (v/c) e are the Duhem-Gibbs (or Euler) relation, respective≡ rest frame fields.× For≡ reasons− that× will become M 3 M P ∂( u0 d r)/∂V = u0 + Ts + (ξ0)αρα clear soon, we shall refer to this set of equations as the ≡− − “2-field” theory. R = uM + Ts + ξ ρ + v gM. (30) − α α · In the low-resolution theory of the next chapter, we Combining Eq (29) with (30) yields shall be dealing with a dense system containing hidden charges and dipole moments. This entails coarse-graining ∇P = s∇T + ρ ∇ξ + gM∇v . (31) α α j j operations, of which the most obvious result is the ap- Inserting this and the first of Eq (24) in (26), we find pearance of four fields, E, D, B, H, replacing the two here. For the remaining of this chapter, however, we ρ d v = ρ (e + v b/c) dt e × shall ignore this complication, while considering the less (s∇T + ρ ∇ξ + gM∇v ), (32) obvious, but not less important changes. − α α j j 9

Of these, the most relevant one is that we no longer Eq (36) to find that we have already made the right choice have an explicit expression for the field energy uF; more- of variables for the “2-filed” theory. What is more, given over, it is no longer possible to distinguish uF from uM, the cogent, one-to-one relationship between the thermo- such that the former only depends on the field variables dynamics and hydrodynamics, the associated set of dif- and the latter only on the hydrodynamic variables. (We ferential equations also possesses the right structure of may of course expand uF in the low-field limit, and re- the low-resolution theory. This is fortunate, as all there 1 2 tain only the lowest order term, say 2 B /µ. Although is left to do is to find out how the two fields turn into four. 1 2 That is, we may now construct the genuine low-resolution this resembles the microscopic expression, 2 b , the coef- ficient µ is a function of hydrodynamic variables.) Only theory by requiring it to reduce to the present one in the utot = uF + uM retains a clear-cut significance, because 2-field limit, E, D e and H, B b, (implying also → → it is conserved. This is relevant, as the form of uF was an E0, D0 e0 and H0, B0 b0). → → important input of the electromagnetic part of our the- One point remains to be settled: Given the algebraic ory, cf the last paragraph of section IIA2. On the other identity, hand, this is not an entirely new circumstance, as we did tot tot M u˙ = T s˙ + ξ ρ˙ + v g˙ + e0 ˙e + b0 b˙ , (39) also loose the knowledge about the explicit form of u α α i i · · when going over to the dense, second system, cf Eq (28). the “2-field” theory must be given only in terms of the So we shall proceed in two steps. First, we reconsider quantities in this equation. Unfortunately, the two fluxes, the hydrodynamic theory of a neutral system to under- Eqs (33, 34), do not seem to conform to this rule and stand its derivation, especially how the ignorance about contain gM. This is alright at present, because gM = the energy was overcome. Next, this method is general- ρv is given in terms of two quantities that do appear ized to include the e, b-fields. The result will be a low- in Eq (39), but it seems to spell disaster for the low- resolution Maxwell theory — except that two instead of resolution theory, because gM is without a clear meaning four fields are considered, or the 2-field theory. there. In equilibrium, the entropy s is a function of all the Similar to the case for utot, there is no unambiguous conserved quantities in the system, because these are division of gtot into gM and gF in systems with hid- the only ones that do not change when the system M M den charges (although it is being attempted over and is closed. In our case, the variables are u ,gi ,ρα. over again with predictably rather contradictory results). Given some (usually satisfied) mathematical properties gM M M M M This is because lacks any physical significance: First, of s(u ,gi ,ρα), we may rewrite this as u (s,ρ,gi ), or gM is devoid of its significance as the conserved material Eq (29). [Note that, at this stage, we do not yet know M M M momentum density of a neutral system. Second, without the relation between g and v ∂u /∂g .] M i ≡ i a uniquely defined u and the associated energy current As remarked in the last section, the hydrodynamic QM, neither is gM = QM/c2 useful as a work-around. equations are given in terms of the thermodynamic vari- M (Not to mention that this equality is invalid, since ℓ ables and the associated conjugate variables. This is is no longer conserved). The same holds for gF. Never- closely related to the algebraic identity given by the time theless, one clear-cut definition of gM does survive: Be- derivative of Eq (29), cause of the algebraic equivalence between Eqs (35) and u˙ M = T s˙ + ξ ρ˙ + v g˙ M, (37) (36), we may start from the second expression to obtain α α i i the first, defining gM gtot e b/c as a shorthand. ≡ − × which the equations of motion (24,25,26) must satisfy, Clearly, this guarantee that the two fluxes, Eqs (33, 34), are still given by the quantities of Eq (39), even if gM is ∇ M ∇ ∇ M v Q = T (sv)+ ξα (ραv)+ vi j Πij . (38) no longer given by ρ . · · · ∇ Finally, a few remarks about the pressure. While T This is a very confining expression, and although not and ξ, as indicated by Eq (36), remain well defined as all that obvious, does uniquely determine the fluxes QM conjugate variables, and hence retain their place in the M and Πij with the help of some auxiliary thermodynamic low-resolution theory of the next section, the pressure considerations. (This is frequently referred to as the hy- does not. [This is the reason we have chosen to elim- drodynamic standard procedure.) So the hydrodynamics inate it from Eqs (32,34).] One might think of tak- is indeed determined once the thermodynamics is; there ing P ∂ utotd3r/∂V , instead of Eq (30), only this ≡ − 0 is an one-to-one correspondence between them. derivative isR ambiguous and ill-defined: A dense system Given the explicit form of QM, the equality gM = ρv in the presence of field is anisotropic, so the energy not may now be established via gM = QM/c2. Had QM only depends on the volume, as implied by this definition, been different for some reasons, so would gM, but the but also on the shape. And the appropriate quantity to M M M hydrodynamic theory – in terms of gi and v ∂u /∂gi deal with here is not the pressure, but the stress tensor. independently – would remain formally unchanged.≡ In isotropic liquids, the pressure encompasses many Including charges and fields, the conserved quantities concepts that we find convenient, even intuitive: as the tot tot are u ,ρα and g . So the entropy will depend on surface force density, as the momentum current, as a tot tot them and the two fields, s(u ,ρα, g , e, b). Again, this quantity that is continuous across interfaces, and as a tot tot is equivalent to u (s,ρα, g , e, b) – compare this to function of T and ρ. Hence there is widespread reluc- 10 tance to abandon the pressure at finite fields. Unfortu- Poynting vector is obtained by inserting the equations of nately, though there are numerous ways to generalize the motion (21, 24, 41), for v = 0, into Eq (42), resulting in tot tot tot pressure that will preserve some of these properties, none u˙ 0 = Q0 , with Q0 = cE0 H0. covers all. So one may either define many different pres- Knowing−∇ · the rest frame quantities,× it is not difficult sures – an approach we eschew as it requires exceeding in principle to deduce the corresponding ones for the lab care and has led to considerable confusion in the litera- frame of a moving liquid, v = 0, although this does entail ture – or face up to the Maxwell stress tensor, as we shall some algebra. Fortunately,6 having established the 2-field do here. limit, we have prepared ourselves a shortcut. (The re- sults of the dilute systems of course display the correct transformation behavior.) First, Eqs (36,42) leave little C. Hydrodynamic Maxwell Theory choice other than

dutot = T ds + ξ dρ + v dgtot The theory we are going to consider, and the expres- α α · sions obtained hereby, are fairly general: They are valid +E0 dD + H0 dB. (43) for arbitrarily strong fields and nonlinear constitutive re- · · lations; the medium may be moving, and the electromag- [The algebraically minded reader may prefer to deduce netic field may depend on time. But we do assume the this expression directly. The appendix in section V is validity of local equilibrium, which needs the character- specifically written for him.] Second, since the rest frame fields are E0 = E + (v/c) B and H0 = H (v/c) D, istic time τ to be established. Therefore, all frequencies × − × are confined to ωτ 1. (We emphasize that the theory we may rewrite Eq (43) as is only complete after≪ the dissipative terms have been in- dutot = T ds + ξ dρ + v dgM + E dD + H dB, (44) cluded. Although the terms given here retain their form α α ·M tot · · off global equilibrium, at finite frequencies, we need to g g D B/c, (45) ≡ − × exercise caution when applying them alone.) M To derive the general theory, we need, as in the pre- Eq (44) is the generalization of Eq (35), with g now vious two cases, the input of the Maxwell equations and given by Eq (45). As discussed, this does not imply a field contribution of D B/c, though it does tells us some information on the energy content of the field. The M × macroscopic Maxwell equations are, that this new g replaces the old one in the fluxes of Eqs (46,49) below. Note that all conjugate variables are ∇ D = ρe, ∇ B =0, (40) now functions of all thermodynamic variables, eg T is · · a function also of D, B, and conversely, H depends on D˙ = c∇ H ρ v, B˙ = c∇ E. (41) 0 × − e − × s,ρ. Taking D,B as the variables, the first two are constraints, The rest frame Poynting vector cE0 H0 and Eq (33) × and the next two their equations of motion. If E and H makes it unambiguous that the energy flux, and hence are not specified, Eqs (41) do not contain any informa- the momentum density, now have the form, tion beyond charge conservation, as they may actually be tot M ∇ Q = (Ts + ξαρα + v g )v + c E H, (46) derived fromρ ˙e + (ρev) = 0 and Eqs (40). Especially, · × no restriction whatever· is implied for the two “fluxes” E gtot = ρv + E H/c, (47) × H and . M 2 where [Ts + ξαρα + v g ]/ρc 1 is again neglected Taking the temporal derivative of the second constraint tot · ≪ yields ∇ B˙ = 0, implying there is a vector field cE in g . (Being a term of zeroth order in the velocity, · ˙ E H/c may not be neglected with the same argument.) such that the divergence-free field B may be written as × B˙ = ∇ ( cE). Similarly, deriving the first constraint To derive the stress tensor, the last unknown expres- ∇× ˙− sion of the low resolution theory, we again take the tem- yields (D+ρev) = 0, so this divergence-free field must tot · poral derivative of Eq (43),u ˙ = T s˙ + ξαρ˙α + and relate to a field cH, such that D˙ + ρ v = ∇ (cH). e insert the respective equations of motion, Eqs (21,24,· · · 41) The temporal Maxwell equations (41) are given× content and Eq (46), to arrive at only by taking E and H as thermodynamic conjugate tot D tot B M variables, ∂u /∂ and ∂u /∂ , respectively. More ∇ [(Ts + ξαρα + v g )v + c E H] (48) precisely, we assert that, in the rest frame, the thermo- · · × = T ∇ (sv)+ ξ ∇ (ρ v)+ v Πtot dynamic energy has the form · α · α i∇k ik E0 (c∇ H ρev)+ cH0 ∇ E. tot − · × − · × du0 = T ds + (ξ0)αdρα + E0 dD0 + H0 dB0. (42) · · This equation is satisfied by the stress tensor It should not come as a surprise that we would need F tot tot M this as an input, since the field dependence of u was Πik = Πki = gi vk EiDk HiBk + tot − M− also an input, cf the last paragraph of section IIA2. ( u + Ts + ξαρα + v g + E D + H B)δik(49). (Some textbooks seem able to “derive” the expression − · · · tot E dD + H dB from the Maxwell equations, but this ev- Although this calculation only yields kΠik , so one can idently· contradicts· our simple consideration above.) The always add a term ǫ A (with ∇A arbitrary), the kmn∇m in in 11 requirement that it must reduce to Eq (34) in the 2-field are actually derived by integrating the differential equa- limit eliminates this ambiguity. tions across the interface, yielding Much information is contained in Eq (48), and one D , B , E , H = 0 (53) could indeed have derived the complete hydrodynamics △ n △ n △ t △ t directly from it , without the detour over the dilute sys- tem [7]. The path chosen in this article highlights the (provided surface charges and currents are absent). The physics and minimizes the algebra. [To understand how subscripts n and t denote the components normal or tan- gential to the interface, and A A A denotes stiflingly restricting Eq (48) is, just try out a few terms △ ≡ L − R tot the discontinuity of any quantity A, from left to right of not in Πik .] Consider the rotational invariance of the energy utot the interface, toward positive spatial coordinates. Note that since the integration has to take place in a frame in to see that Πik is symmetric: Rotating the system by an infinitesimal angle dθ, the scalars are invariant, which the interface is stationary, we have vn = 0 in all tot the boundary conditions derived in this section. du , ds, dρα = 0, while the vectors change according to dgM = gM dθ, dD = D dθ, dB = B dθ. Inserting The results of the integration are better termed “con- these into Eq× (44) yields, × × necting conditions”, since they connect the behavior of the variables on both sides of the interface. This is pos- M sible because the Maxwell equations are valid also on the v g + E D + H B =0, (50) × × × other side of the boundary. In contrast, mathematical boundary conditions make no supposition on the validity where the left side is equal to ǫ Πtot. (For vanishing ijk kj of the differential equations outside the considered region fluid velocity and linear constitutive relations, v 0, – in a sense, they shield the region from outside influence. ǫE = D and µH = B, the symmetry of the stress tensor≡ An analogous integration of the energy and momentum is obvious, otherwise it is not.) conservation, Eqs (52), yields Aside from the dissipative terms that we have consis- − − tently neglected, the expressions of this section represent Qtot =0, Πtot =0, Πtot = α(R 1 + R 1). (54) the complete and closed hydrodynamic theory of dense △ n △ tn △ nn 1 2 systems that are charged or subject to external fields. The first two connecting conditions do not contain much tot tot The independent variables are: s,ρα, g , D, B, with u independent information: Assuming v = 0 for both sides gtot and given by Eqs (44) and (47), respectively. The of the interface, and inserting the expressions for Qn, Πtn attendant equations of motion are the Maxwell equations from Eqs (46,49), we find them to be satisfied automat- Eqs (40,41), and ically in equilibrium if the connecting conditions of the fields are: Defining ni and ti as unit vectors that are ρ˙ + ∇ (ρ v)=0, s˙ + ∇ (sv)=0, (51) tot tot α α normal and tangential to the interface, Πtn = Πik tink, tot · tot tot · tot u˙ + ∇ Q =0, g˙ + Π =0, (52) Ht = Hiti, Et = Eiti, Bn = Bknk, Dn = Dknk, and · i ∇k ik with Ht, Et, Bn and Dn continuous, cf Eqs (53), we tot tot tot where Q and Π are given by Eqs (46,49). have Qn = 0 and ik △ In comparison to the considerations in available text- Πtot = (H B + E D )=0. (55) books, the main advantage of the above derivation is △ tn −△ t n t n the proof of consistency of the equations of motion with −1 −1 general principles: Especially thermodynamics, transfor- Being a generalization of P = α(R1 + R2 ), the third △tot tot mation behavior, and conservation laws. At the same boundary condition for Πnn = Πik nink is rather more time, the range of validity of these equations are clarified. useful and will be employed below repeatedly. (α > 0 is the surface tension, and R1, R2 the principle radii of The advantage does not lie so much in the additional, −1 −1 frequency and velocity dependent terms, which are fre- curvature. The term α(R1 + R2 ) is the equilibrium quently small – although this will change once dissipative surface current for the momentum density, a singular sink terms are included. or source for the bulk current.) This completes the derivation of the hydrodynamic theory. Strictly speaking, this is all we need to predict the behavior of polarizable systems, as these partial differen- D. Boundary Conditions tial equations will deliver unique solutions if a sufficient number of appropriate initial and boundary conditions To solve the derived set of hydrodynamic equations, are provided. This renders our initial question concern- boundary conditions are needed. In mathematics, boun- ing the force of electromagnetic origin, although as yet dary conditions represent information that is extrinsic to unanswered, seemingly irrelevant. However, physics does the differential equations. They are provided to choose not comprise solely of mathematics, and clear thinking one special solution from the manifold of all functions about the valid expressions for the electromagnetic force satisfying the differential equations. In physics, circum- will give us considerable heuristic power in prediction stances are frequently different. An example are the — without having to solve the set of partial differential Maxwell equations, for which the boundary conditions equations each and every time. 12

III. STRESS, FORCE AND PRESSURE where ρ1 is the density of magnetic particles, ρ2 that of the fluid, and ρ = ρ1 + ρ2 the total density. (A ferrofluid To find an expression for the force in the presence of is a suspension of magnetic particles, typically of 10 nm fields, let us first remind ourselves of its form without diameter. Note that ρ1 is the mass of particles over the fields, a quantity that revolves around the pressure: total volume. Its variation does not arise from compress- ing each particle individually, but from increasing the M Its gradient, ∇P = s∇T + ρα∇ξα + g ∇vj , is the number of the particles in a given volume element. This • j bulk force density, the quantity that accelerates a appears to be a recurrent misunderstanding in the fer- neutral volume element in Eq (32). ∇P vanishes rofluid literature.) It is useful to introduce a number of in equilibrium if we neglect gravitation. potentials, The pressure P itself is a surface force density, tot M F (T,ρα, vi,Bi,Di)= u Ts gi vi, (58) • which contributes to force equilibrium for instance − − F˜(T,ρ , v , H , E )= F H B E D , (59) in hydrostatics. P remains finite in equilibrium. α i i i − i i − i i G(T, ξα, vi,Bi,Di)= F ξαρα, (60) Since the Maxwell stress Πtot is the generalization of the − ik G˜(T, ξ , v , H , E )= F˜ ξ ρ . (61) pressure for polarizable media, we expect its gradient to α i i i − α α be a bulk force density, see section III A, and a quan- The gradient of G˜ is tity that relates to the stress itself to be a surface force density, see section III D. Presumably, as in the field M kG˜ = s kT ρα kξα g kvi (62) free case, the bulk force density vanishes in equilibrium ∇ − ∇ − ∇ − i ∇ B H D E . if gravitation is neglected, while the surface force density − i∇k i − i∇k i remains finite. Writing the Maxwell stress, Eq (49), as Generally speaking, the magnetic ponderomotive force is up to five orders of magnitude stronger than the elec- Πtot = Gδ˜ + gMv H B E D , (63) tric one. This is connected to the fact that their re- ik − ik i k − i k − i k spective, easily attainable values are similar in SI units: we find Eˆ 107V/m, Hˆ 107A/m (ie E 30, H 104, in ≈ 3 ≈ ≈ ≈ tot J/m ). Also, both susceptibilities are similar in mag- kΠik = s iT + ρα iξα + Bk( iHk kHi) (64) 4 ∇ ∇ ∇ M∇ − ∇ M pnitude, and do not usually exceed 10 , hence we have +Dk( iEk kEi) ρeEi + g ivk + k(g vk) 2 2 5 k i µ0Hˆ ǫ0Eˆ 10 . In addition to the greater ease and ∇ − ∇ − ∇ ∇ safety≈ of handling,× this frequently makes magnetic fields which, in addition to the Maxwell equations (41), are to the preferred ones. In almost all the formulas of this and be inserted into the momentum conservation, the next chapter (except in sections III A and III E), the tot tot g˙ + Π = ρgez. (65) electric terms are completely analogous to the magnetic i ∇k ik − ones, and the former may be obtained from the latter simply by employing the replacements [We shall from now on include gravitation, where g is the acceleration of gravity, and the unit vector ez points B D, H E, M P. (56) upward.] The result is → → → d For linear constitutive relations (abbreviated hereafter ρ [v + (E H D B)/cρ]= ρe(E + v B/c) dt × − × × as lcr), they imply µ ǫ, χm χ. To render the ∇ ∇ M∇ (s T + ρα ξα + gj vj ) ρgez. (66) formulas simple, we shall→ therefore→ usually display only − − the magnetic terms. Compare this to Eq (32) and register the great similar- ity — but do also remember that the temperature T and chemical potentials ξα are now functions of the fields. All A. The Bulk Force Density d terms except ρ dt v may be taken as various force densi- ties: The frequency-dependent, so-called Abraham-force, In this section, we shall calculate Πtot and look for ρ d (E H D B)/cρ, not usually a large term, the ∇k ik − dt × − × all terms in the momentum conservation that alter the “fake” Lorentz force, ρe(E + v B/c), only significant if d × acceleration ρ dt v of a volume element, starting from the the system is macroscopically charged, ρe = 0, and the energy density Eq (44), bulk force density 6

tot M bulk M du = T ds + ξ dρ + v dg + E dD + H dB. (57) f = (s∇T + ρα∇ξα + g ∇vj ) ρgez, (67) α α · · · − j − As discussed in the last chapter, the subscript α enu- which includes both the gravitational and the electro- bulk tot merates the conserved densities of the system. If there magnetic force. Note f = kΠik ρgez in the are more than one, a summation over α is implied: neutral, stationary limit, for v, −∇∇ E, ∇− H = 0, cf bulk × × ξαdρα = ξdρ for a one-component fluid, and ξαdρα = Eq (64). f is the proper macroscopic, coarse-grained ξdρ+ ξ1dρ1 for a two-component one such as ferrofluids, force, valid as long as the hydrodynamic, macroscopic 13

−2 in Maxwell theory is. Together with the Abraham force, the osmotic compressibility, κos = ρ1 ∂ρ1/∂ξ1 , which is it accounts for the difference between the true and the larger by around 6 orders of magnitude than the com- “fake” Lorentz force discussed in the Introduction. It pressibility of an ordinary liquid, cf section IV A. consists only of thermodynamically well defined quanti- ties, either variables or conjugate variables of Eq (57). This is no longer the case if we follow a widespread con- C. The Maxwell Stress in Equilibrium vention to write f bulk as a sum of the zero-field pressure gradient and the ponderomotive (or Kelvin) force, as this We consider the simplifications that occur for the introduces thermodynamically ill-defined quantities and Maxwell stress tensor if the system is stationary (v 0) unwelcome ambiguities, cf section III F 3. In equilibrium, and in equilibrium. [Starting from this section, we shall≡ we have no longer display the electric terms explicitly, E, D = 0. And the subscript 0 denoting rest frame quantities will ∇T, ∇ξ1 =0, ∇ξ = gez. (68) − also be eliminated.] Inserting these in Eq (67), we find, for stationary fluids bulk in equilibrium, f = 0. 1. Total Equilibrium The considerations of the last two chapters make abun- dantly clear that the conserved momentum density is tot With the equilibrium conditions as given in Eqs (68), the sum of material and field contribution, g = ρv+ ˜ E H/c, and that the corresponding flux is the Maxwell G is nonuniform only due to the inhomogeneities in the tensor.× Nevertheless, in the context of condensed matter, fields, both gravitational and electromagnetic, cf Eq (62). because ρv E H/c, the second term may usually be This we may utilize to obtain a more handy expression neglected —≫ and× since D B and E H are of the same for the stress. Inserting Eq (68) in (62) and integrating order of magnitude, so is× the Abraham× force. [Taking ρ the resulting expression from 1 to 2, two arbitrary points as 1 g/cm3, v as 1 cm/s, H as 104 and E as 30, both in the medium, we have in J/M2, (ie Hˆ = 107A/m, Eˆ = 107V/m,) we have ∆(G˜ ρgz¯ + B dH )=0, (71) − Eq i i ρvc/EHp 3000.] R ≈ where ∆ denotes the difference between the two points of the quantity behind it, eg ∆G˜ G˜2 G˜1. The first in ≡ 2 − B. The Incompressible Limit the above equation is from ∆G˜ = G˜dr ; the mag- 1 ∇k k 2 R netic term is from the integral 1 (Bi kHi)drk: Writing It is noteworthy that the incompressible limit in fer- 2 ∇ it as B dH , we note that itR is in fact a difference of rofluids does not implyρ ˙, ∇ρ = 0, and hence ∇ v = 0. 1 i i · purelyR local quantities, with no reference to the path: This is because incompressibility means the constancy of 2 2 1 BidH = BidHi BidHi ∆ BidHi, where the two local, actual densities, ρM of magnetic particles 1 0 0 0 denotes a (possibly− virtual) spot≡ of vanishing field, and ρF of the fluid matrix (ie ρ1 = ρM and ρ2 = ρF , R R R R with the averaging taken over a volumeh i containing manyh i Hi = 0, and 1, 2 respectively the field values at the spa- particles). Yet since the particles are usually denser than tial points 1, 2. The integration is unambiguous because it is to be carried out in equilibrium, for constant T, ξ1 the fluid, ρM 5ρF, an increase of particle concentration ≈ and ξ +gz, or for constant T , ξin +γgz in the incompress- will also increase the total density ρ = ρ1 + ρ2. More 1 ible approximation. To emphasize this, the subscript quantitatively, ρ1/ρM+ ρ2/ρF = 1 because ρ1/ρM is the Eq fraction of volume occupied by the particles, and ρ /ρ is added. With Mi Bi Hi, we may also write thisR 2 F 1 2 ≡ − that occupied by the fluid. Taking ρM and ρF as constant term as 2 H + EqMidHi. The gravitational term comes 2 in the incompressible limit, we have from integratingR the chemical potential, ρ ξdr = − 1 ∇k k 2 2 1 R g 1 ρdz = g 0 ρdz g 0 ρdz = g∆(zρ¯), where zρ¯(z) dρ = γdρ1, γ =1 ρF/ρM. (69) z − − R ρdz. R R ≡ 0 in Eq (71) states the constancy in equilibrium of the Inserting this expression into ξdρ+ ξ1dρ1, we have ξ1 dρ1 R in quantity in the bracket, call it K, with ξ1 = ξ1 + γξ, and the modified equilibrium condi- − tions K G˜ ρgz¯ + 1 H2 + M dH . (72) − ≡ − 2 Eq i i in ∇T =0, ∇ξ = gγez. (70) R 1 − This enables us to write the stress tensor, Eq (63), as Ferrofluids may frequently be approximated as being in Πtot = (K + 1 H2 + M dH ρgz¯ )δ H B , (73) ij 2 Eq i i − ij − i j the incompressible limit considered here, because the R variations of ρF and ρM, say as a result of field inho- the announced handy expression. As we shall see, it mogeneities, are usually negligible. The particle density is rather useful, and even includes the information con- ρ1, on the other hand, may vary greatly, and so does ρ tained in the magnetic Bernoulli equation [2], see sec- tot via Eq (69). The parameter controlling this behavior is tion III F 2. More generally, we may calculate Πij for 14 an arbitrary point within the medium — if we know the Inserting Eq (78) into the third of Eqs (54), assum- field everywhere and the value of the constant K (usually ing atmosphere on one side, employing Eq (74) while ne- via the boundary conditions to be discussed below). glecting the gravitational termρgz ¯ of the atmosphere, If the system under consideration (say the atmosphere) the boundary condition reduces to tot M is non-magnetic, Mi = 0, Bi = Hi and u = u + 1 2 1 2 H hold. Inserting these into Eq (72) and employing K + MidHi + Mn ρgz¯ 2 Eq 2 − Eqs (30), we have R −1 −1 = Patm + α(R1 + R2 ), (79) K =ρgz ¯ + P , (74) atm a useful formula that we shall frequently refer to below. with K clearly the pressure at z = 0. If the system is magnetic, we may still write G˜(T, ξα, Hi) = G˜(0) + 2. Quasi-Equilibrium G˜em, where the first denotes G˜ for vanishing H, while the second the contribution from the field, The considerations of the last section is not confined 1 2 G˜em = B dH = H + M dH , (75) to total equilibrium, and may be generalized to include − Eq i i 2 Eq i i R R quasi-equilibria. Establishing equilibrium with respect and identify to the distribution of magnetic (or electric) particles is a slow process, because mesoscopic particles diffuse much K =ρgz ¯ G˜(0). (76) − more slowly than atomic ones. Depending on the field gradient, the relevant volume and the size of the par- For obvious reasons, G˜(0) is frequently referred to as the ticles, this may take days, even weeks [4, 8]. For a “zero-field pressure”. This is unfortunate, as G˜(0) G˜ ≡ rough estimate, we equate the Stokes with the Kelvin G˜em is a function of T,ξ,ξ1 – these are the natural − force to calculate the velocity v with which a magnetic variables of G˜, and the value of the integral in Eq (75) 3 2 particle moves: 6πηRv = (4πR /3) χµ0 Hˆ /2. Tak- explicitly depends on this choice of variables. On the ing the particle radius as R = 10nm,· the∇ viscosity as other hand, if the zero-field pressure were a physically η = 10−3kg/ms, the susceptibility as χ 1, the field as ≈ sensible quantity, it should remain the same whichever Bˆ = 0.1T, and the field gradient Bˆ as 1T/mm, the variables it is taken to depend on. It does not, of course, −3 ∇ ˜ velocity is around 10 mm/s, and the time the particles as it is equal to G(0). This renders the associated concept needs to achieve equilibrium is τ 1mm/v 103s. On of pressure ambiguous, ill defined, and best avoided, see the other hand, particles 102 times≈ larger, of≈ the size of section III F for more details. 1µ, lead to velocities 104 times larger, and a characteris- Combining Eq (62) and (75) – ignore gravitation and tic time of 10−1s. For time scales much smaller than τ, the electric field – we find after a uniform ferrofluid is brought into contact with an inhomogeneous magnetic field, the following conditions ∇ BidHi = Bi∇Hi. (77) Z hold instead of Eqs(68),

For a function of one variable, this relation always holds: ∇ρ1 =0, ∇T =0, ρ1∇ξ1 + ρ∇ξ = ρgez. (80) Defining F = f(g)dg, we have ∇F = (dF/dg)∇g = − f∇g. For a functionR that depends on more variables, This is because both the diffusion of heat and the estab- this relation holds only if the additional variables are lishment of mechanical equilibrium are in comparison fast spatially constant. In the present case, B does depend on processes. The last of Eq (80) states f bulk = 0, cf Eq (67), additional variables, T , ξ1, ξ + gz. Fortunately, all three which is due to the density ρ quickly turning slightly are constant. [B may also, as in next section, depend on nonuniform to compensate for the field inhomogeneity. ∇ ˜ ∇ constant T , ρ and ρ1 — which is the reason the subscript Inserting Eq (80) in (62), we find G = ρgez Bi Hi, lcr an expression that we can integrate directly, from− a point in Eq is eliminated from Eq (77).] Considering , we 1 2 1 to a point 2, if we approximate the variables T , ρ, ρ1 findR ∇ BidHi = ∇( µH ), which is equal to Bi∇Hi if 2 as constant, yielding µ (sayR as a function of T ) is constant. Turning our attention now to boundary conditions, ˜ tot ∆(G ρgz + uniBidHi)=0, (81) we find that the expression Πnn [which occurs in the − boundary condition Eqs (54)]△ may be written as R where the subscript uni indicates that the integral is tot 1 2 evaluated at constant T , ρ, and ρ1. Again calling the Π = (K + MidHi + M ρgz¯ ) (78) R △ nn △ Eq 2 n − spatially constant quantity in the bracket K, the stress R − 1 2 tensor is in equilibrium. Two remarks: (i) The equality ( 2 H 1 2 1 2 1 2 △ − HnBn) = ( 2 Mn + 2 Ht 2 Bn) was used to derive tot 1 2 △ − Πij = (K + H + MidHi ρgz)δij HiBj , (82) Eq (78), where (H2), (B2 ) vanish, cf Eq(53). (ii) If 2 uni − − △ t △ n R appropriate, substitute Eq with uni, andρ ¯ with ρ, cf where the subscript of the integral and the lacking bar Eq (82) below. R R over ρ are the only differences to Eq (73). Integrating as 15 here the magnetization while holding constant T and ρα D. Equilibrium Surface Force yields the electromagnetic contribution to the free energy,

1 2 Given a solid, polarizable body, or one that is (al- F˜em(T,ρ )= B dH = H + M dH , (83) − α uni i i 2 uni i i though not polarizable) submerged in ferrofluid, we may R R so K may be written as lift the body off the ground electromagnetically, against the gravitational force, or balance it against some elastic K = ρgz [F˜(0) ξ ρ ], (84) − − α α force exerted by a string or spring. The same applies to while Eq (74) still holds for nonmagnetic media. a vessel containing ferrofluid. All these happen even in equilibrium, when the bulk force density, Eq (67), van- Note that Mi(T,ρ1, Hi) is the appropriate function for in ishes. This force is evaluating uni MidHi, while Mi(T, ξ1 , Hi) is the proper one for EqRMidHi – both in the incompressible approxi- elm 1 2 = [ M + MkdHk] dA (89) mation.R The first is given by an experiment that quickly F △ I 2 n Eq R measures the magnetization while varying the external = [ MtdHt + MndBn] dA, (90) field in a closed system, such that ρ1 stays constant – △ I Eq Eq though contact with a heat bath is necessary to main- R R in with the electric counter terms as usual given by Eq (56). tain the temperature. The latter, Mi(T, ξ1 , Hi), is to now specifically denotes: internal - external, and the be measured in an open system that is connected to a △ particle reservoir, which itself is not subject to a vary- two integrals are to be taken along the surface of the in body, one right inside and the other right outside of it. ing field, so its chemical potentials ξ1 remains constant. Increasing the field of the system now to measure the As in the last section, Eq is to be substituted by uni, magnetization, magnetic particles will enter the system andρ ¯ by ρ, where appropriate.R R from the reservoir, resulting in a stronger magnetization Before deriving Eq (89), we shall first consider the ram- than in the closed case. ifications of this formula in two examples: (i) A magne- Either experiment is sufficient to determine both mag- tizable solid body (or a nonmagnetic vessel containing netizations, as one can be calculated from the other. This ferrofluid) in atmosphere, (ii) a nonmagnetic solid body is accomplished by employing the thermodynamic rela- in ferrofluid. tion, In the first example, the external magnetization is zero. Employing the Gauss law, we change the surface integral ∂M ∂M ρ2 ∂M 2 ∇ 1 of Eq (90) into a volume integral: With EqMtdHt = = + −1 , (85)  ∂H  in  ∂H  κos  ∂ρ1  ∇ ∇ ∇ R ξ1 ρ1 H Mt Ht and EqMndBn = Mn Bn, cf Eq (77), the −1 2 electromagneticR force is where the inverse osmotic pressure is given as κos = ρ1 in (∂ξ1 /∂ρ1)H . Eq (85) may be verified by combining a elm ∇ ∇ 3 × = (Mt Ht + Mn Bn)d r. (91) thermodynamic identity with a Maxwell relation, F Z ∂M ∂M ∂M ∂ρ Consider a plate with the field gradient normal to its = + 1 , (86)  ∂H  in  ∂H   ∂ρ1   ∂H  in surface: If the field is predominantly tangential to its ξ1 ρ1 H ξ1 3 surface, the electromagnetic force is Mk∇Hkd r; if the ∂ρ ∂B ∂M ∂ρ 3 1 1 field is normal to the plate’s surface,R it is Mk∇Bkd r. = in = in . (87)  ∂H  in ∂ξ   ∂ρ  ∂ξ  ξ1 1 H 1 H 1 H So interestingly, the magnetic force densityR interpolates between both forms of the Kelvin force as discussed in [We are assuming that M H, so all M and H here are k the introduction – and in greater details in section III F 3 to be understood as the respective magnitude. In addi- below – though with the difference that these here are tion, T is held constant and ρ is taken to satisfy the in- exact results, as no assumption was made with respect compressibility condition Eq (69) throughout.] For lcr, to the constitutive relation, especially the density depen- Eq (85) reduces to dence of the magnetization (or susceptibility). m 2 2 If the geometry is more complicated than in a plate, m in m ∂χ H say if the magnetizable body is an ellipsoid, it is less clear χ (T, ξ1 ,ρ)= χ (T,ρ1,ρ)+ ρ1 −1 , (88)  ∂ρ1  κos what the normal and tangential component of M, H and showing that the difference between the two susceptibili- B are in the bulk, and Eq (91) appears ambiguous. For- ties is of higher order in the field, and may be neglected if tunately, this does not matter, because the integral of one strictly adheres to lcr. On the other hand, this cal- Eq (91) has (by virtue of the Gauss law) a unique value, culation also shows when the second term may no longer as long as MtdHt and MndBn are given at the sur- be neglected: Assuming for an order of magnitude es- face, cf EqR (90), and the fieldsR Mt, Ht, Mn, and Bn being m m −1 continuous in the bulk. [An analogous situation is given timate that ρ1(∂χ /∂ρ1) χ 1, and taking κos 2 3 ≈ ≈ ′ 10 Pa (see the calculation in section IV A below), by the simple integral 1 fdx, F = f, and F (1), F (2) ≈ ˆ 2 R we find that a field of H 300 Oe suffices to render fixed. Then 1 fdx = F (2) F (1) is unique irrespective 2 −1 ≈ − H /κos 1. of how f variesR in the interval between 1 and 2.] ≈ 16

tot Turning our attention now to the second system, a Inserting Eq (78) for Πnn, the Gauss law is em- non-polarizable body submerged in ferrofluid, we take ployed to find KdA△ = Kd3r = 0. As △ i △ ∇i the internal magnetization as zero and find a result, only termsH that dependR explicitly on fields are left. From these, the gravitational force is found grav ∇ 3 elm 1 2 A to be = gz ρ¯dA = ( ρgz¯ )d r = = [ 2 Mn + EqMkdHk] d (92) F 3 − △ − △ F − I geˆz ρ d r, whileH the electromagneticR force is found R − △ to be givenR by Eq (89). = [ MtdHt + MndBn] dA. (93) − I Eq Eq Because Eq (94) is valid for any interface, and not R R confined to the considered geometry, so are the formu- It may also here be useful to employ the Gauss law for a las Eqs (89), (90), (96), and (97). Clearly, there is an conversion of the surface integral into one over the volume electromagnetic surface force density of the given form of the body. However, as the physical fields in the volume whenever the Maxwell stress is discontinuous, ie when are not continuous with their respective surface values, the magnetization (or polarization) changes abruptly. virtual fields have to be defined which do. They need not To confirm this, consider a further example: Ferrofluid be healthy fields, may violate the Maxwell equations, or contained in a nonmagnetic vessel, and the whole system have a non-physical susceptibility. hung on a string in atmosphere. It is slightly more com- We now derive Eqs (89) by first considering a solid plicated geometry, as there are two interfaces to consider: body submerged in fluid and attached to a string. Both ferrofluid-solid, and solid-air. Each has a boundary con- s a the solid and fluid may be magnetizable — though differ- dition of the type of Eq (94). We have σij = σij at the tot ently, with the Maxwell stress Πij respectively denoted solid-air interface, because the Maxwell stress is contin- s f s a as Πij and Πij . One of the boundary conditions at the uous there, Πij = Πij . An integration yields, as before, interface is then σs dA = σa dA = elast. The boundary at the nn i nn i −Fi ferrofluid-solid interface is Πff = Πs + σs , Integrating s s f f H H ij ij ij σij + Πij = Πij + σij , (94) over the interface, we find (Πff Πs ) dA + elast = 0, ij − ij i Fi or again that Eq (96), andH therefore Eqs (97) and (89) where σs ∂utot/∂u is the elastic stress tensor of the ij ≡ ij to be valid. solid, given by deriving the thermodynamic energy with In the literature [2, 5], instead of Eq (89), the electro- respect to the elastic strain u . In equilibrium, σs ij ∇j ij magnetic force is usually given as = 0. Being a liquid, σf 0 except where the surface ij ≡ integral cuts across the string, which also has elasticity elm = (HiBj BkdHkδij )dAj . (98) we need to account for. Same as Eqs (54), this boundary F I − condition states the continuity of the total momentum R current σij + Πij . [The surface tension α is irrelevant for This expression may also be obtained by integrating over s s s s 3 a solid surface of given shape and therefore eliminated.] Eq (94). First note (σ +Π )dAj = j (σ +Π )d r ij ij ∇ ij ij Each of the four terms of Eq (94) stands for a sur- = 0, because the integrand,H as the totalR momentum flux face force density. So the integral over the closed surface density, has vanishing divergence in equilibrium (neglect, of the solid body must yield the equation of force equi- for simplicity, gravitation). Writing the force equilib- f f elast librium, between the gravitational, electromagnetic and rium as (Πij + σij )dAj = 0, again identifying i = f elm f F elastic force, σij dAH j , we find i = Πij dAj by comparison − F − f elast grav elm withH Eq (95). Now, inserting EqH (73) for Πij , and be- i + i + i =0. (95) F F F cause KdAj = 0, we obtain Eq (98). Although algebraically equivalent, Eqs (89) and (98) The first term of Eq (94) vanishes, because σs dA H ij i may behave quite differently when being evaluated for a = σs d3r = 0. The last term yields the elasticH force ∇j ij concrete geometry. For instance, considering the force on exertedR by the string, which is given by the normal com- elast f a plate starting from Eq (98), it is not easy to find the ponent, i = σnndAi, because the string cannot result of Eq (91). This is because Eq (98) is a rather non- F − f sustain a shear stress,H σtn = 0. Clearly, this implies local expression, which does not even explicitly depend on the susceptibility of the magnetizable body. grav elm tot The above lengthy discussion should not obscure the i + i = Πij dAj . (96) F F I △ point that these equilibrium surface forces are no more tot tot tot than interpretation and visualization of boundary con- Because Πij dAj = Πij nj dA = δik Πkj ditions, especially Eqs (78, 94). In fact, when consider- △ △tot △tot nj dA =H (titk + nink) HΠkj nj dA = (nHi Πnn + ing the experiments in Chapter IV, we shall usually em- tot △ tot △ ti Π ) dAH, and because Π vanishesH identically, cf ploy the boundary conditions, to solidify the algebraic, △ tn △ tn Eq (55), this reduces to safe, albeit slightly more tedious approach. But we shall also frequently point out how the surface forces consid- elm grav tot ered here would have yielded the same results. The lat- i + i = Πnn dAi. (97) F F I △ ter serves to demonstrate that these forces represent a 17 heuristic concept of considerable power, and to give fur- ther reassurance, if one is needed, of the validity of the formulas in this section.

E. Thermodynamic Derivation of the Stress

It is worthwhile to rederive our central result, the Maxwell tensor, Eq (63), via a more direct path, though with a rather narrower focus and ensured range of va- lidity. This is done in the form of an expanded Lan- dau/Lifshitz consideration, separately for the electric and + − magnetic contributions. We start from the energy ex- FIG. 1: Metal plates at Sx and Sx with constant charge, + pression Eq (57), or from one of its potentials, Eqs (58, dielectric medium sandwiched between them. Displacing Sx 59, 60, 61), and shall take pains in eliminating the flaws by δx or δz respectively decompresses and shears the medium. mentioned in the introduction. Deforming an isolated system at given entropy, mass xz, Az = xy, respectively; and the volume is V = xyz. and electric charge, the energy change is ± Taking the two metal plates as Sx , the electric fields tot tot E,D are along eˆx, see Fig 1. (We neglect the small stray δU δ u dV = ( Π δri)dAk, (99) ≡ Z − I △ ik fields at the edges.) The capacitor is placed in a vacuum, so there is neither electric field nor material outside. where the energy density utot is given by Eq (57) with + We now successively displace the three surfaces Sx , v + + = 0. Notations: δri is the infinitesimal (or virtual) Sy , Sz , in all three directions, δri = δx,δy,δz (which displacement of the surface, and dAk nkdA with dA is why the capacitor has to be finite), while holding con- ≡ the surface element and nk the surface normal pointing stant the quantities: entropy sV , masses ραV , and elec- outwards. The validity of this equation is connected to tric charges q = DAx (from ∇ D = ρe). Because of the tot ± · Πik being the electromagnetic force density needed simple geometry, Eq (100) holds and will be evaluated. −△to deform the system, see Eq (96). All gravitational + First, take the surface to be displaced as Sx . When terms are neglected in this section. the displacement is δx, we have If systems are considered for which the external stress tensor vanishes, because neither material nor field lies δV = Axδx, δs/s = δρα/ρα = δx/x, δD =0; (101) tot − outside the considered volume, we may substitute Πik for tot and we have δV , δs, δρα, δD = 0 if the displacement is Πik in Eq (99). Only these systems will be considered + △ tot δy or δz (implying a shear motion of Sx ). Inserting all below. In the absence of fields, Πik Pδik with P uniform in thermodynamic equilibrium,→ so δU reduces three into Eq (100), we obtain to P δridAi = P δV , or Eq (30). Note that if the tot tot tot tot − − tot tot Πxx δx = (Ts + ξαρα u )δx, Πyx = Πzx =0. (102) projectionH of the surface force Πnn = Πik nink is positive − (ie positive pressure if without field), the volume tends If the surface is S+ and the displacement δz, we have tot z to expand, if Πnn < 0, it tends to contract. δV = A δz and δs/s, δρ /ρ , δD/D = δz/z. If the tot tot z α α − In simple geometries, if u and Πik δri are uniform, displacement is δx or δy, we have δV , δs, δρα, δD = 0. Eq (99) reduces to Hence δU = δ(utotV )= utotδV + V δutot = A Πtotδr . tot tot k ik i Πzz δz = (Ts + ξαρα + ExDx u )δz, (103) − (100) − tot tot and Πxz = Πyz = 0. As the directions eˆy and eˆz tot tot Since u is known, we shall evaluate AkΠik δri while are equivalent, we know without repeating the calcula- + tot tot taking Ak and δri to point in all three directions, per- tion that a displacement of Sy yields Πzz = Πyy and tot tot pendicular and parallel to the field, and hereby obtain all Πxy = Πzy = 0. (The term ExDx is a result of the metal nine components of Πik. plates being squeezed, compressing the surface charges, δq/q = δD/D = δz/z. The compressibility of the metal is taken to be infinite.− Otherwise, it would contribute an 1. The Electric Part additional term in the stress tensor that we are not in- terested in here.) We consider a parallel-plate capacitor that is filled with These considerations have yielded all nine components tot a dielectric medium. Denoting the three linear dimen- of Πik for a special coordinate system. Because the stress sions of the capacitor as x,y,z, with x y,z, the six tensor of Eq (63), for D, E eˆx and B, v = 0 produces ± ± ± ≪ k surfaces Sx , Sy , Sz (with the outward pointing nor- exactly these components, it is the correct, coordinate- mal eˆ , eˆ , eˆ ) have the areas A = yz, A = independent expression. ± x ± y ± z x y 18

This conclusion may appear glib, but is in fact quite solid. Consider first a vector: If two vectors are shown to be equal in a special coordinate system, we know that they remain equal in any other system – as long as we are sure that they are indeed vectors. The same also holds for tensors. (The careful reader may notice an ambigu- ity with respect to the off-diagonal part, as both EiDk and EkDi yields the same nine components derived here. Fortunately, there is no difference between these two ex- pressions, because E D = 0 for B, v = 0, cf Eq (50).) This concludes a thermodynamic× derivation of the elec- tric part of the Maxwell stress tensor. tot Substituting the dielectric medium with vacuum, Πxx , Πtot reduce to e2/2, respectively, implying a tendency zz ∓ to contract along eˆx and expand along eˆy and eˆz — as it should in the considered case: The differently charged plates want to come closer, while the charge in each plate FIG. 2: A magnetic system with constant current J ⊥ xˆ, would like to expand. If the medium is one with negligible enforced by an external battery, not shown. Again, it is de- tot tot 2 + susceptibilities, Πxx , Πzz are given as P e /2, so the formed by displacing say Sx along xˆ or zˆ. same electric force must now contend with∓ the pressure in deforming the system. Consider the same capacitor, now held at a constant the isolated electric one, with all equations – both the dis- voltage φ. The modified system must lead to the same played Eqs (100,101,102,103) and the others around these stress tensor, because the stress tensor is a local expres- – remaining valid after the displacement D B, E H sion and may depend only on the local field. The cal- has taken place. (When considering compressional→ → dis- culation is rather similar, though one needs to replace placements, the elastic energy of the metal sheet is again utot in Eq (100) with the potentialu ˜ utot E D, neglected.) as the system is no longer electrically isolated.≡ − And· the We shall now consider the rod being deformed at con- constraint for E as the new variable is Ex = φ (which re- stant current and temperature, so the energy needed to places DAx = q). Connecting the capacitor in addition deform the system is to a heat bath necessitates the employment of the free tot ˜ ˜ ˜ tot energy, F˜ = u Ts E D, and changes the constraint δ(F V )= F δV + V δF = AkΠik δri, (104) − − · ˜ − (from constant sV ) to δT =0. (F is the potential used δF˜ = sδT + ξαδρα B δH, (105) in [5].) For the explicit calculation please cf the mag- − − · netic case below, Eqs (104, 105, 106, 107, 108), with the where F˜ = utot Ts H B. We again successively replacement B D, H E implemented. The final − − ·+ + + → → displace the three surfaces Sx ,Sy ,Sz , in all three direc- result is the same, given by Eq (63). tions, δri = δx,δy,δz, while holding constant the quanti- ties: temperature, masses, and the current, ie under the conditions, δT = 0, δ(ραV ) = 0 and δ(Hx)=0. + 2. The Magnetic Part The first surface to be displaced is Sx . When the displacement is δx, we have To obtain the magnetic part of the stress tensor, con- δV = Axδx, δH/H = δρα/ρα = δx/x; (106) sider a rod along eˆx, of square cross section, made of − a magnetizable material and placed in a vacuum. The and we have δV , δρ , δH = 0 if the displacement is δy surfaces S±,S± are covered with a sheet of metal that α y z or δz (implying a shear motion of S+). Inserting these carries a current J eˆ . With A A , A , the mag- x ⊥ x x ≪ y z into Eq (104), we obtain netic field will be essentially along eˆx and confined to the interior of the rod, see Fig 2. So again, there is neither tot tot tot tot Π δx = (Ts + ξαρα u )δx, Π = Π =0. (107) field nor material outside. xx − yx zx If the system is isolated, the metal needs to be su- + If the surface is Sz and the displacement δz, we have perconducting to sustain the current, and the constraint δV = A δz, δH = 0, and δρ /ρ = δx/x. If the z α α − on the variable B during a deformation is constant flux, displacement is δx or δy, we have δV , δρα, δH = 0. BAx = Φ. (Compare this to the isolated electric case Hence with DAx = q.) If the current J is held constant by a ∇ e tot tot battery, the constraint is Hx = J/c (from H = j /c Πzz δz = (Ts + ξαρα + HxBx u )δz, (108) and analogous to Ex = φ), and the attendant× potential − tot tot tot isu ˜ u H B. and Πxz = Πyz = 0. Since the directions eˆy and eˆz are ≡ − · tot tot tot tot The calculation of the isolated magnetic case repeats equivalent, we have Πzz = Πyy and Πxy = Πzy = 0. 19

This consideration has yielded all nine components of F. Ambiguous Notations tot Πik for a special coordinate system. Because the stress tensor of Eq (63), for B, H eˆ and D = 0 produces these k x In this section, we shall consider some frequently em- components, it is the correct, coordinate-independent ex- ployed expressions and notations that we shall see are pression. This concludes the thermodynamic derivation rather misleading, and therefore perhaps best avoided. of the magnetic part of the Maxwell stress tensor. On the other hand, many of us have so accustomed Although we have only employed the potentials F and ourselves to this notation that we tend to think along F˜ above, it should be clear by now that we could just the categories it provides. But even then, or especially as well have employed G and G˜ from Eqs (60,61), as- then, one should welcome the opportunity to realize all suming that the polarizable medium is connected to par- its pitfalls which, as discussed in the introduction, lie ticle reservoirs, which keep the chemical potentials ξα mainly in the ambiguity of the zero-field pressure, and constant. This changes the constraint from δρα/ρα = of M dH. The expressions one arrives at are for in- · δx/x, δy/y to simply δξα = 0. The derived expres- stance (P + 1 H2)δ H B for the stress, or M H for − − R 2 ik i k i i sion for the stress tensor remains unchanged. the Kelvin force. Scrutinizing− their derivation, we∇ shall As a stand-alone, the thermodynamic consideration of in addition conclude that these formulas are only valid the last two sections gives us a fairly clear idea on the for small susceptibility χm 1, or M H — a range tot ≪ ≪ form of the macroscopic Maxwell stress tensor Πik , in of validity dramatically smaller than taken for granted equilibrium, for v 0, and with either the electric or the usually. So if χm is of order unity, and M H, as is magnetic field present.≡ The most important information frequently the case in strongly polarizable systems≈ such tot tot it withholds is that about gi , without which Πik is as ferrofluids, these formulas are invalid. not unambiguously defined at finite frequencies. Also, tot gi is needed to complete and close the set of differen- tial equations given previously, which alone is capable of 1. Different Zero Field Pressures providing a consistent and comprehensive description of polarizable systems. In addition, relying solely on this We shall take an approach here that is somewhat consideration, one would perhaps need to be more care- broader than usual, and simultaneously work with differ- ful with the stray fields, especially when thinking about ent potentials. This will lead to rather flagrant appearing the possibilities of terms such as E , H . ∇i j ∇i j contradictions, the understanding of which should lend us Rosensweig has also considered and derived the mag- a sharpened view of the adopted notation. To simplify netic part of the stress tensor in [2]. His result is the same the formulas, we shall neglect gravitation, and consider in as the one here, though his algebra is rather more com- the incompressible limit of ferrofluids, ρα, ξα ρ1, ξ1 , in → plicated. More problematically, one of his basic, starting cf section III B. (Replacing ρ1, ξ1 with ρ, ξ yields the assumptions does not hold up: His geometry is a slab results for a one-component, compressible liquid.) tot with Ay, Ax Az and current-carrying wires along the We start by separating the energy u (s,ρ1, B), the ± ≪± surfaces Sy ,Sz , see his Fig 4.1. The winding of the wires free energy F (T,ρ1, B), and the potentials F˜(T,ρ1, H), is oblique, the currents flow along eˆy in the two larger in ˜ in ± ± G(T, ξ1 , B), and G(T, ξ1 , H), cf Eqs (58 - 61), into their plates Sz , but has a component along eˆx in the two ± ± zero-field and electromagnetic contributions, narrow side walls Sy – take them to be along mˆ , a ± tot vector in the xz-plane. Rosensweig maintains that the u = u(0) + uem(B), F = F (0) + Fem(B), (109) resultant field is uniform and perpendicular to the sur- face given by the winding, ie by eˆ and mˆ . y with analogously defined Gem(B), F˜em(H), G˜em(H). Ad- Unfortunately, the field is neither uniform nor mainly hering to convention, we write the potentials that are oblique, rendering large parts of the ensuing considera- functions of B as tion invalid. First the qualitative idea: If the two much ± 1 2 larger plates S were infinite, the field would be strictly uem, Fem, Gem = H dB = B M dB, (110) z · 2 − · parallel to eˆx. This basic configuration should not change R R much if the plates are made finite, and supplemented with and the potentials with tilde that are functions of H as the two narrow side walls S± – irrespective of the cur- y ˜ ˜ 1 2 rents’ direction there. This argument is born out by a Gem, Fem = B dH = H M dH. (111) − · − 2 − · calculation to superpose the fields from various portions R R of the currents. First, divide all currents along mˆ into Note that the respective integral is to be taken at con- ˜ two components, along eˆ and eˆ . Next, combine± the stant s,ρ1 for uem, at constant T,ρ1 for Fem, Fem, and ± z ± x in ˜ first with the currents along eˆ , such that the four sec- at constant T, ξ1 for Gem, Gem. So uem, Fem, Gem are ± y ˜ ˜ tions of the four surfaces form a closed loop at the same in fact not equal, neither are Gem, Fem. The situa- x-coordinate. The resultant field of all loops is clearly tion is similar to evaluating T ds when considering a the main one, and strictly along eˆx. The leftover cur- Carnot process: Since T = TR(s,ρ), we need to specify ± what the density ρ does when s varies. [We have al- rents are those at Sy along eˆx and their total effect is ± 1 2 a small dipole field. ready introduced G˜em = H + MidHi and F˜em = − 2 Eq − R 20

1 2 2 H + uniMidHi in Eqs (75, 83), given each a specify- thermodynamic expressions do not depend on the vari- ing index,R and discussed why the difference between these ables chosen. This stems from the fact that P is not a expressions is irrelevant as long as lcr holds, cf Eq (88).] thermodynamically defined quantity in the presence of To evaluate the stress Πik, Eq (63), in terms of these fields: It does not appear in Eq (44), as all bona fide potentials, we start with its diagonal part G˜, given re- thermodynamic variables do. Since Eq (49) or (63) con- spectively as − sist only of quantities which do appear in Eq (44) – both tot with or without field – the stress tensor Πik given there ∂ ∂ tot 1 2 G˜ = (s + ρ1 1) u = P (s,ρ1)+ H holds for any set of variables. These are good reasons to − ∂s ∂ρ1 − 2 1 2 ∂ ∂ make this the expression of choice. M + dB (1 s ρ1 )M, (112) − 2 · − ∂s − ∂ρ1 Frequently, a further approximation is employed for ∂R 1 2 G˜ = (ρ1 1) F = P (T,ρ1)+ H the diagonal part of the stress. In dilute, one-component − ∂ρ1 − 2 1 2 ∂ systems, the magnetization is usually proportional to M + dB (1 ρ1 )M, (113) − 2 · − ∂ρ1 the density, or M = ρ(∂M/∂ρ). Similarly, in magnet- ∂ R 1 2 G˜ = (ρ1 1) F˜ = P (T,ρ1)+ H ically dilute ferrofluids, we may assume that the mag- − ∂ρ1 − 2 ∂ netization is proportional to the particle density ρ1, + dH (1 ρ1 )M, (114) · − ∂ρ1 M = ρ1(∂M/∂ρ1). Inserting this into Eqs (113) and inR 2 G˜ = P (T, ξ )+ 1 H + M dH, (115) (114), we find, respectively, − 1 2 · R where depending on the variables, the zero-field pres- G˜ = P + 1 (H2 M 2), G˜ = P + 1 H2. (117) ∂ ∂ 2 2 sures are P (s,ρ1) = (s + ρ1 1)u(0), P (T,ρ1) = − − − ∂s ∂ρ1 − ∂ in ˜ The difference between these two expressions are not due (ρ1 1)F (0), and P (T, ξ1 )= G(0), cf Eq (30) with ∂ρ1 − − to a different set of variables, as P is a function of T and u(0) = uM, v = 0. These four differing G˜, and with i ρ in both cases. So we are dealing with a different pitfall tot ˜ 1 them the attendant stress Πik = Gδik HiBk cer- here, one that we shall discuss in details in section III F 3 tainly look disturbingly contradictory.− Yet− all must be in below. The gist of it is: When we assume M ρ1, this equal. Equating Eq(114) with (115), we find P (T, ξ1 )= is meant as an approximation, implying the∼ neglect of P (T,ρ1) ρ1∂M/∂ρ1 dH, forcing the conclusions that 2 3 − · square and higher order terms ρ1,ρ1 . But consis- the zero fieldR pressure P is a quantity that depends on tency then dictates that we must∼ neglect· ·all · higher order the variables chosen, and that their difference is of or- 2 2 terms, including M ρ1. This implies especially that der magnetic field squared. This means especially that the dilute limit is only∼ given if M H and χm 1 only one of the pressures may be field-independent under 1 2 ≪ ≪ hold. Therefore, the term 2 M in Eqs (117) must be given circumstances. neglected, and the popular form for the Maxwell stress The pressure all textbooks (including [2, 5]) take to be 1 2 tensor, Πik = (P + 2 H )δik HiBk, is to be taken with field independent is P (T,ρ), equivalent to P (T,ρ1) in our a large grain of salt, as it is− valid only for M H, and case. This is the pressure that one would measure in the quite useless if M H, or χm 1. ≪ absence of field, for given temperature and density. The ≈ ≈ tot gradient of the remaining, field dependent terms, j [Πij P (T,ρ )δ ], cf Eq (114), are then interpreted∇ as the 1 ij 2. Magnetic Bernoulli Equation −electromagnetic force, referred to as ponderomotive or Helmholz force. But all this is obviously only valid if the temperature and density are indeed kept constant when The magnetic Bernoulli equation by Rosensweig is a the field is switched on. Under adiabatic circumstances, very useful relation. It has been extensively employed when s is kept constant instead of T , the pressure P (s,ρ1) in his book [2], and in the literature on ferrofluids. We is the one that is field independent. Consequently, the shall include the variation of concentration, which he did above Helmholz force is no longer the correct expression not consider, and in addition, free this relation from the for the electromagnetic force. Rather, one must take the ambiguous notation of the last section III F 1, in which field dependent terms from Eq (112) instead. it is given. The point is, the information in the magnetic The ambiguity of P arises from the fact that the de- Bernoulli equation is contained in Eq (71). Combine it pendent variables change when the field is switched on. with Eq (115) to yield in ˜ For instance, the chemical potential, ξ1 = ∂F (0)/∂ρ1+ in ˜ in in in ∆[P (T, ξ1 )+¯ρgz]=0, (118) ∂Fem/∂ρ1 = ξ1 (0)+∆ξ1 , changes by the amount ∆ξ1 = ∂ ∂F˜em/∂ρ = dH M. Therefore − · ∂ρ1 an expressions of the magnetic Bernoulli equation if the R in in ∂P in system is in equilibrium with respect to particle distribu- P (T, ξ1 )= P (T, ξ1 (0))+ in ∆ξ1 ∂ξ1 tion. (Note that of all the MidHi in Eqs (112-115), only ∂ = P (T,ρ1) dH ρ1 M, (116) that in Eq (115) has theR same variables as EqMidHi, − · ∂ρ1 R hence these two expressions cancel each other.)R Before explaining the difference between Eqs (114) and (115). arriving at total equilibrium, and as long as the con- The ill behavior of P takes one by surprise, as proper centration is uniform, we may start from Eq (81) and 21 combine it with Eq (114). The result is may be obtained by essentially the same derivation. We start by defining a slightly different susceptibility: ∂ ∆[P (T,ρ1)+ ρgz ρ1 MidHi]=0. (119) M mB − ∂ρ1 =χ ˜ . Although different from the more usual con- R vention of Eq (124), the new susceptibility is undoubt- Substitute ρ for ρ1 to arrive at the expression as given edly physically equivalent to the old one, and we have by Rosensweig. no a priori reason to prefer either. Both susceptibili- The velocity dependent terms in the original magnetic ties are related viaχ ˜m = 1 1/µ = χm/(1 + χm), or Bernoulli equation have not been included here, because d˜χm = dχm/(µ)2. So Eq (123)− may be rewritten as considerations of mass currents in ferrofluids need to in- P 1 2 m 1 2 m clude viscosities. Besides, some of the velocity dependent f = ∇( B ρα∂χ˜ /∂ρα) B ∇χ˜ . (126) 2 − 2 terms in the stress tensor, Eq (49), are missing in [2]. This time, assumingχ ˜m is proportional to one of the densities, we obtain Eq (125). 3. Kelvin and Helmholtz Force Which is the correct one, Eq (124) or (125)? Since Eq (123)and (126) are algebraically equivalent, the differ- ence must stem from the two assumptions, χm ρ and As in section III F 1 above, we may separate out the χ˜m = χm/(1 + χm) ρ. Reviewing the above∼ deriva- zero-field pressure P from the hydrodynamic bulk force tions, it is obvious that∼ if one of the two assumptions density of Eq (67), f bulk = s T + ρ ξ . (We again α α were strictly correct, the other would be wrong, and only neglect gravitation, and consider∇ a stationary∇ medium, the associated force expression is applicable. Generically, v = 0.) The remaining terms are referred to as the pon- however, both χm orχ ˜m are power series of ρ. Assuming deromotive force f P, either susceptibility is linear in ρ is only an approxima- bulk P tion. And consistency dictates that all quadratic terms f ∇P (T,ρα) f , (120) P ≡ − are then to be discarded. This implies (i) in the dilute f = (∂Fem/∂T )∇T ρ ∇(∂Fem/∂ρ ) (121) − α α approximation, both susceptibilities may only retain the = ∇(Fem ρα∂Fem/∂ρα) Hi∇Bi. (122) term linear in ρ and are therefore equal; (ii) all other − − terms quadratic in the two susceptibilities are also to be ∇ ∇ [The relation Fem = (∂Fem/∂T ) T + (∂Fem/∂ρα) discarded, as (χm)2, (˜χm)2 ρ2 0. We conclude: the ∇ ∇ · · ρα + Hi Bi was used for the last equal sign.] Kelvin force is only valid for∼χm → 1 or M H, to lin- In this section, we shall only employ the two poten- m ≪ ≪ ear order in χ or M. But then Mi∇Bi and Mi∇Hi are tials F˜(T,ρα, Hi) and F (T,ρα,Bi). So the zero field equally valid. The same of course also holds for Pi∇Di pressure depends on temperature and densities, and the and Pi∇Ei in the electric case. derived ponderomotive force is valid only under isother- The following example aims to illustrate this conclu- mal condition of constant densities, cf the discussion of sion [12]. Consider a thin slab of ferrofluid exposed section IIIF1. to a homogeneous, external magnetic field B, oriented We now proceed to derive the Kelvin force by in- normal to the slab. An enforced temperature gradi- corporating some specific simplifications. For lcr, 1 2 m ent within the slab ensures an inhomogeneous suscep- ∂Fem/∂T = 2 H ∂χ /∂T , and similarly ∂Fem/∂ρα, so tibility, χm(T ). The internal B-field is uniform, but −P f can be cast as not the internal H-field, as H = B/(1 + χm). The ponderomotive force of Eq (125) is zero in the given f P = ∇( 1 H2ρ ∂χm/∂ρ ) 1 H2∇χm, (123) 2 α α − 2 circumstance, but not that of Eq (124), which yields M ∇H = χmH2(∇χm)/(1 + χm), an apparent con- usually referred to as the Helmholtz force in the litera- i i − ture. If the system is magnetically dilute, we have χm ρ tradiction. m ∼ A proper analysis of the force should start from the in a one-component gas, and χ ρ1 for dρ = γdρ1 in ferrofluids (with no additional dependence∼ on ρ). Both simple expression of Eq (67). If not, the contribution of the zero-field pressure, P (T,ρ1), in the presence of a imply ρα∂χ/∂ρα = χ. Then the ponderomotive force reduces to 1 χm∇(H2), or temperature and concentration gradient, needs to be in- 2 cluded. In addition, one must include higher order terms 2 P m in the density, takeχ ˜ = αρ + βρ , and χ =χ/ ˜ (1 χ˜) f = Mi∇Hi, M = χ H. (124) αρ+(β+α2)ρ2. Inserting these respectively into Eq− (123)≈ Assuming in addition a static field: ∇ H = 0, the and (126), we find, for both cases and a constant B-field, force f P may also be written as (M )H×, a form one i i f P = 1 B2∇[βρ2]. (127) often encounters. ∇ 2 This seems a satisfactory derivation of the Kelvin force, Assumingχ ˜ ρ or χ ρ to hold strictly is equivalent as both the linear constitutive relation and the propor- to taking β =∼ 0 or β ∼= α2, resulting respectively in tionality to the density are frequently valid approxima- f P = 0 and f P = χB2 −χ, or equivalently, M ∇B or − ∇ i i tions. Unfortunately, the obviously different expression Mi∇Hi. Next we go on to consider nonlinear constitutive rela- P ∇ f = Mi Bi (125) tions, and shall convince ourselves that the two Kelvin 22 force expressions Eqs (124,125) remain valid – though rewrite the two thermodynamic derivatives, the second as m (in a generalization of the above conclusion) only to lin- [∂ξ/∂H]ρ = [∂B/∂ρ]H = [∂M/∂ρ]H = H[∂χ /∂ρ], ear order in the magnetization and polarization. We in- where the last− equal sign is− valid only for lcr− . The first sert Eq (110) in Eq (122), and equate ∂Fem/∂ρα with may be approximated: Without field, the inverse isother- − (∂M /∂ρ )dB , to yield − mal compressibility ρ2[∂ξ/∂ρ] = ρ[∂P/∂ρ] κ 1 is i α i T T ≡ T R usually a large enough quantity that one may neglect the P field related corrections, which is 1 (ρH)2(∂2χm/∂ρ2) f = Mi∇Bi + ∇ (ρα∂Mi/∂ρα Mi)dBi. (128) 2 Z − for lcr. Employing this approximation− and assuming lcr, we integrate Eq (132) to yield the variation in den- Assuming that ρ ∂M /∂ρ = M at constant B – equiv- α i α i sity in response to the gradient of magnetic field strength alent to ρ ∂χ˜m/∂ρ =χ ˜m for lcr — the integral van- α α and the gravitational potential, ishes, and we retain Eq (125). If we start from F˜em = B dH to evaluate ∆ 2 ∆ 1 2 m − · ρ = ρ κT ( 2 H ∂χ /∂ρ gz), (133) R − P f = (∂F˜em/∂T )∇T ρ ∇(∂F˜em/∂ρ ) (129) − α α where the boldfaced ∆ denotes (as before) the differ- = ∇[F˜em ρ ∂F˜em/∂ρ ]+ B ∇H (130) ence of any quantity at two different points in the liquid, − α α i i ∆A A2 A1. The electric terms may be obtained, as = Mi∇Hi + ∇ (ρα∂Mi/∂ρα Mi)dHi, (131) − usual,≡ according− to Eqs (56). (Frequently, this effect — P R we find f = Mi∇Hi if ρα∂Mi/∂ρα = Mi for constant referred to as the electro- or magnetostriction — is cal- H. culated using the ponderomotive force of section III F 3. The above calculation shows that there is no difficulty at all to avoid the ponderomotive force, and the ambiguity IV. EXPERIMENTS associated with its notation.) Electrostriction has been verified [13], using the refractive index to measure the Having been derived from thermodynamics and con- density change. servation of total energy, momentum and angular mo- If the fluid has two components – such as when it mentum, the theory presented in the preceding chapters is a solution or suspension – we need to consider both is fairly general, and valid for all magnetizable and polar- chemical potentials, ∇ξ1 = 0 and ∇ξ = gez. More − izable liquids, from single-component paramagnetic fluid conveniently, however, in the incompressible limit of sec- to ferrofluids, and for their respective electric counter- tion III B, we may consider Eqs (70) alone, parts. In the case of ferrofluids, which are suspensions in in of ferromagnetic particles, one is tempted to think that ∇ in ∂ξ1 ∇ ∂ξ1 ∇ ξ1 = ρ1 + H = gγez, (134) the properties of the particles are important, especially  ∂ρ1 H  ∂H ρ1 − their orientation (given by the magnetic moment), and in −1 ∇ρ1 = [∂ξ /∂ρ1] ([∂M/∂ρ1] ∇H gγez) . (135) their internal angular momentum. This is indeed true 1 H H − if a high-resolution, mesoscopic account of the system is in (Note that ∂M/∂ρ1 is connected to ∂ξ1 /∂H via a the prescribed goal. However, on a coarser scale, with Maxwell relation.) To our knowledge, field-induced equi- many particles per grain and relevant for most experi- librium variation in the solute or particle density has not ments, the theory derived above is quite adequate, even yet been measured in any two-component liquids. This uniquely appropriate for being not unnecessarily detailed is unfortunate especially in ferrofluids, where the vari- and complicated. ation in particle density should be rather pronounced. As in the last chapter, we shall only display the mag- Quantitatively, this experiment yields the thermody- netic terms, accounting for magnetic effects, as the anal- in 2 −1 namic derivative ∂ξ1 /∂ρ1 = [(ρ1) κos] , with κos the ogous electric ones are easily obtained via Eqs (56). osmotic compressibility. Since this is a diagonal deriva- tive, its significance in characterizing the ferrofluid ranks with that of the compressibility, specific heat and mag- A. Field Induced Variation in Densities netic susceptibility. Let us estimate the magnitude of this effect: Writing We consider the change in densities, ∆ρ and ∆ρ1, of m 1 ˆ 2 Eq (135) as ∆ρ1/ρ1 = κos(ρ1∂χ /∂ρ1)∆(µ0 2 H ), we a magnetizable liquid in equilibrium, from a region of m −3 approximate (ρ1∂χ /∂ρ1) 1, estimate κos 10 /Pa, low (or no) field to one of high field. In equilibrium, ≈ − ≈ and find ∆ρ /ρ 0.1 for Bˆ = 10 2T. [The value for Eqs (68) holds. For a one-component liquid, they reduce 1 1 κ is obtained by≈ considering a ferrofluid with 10% of to ∇T = 0 and os its volume occupied by magnetic particles of the radius 3 ∂ξ ∂ξ r = 10nm, so the particle density is n1 = 0.1/(4πr /3). ∇ξ = ∇ρ + ∇H = gez, (132) ∂ρ ∂H  − Approximating these particles as ideal gas, the inverse H ρ −1 osmotic compressibility κos is equal to the osmotic pres- −3 where the chemical potential has been taken as a func- sure, Pos = n1kBT , so κos = 10 /Pa if T = 300K.] tion of ρ,T and the field magnitude H. It is useful to Compare this with tiny change of the total density, 23

∆ρ/ρ = 5 10−8 at the same field, cf Eq (133), a result of the small· total compressibility, κ =5 10−10/Pa. ·

B. Current Carrying Vertical Wire

We consider a vertical wire that goes through a dish filled with ferrofluid. Feeding the wire with electric cur- rent will drag the ferrofluid toward the wire (located at r = 0 and along z in cylindrical coordinates). The sur- face of the ferrofluid column is given by z(r), with z at which the radius diverges (obtained by extrapolation) as the origin, ie z( ) = 0. The boundary condition Eq (79) is evaluated for∞ two points, z(r) and 0. Because the mag- FIG. 3: A U-tube filled with ferrofluid with only the left arm P netization vanishes and the curvature radii diverge at 0, subject to a field. The pressure atm outside the ferrofluid is constant. Depending on the orientation of the field, the level the attendant result is K = Patm. Inserting this into the difference z2 − z1 is given by Eqs (138) and (139). boundary condition at z(r), we obtain

−1 −1 EqMidHi = α(R1 + R2 )+¯ρgz, (136) the boundary condition simply states Patm = K ρgz¯ 1. R Inserting this into the boundary condition for tube− 2, we because M = 0 for the given geometry. The integral n obtain is to be evaluated for H = J/(2πr), and at constant 1 2 T, ξ1 if equilibrium has time to be established. For times ρg¯ (z2 z1)= M dH + M . (137) − Eq i i 2 n much briefer after the current has been applied, we need R to substitute ρ forρ ¯, and uni for Eq, and evaluated Note again the force balance between the electromagnetic the integral at constant T , ρR1, cf theR discussion leading surface force Eq (89), operational at the surface in tube 2, to Eq (81). Neglecting the surface tension, α = 0, and and the gravitational force from the disparity in height. taking the subscript uni lead to the result given in [2]. (As in section III C, the integral is to be evaluated at con- In the spirit of theR last paragraph of section IIID, we stant T and ξ1 if equilibrium has time to be established. remark that Eq (136) may be considered as an expres- For much briefer times after the field has been applied,ρ ¯ sion for force equilibrium, between gravitation, surface is to be replaced by ρ, and the integral by uni, evaluated tension and the magnetic surface force. at constant T,ρ1.) R m For lcr, Mi = χ Hi, the left hand side reduces to If the field is either predominantly tangential to the 1 m 2 2 χ H , and we are left with a quadratic, hyperbolic liquid surface (H = Ht, B = Bt and Hn,Bn = 0), or 2 2 m 2 profile of the interface, 8π ρ¯ gz = J χ /r if the surface predominantly normal to the liquid surface (H = Hn, tension α is neglected. The effect of α is more impor- B = Bn and Ht, Bt = 0), we have respectively, tant for weak currents, J small. It may be neglected in 1 2 m gρ¯(z2 z1)= M dH = H χ (138) any case for z 0, where both curvature radii are large − Eq i i 2 enough to be ignored.→ For z large, one curvature radius is R 1 2 m gρ¯(z2 z1)= EqMidBi = 2 H χ µ, (139) simply r, and the other . So this part of the ferrofluid − ∞ 2 2 m 2 R column is accounted for by 8π (¯ρgz+ α/r)= J χ /r , with the last equal signs in both equations only valid for − with the term r 2 being asymptotically (r 0) the lcr. ∼ → dominant one. In between, where the actual bend from Next we consider the quantity that a pressure gauge the horizontal to the vertical takes place, both curvature measures in a ferrofluid exposed to a magnetic field. As radii are finite and need to be included for an under- emphasized, it is not the ordinary pressure, yet as it will standing of the surface – note, however, that they have give some value, the question is what this is. Think of different signs. the gauge as an enclosed volume of air, at the pressure Patm, see Fig 4. One side of this volume is an elas- tic membrane, which is displaced if the external stress C. Hydrostatics in the Presence of Fields tensor deviates from the internal one. A finite displace- ment d stores up the elastic energy kd2/2 per unit area In a system of two connected tubes, with only the sec- of the membrane. (Take the membrane to be stiff, ie ond subject to a magnetic field, we expect the ferrofluid k large and d small, then we need not worry about column to be higher in this tube, as ferrofluid is attracted the pressure change inside.) The elastic energy implies to the region of stronger fields, see Fig 3. To calculate a force density kd, rendering the boundary condition tot the level difference, we employ the boundary condition across the membrane as Πnn = kd, or via Eq (79), △1 2 Eq (79) for the (flat) liquid-air interface in both tubes. K Patm + MidHi + M ρgz¯ = kd. (d is taken − Eq 2 n − Since the field vanishes in the first tube (denoted as 1), to be positiveR when the membrane protrudes into the 24

The calculation is already given in section III D. Bal- ancing the gravitation with magnetic force, we have elm + grav = 0, where grav = V (ρs ρf )geˆ , and F F F − − z elm is given by Eqs (89, 90), and especially by Eqs (92, F93) if the solid is completely nonmagnetic. (Again, de- pending on the time scale, the integral uni may be the appropriate one, cf section III C.) R Because scrap separation is an equilibrium phenome- non, we may compare energies instead of computing forces. This is a simpler and more qualitative approach to understand the behavior of polarizable systems. Con- sider for instance the fact that ferrofluid is attracted to FIG. 4: A vessel filled with ferrofluid is subject to a field. the region of high magnetic field. Take the field B as The pressure inside the ferrofluid changes with height and given, the field energy per unit volume is B2/2 in vacu- the field strength, in response to which the elastic membrane um (or air), and B2/(2µ) in ferrofluids. Since µ > 1 of a pressure gauge (right bottom) yields. The corresponding in any paramagnets, the second expression is smaller. displacement d is given by Eq (140). Given the choice, a volume element of ferrofluid will therefore always occupy a region with the highest possi- ble field, to reduce the field energy. Conversely, a small, gauge.) We have d = 0 as long as the external stress non-magnetizable object, submerged in ferrofluid, will on tensor is the same as the internal one, ie in the atmo- the other hand tend to occupy the region of lowest field sphere down to the liquid surface of tube 2, and also just strength. If a difference in height is involved, all these below the surface – take this as point 2. Moving down of course happen only as long as the gain in field energy the liquid column, to an arbitrary point 3, the membrane is larger than the loss in gravitational energy levitating moves the distance d to maintain force equilibrium. Em- the object. (If the field H is given instead of B, we need ploying the above boundary condition for both points, to consider F˜ F HB. And again, it is larger in and subtracting one from the other, we find em em vacuum then in the≡ ferrofluid:− H2/2 > µH2/2.) − − kd = ∆( M dH + 1 M 2 ρgz¯ ), (140) Eq i i 2 n − R n n n with ∆A A3 A2. Note ∆Mn M3 M2 , where M3 is the magnetization≡ − at point 3≡ normal− to the pressure n gauge membrane, and M2 the field at point 2 normal to the liquid surface – both components are not necessarily parallel. Eq (140) again states a force balance, between the elastic, magnetic surface and gravitational force. The displacement d is the readout of the pressure gauge: If the field is uniform, and if the membrane of the gauge is parallel to the liquid surface, Eq (140) reduce to ρg(z2 z3) = kd, the zero-field hydrostatic relation; otherwise,− field contributions abound – even if the pres- sure gauge is simply rotated at point 3 in the presence of a uniform field. This amply demonstrates the system’s anisotropy. A further complication is that all fields are the actual ones, distorted by the presence of the pressure gauge – though this is an effect that may be minimized.

D. Magnetic O-Rings and Scrap Separation

In this section, we address the physics of some tech- FIG. 5: Self-levitation of a magnetized body submerged in nical applications: magnetic O-rings, self-levitation and ferrofluid. scrap separation. Consider scrap separation first. An in- homogeneous magnetic field which becomes weaker with Similarly, a permanently magnetized body submerged increasing height may lift non-polarizable bodies sub- in ferrofluid tends to collect as much liquid around merged in ferrofluids off the ground, and hold them at itself as possible, in the space occupied by its stray specific heights which depend on the shape and density field – even at the price of levitating itself off the bot- of the bodies. tom, a phenomenon that is usually referred to as “self- 25

The field is strongest in the middle of the O-ring and decays towards both ends. If ∆P were zero, the force Felm would be the same on both surfaces, and the fer- rofluid stays in the middle of the O-ring. Increasing the pressure on the left (surface 1) pushes the ferrofluid towards right, such that surface 1 is in the region of higher, and surface 2 is in the region of weaker, fields. Equilibrium is achieved when the difference in Felm bal- ances ∆P . The strongest pressure difference maintain- able is when one surface is at the region of highest field, and the other is field-free. Assuming for simplicity that FIG. 6: Magnetic pressure seal: (a) principle and (b) enlarged the magnetic field is predominantly tangential, and that displacement of the ferrofluid plug. lcr 1 m 2 m holds, we have ∆P = 2 χ H . With χ 1, 2 2 5 ≈ H = µ0Hˆ , and taking Hˆ as of order 10 A/m, this pressure difference is about 105N/m2, approximately the levitation” [2]. Frequently, the magnetized body consists atmospheric pressure. of a periodic array of north and south poles, with peri- odicity λ, so the stray field extends one or two λ into the ferrofluid. Levitated approximately that far from the bottom, the body will usually have reached its equilib- E. Elliptical Deformation of Droplets rium position, as no further gain in field energy may be achieved by pushing more ferrofluid below it, and levi- tating itself yet higher, Fig 5. A droplet of ferrofluid exposed to an external magnetic Generally speaking, although energetic considerations field along eˆy is distorted, cf eg [14]. If the field is not too are a useful heuristic tool, the actual minimization of to- strong, it turns from a sphere of radius r to an ellipsoid 2 2 2 2 2 tal energy or free energy is often rather cumbersome, as of essentially the same volume, y /a + (x + z )/b = 1, the field does change considerably when a volume ele- with a,b the semimajor and semiminor axes, b

lcr 1 2 2 Assuming , or u = 2 (D /ǫ + ρv ) with D0 = ǫE0, we have

u v gtot = 1 (D2/ǫ ρv2) v (E H)/c − · 2 − − · × = 1 (D2/ǫ ρv2) v (D H)/cǫ + (v2/c2)(145), 2 − − · × O

and deduce

FIG. 7: A freely suspended ferrofluid droplet elongates along ∂(u v gtot)/∂D the field, the extent of which is given by Eq (143). − · = (D + v H/c)/ǫ = D0/ǫ = E0. (146) × V. APPENDIX Higher order terms (such as one D4 in the energy u) The validity of Eq (43) is shown here directly by trans- do not invalidate this result. The∼ terms in the magnetic forming the rest frame expression. More specifically, we tot field behave analogously. demonstrate ∂(u vg )/∂D = E0, holding s, ρ , v, − α and B constant. We start with In Eq (145), the explicit form of gtot in the lab frame was employed to deduce the lab frame energy, Eq (43), 1 2 from which then the lab frame energy flux, Eq (46), is u = u0(D0 D, B0 B)+ ρv . (144) → → 2 deduced. This may appear as an inconsistency, but is not, tot This pleasingly simple expression is a result of the ac- because with the rest frame expression for g0 given, we already know that the term v is from the rest mass. No cidental cancellation of the terms from the Galilean- ∼ Lorentz transformation with that of the Tailor expansion, detailed information about the energy flux is necessary here. tot 1 2 Acknowledgment: u(D, B)= u0(D0, B0)+2v g + ρv We thank Andreas Engel, Kurt · 0 2 1 2 Sturm and Hubert Temmen for comments, and we are = u0(D0, B0)+2v (E H)/c + ρv · × 2 especially grateful to Hanns-Walter M¨uller for detailed 1 2 = u0(D, B)+ 2 ̺v . criticisms and suggestions.

[*] e-mail: [email protected] (1995); 80, 2937, (1998); see also Phys. Rev. Lett. 74, [1] We shall use the terms “force/force density,” “en- 1884, (1995) and 81, 3223 (1998); Phys. Rev. E50, 2925 ergy/energy density” interchangably. (1994) [2] R.E. Rosensweig, Ferrohydrodynamics, (Dover, New [10] J. V. Byrne, Proc. IEE 124, 1089, 1977 York 1997) [11] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Perg- [3] M.I. Shliomis, Soviet Phys. Uspekhi (English translation) amon, Oxford, 1987) §6, 7, 130 17(2), 153 (1974) [12] W. Luo, T. Du, J. Huang, Phys. Rev. Lett. 82, 4134 [4] E. Blums, A. Cebers, M.M. Maiorov, Magnetic Fluids, (1999); Mario Liu, Phys. Rev. Lett. 84, 2762, (2000) (Walter de Gruyter, Berlin 1997) [13] S.S. Hakim and J.B. Highham, Proc. Phys. Soc. (London) [5] L.D. Landau and E.M. Lifshitz, Electrodynamics of Con- 80, 190, (1962) tinuous Media (Pergamon Press, 1984), §15, 35. [14] J.-C. Bacri and D. Salin, J.Physique-Lettres 43, 179 and [6] S. R. de Groot and P. Masur, Non-Equilibrium Thermo- 649, (1982) dynamics, (Dover, New York 1984) [15] B.M. Bergkovsky, V.F. Medvedev, M.S. Krakov, Magne- [7] K. Henjes and M. Liu, Ann. Phys. 223, 243 (1993) tic Fluids, (Oxford Science Publications, Oxford, 1993), [8] R. Gerber, Transactions on Magnetics, Mag-20, 1159 §3.2 (1984) [9] Mario Liu, Phys. Rev. Lett. 70, 3580 (1993); 74, 4535