Electromagnetic stress and momentum in matter

ELECTROMAGNETIC STRESS AND MOMENTUM IN MATTER

F.N.H. ROBINSON Bell Laboratories, Holmdel 07733, N.J., USA

(~E NORTH-HOLLAND PUBLISHING COMPANY

—

AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters) 16, no. 6 (1975) 31 3—354. NORTH-HOLLAND PUBLISHING COMPANY

ELECTROMAGNETIC STRESS AND MOMENTUM IN MATTER F.N.H. ROBINSON Bell Laboratories, Holmdel 07733 N.J., USA * Received November 1974 Abstract: Whether the momentum density associated with an electromagnetic wave in a refracting medium is solely determined by its intensity or is dependent on the refractive index is perhaps of little practical importance, and probably beyond the power of present experimental techniques to decide. Nevertheless, since this question was first propounded by Abraham and Minkowski. it has continued to attract physicists’ attention and, to judge by current literature, is still regarded as an open and quite possibly insoluble question. This article expounds some of the approaches to its solution that have been proposed, and shows that one of them leads to a complete solution of both the Abraham—Minkowski controversy and the more general problem of describing the distribution of stress in a material medium in the presence of an electromagnetic disturbance. The treatment emphasizes the significance of the electrostrictive and magnetostrictive terms in the stress tensor. These relatively unfamiliar effects are neither small nor unimportant. They express, in macroscopic terms, consequences of the microscopic structure of a medium which are not adequately described by the usual constitutive relations involving only the relative dielectric constant and the relative magnetic permeability. Their presence in the macroscopic theory ultimately reflects the fact that the individual microscopic charges in matter are subject to forces which, though purely electromagnetic, cannot adequately be described in terms of macroscopic fields., even if these fields are modified by the inclusion of a Lorentz local-field correction. As a result they form a crucial link in the connection between macroscopic and microscopic treatments.

Con tents: I. 2. 3. 4. 5. 6.

Introduction The macroscopic electrodynamic equations Microscopic and macroscopic electromagnetism The Helmholtz calculation Manipulating Maxwell’s equations Mechanical conservation laws

315 317 319 322 326 329

7. Gordon’s model 8. Shockley’s model 9. The method of virtual power 10. Discussion References Additional references

334 337 338 341 353 353

Single orders for this issue PHYSICS REPORTS (Section C of Physics Letters) 16, No. 6(1975) 3l3--354. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dli. 15.—, postage included.

*Present (and permanent) address: Clarendon Laboratory. Parks Road, Oxford, England.

j

F.N.H. Robinson, Electromagnetic stress and momentum in matter

315

1. Introduction The form of the electromagnetic stress tensor and the nature and magnitude of the momentum associated with a light wave in a refracting medium have been the subject of controversy for nearly a century. Disagreement has usually focused on whether the momentum density associated with a wave, propagating in a medium of refractive index n, is, as Minkowski suggested, D A B/c2 = n2 E A H/c2 or, as Abraham suggested, it is simply equal to E A H/c2 and therefore always has the same relation to the energy flux vector E A H. Various general and inconclusive arguments have been put forward in favor of each formula and, possibly because the effects involved are small and difficult to observe, there has been no dearth of theoretical treatments, many of them associated with distinguished names. The Abraham—Minkowski controversy is, however, only a part of the more general problem and in this article I shall attempt to set out the solution to the general problem. That this problem still engenders new publications is, perhaps, less surprising if we note that a much simpler electromagnetic problem, the calculation of the torque acting on a magnet immersed in a magnetically permeable fluid, has only very recently been solved by Lowes [19]. It is doubtful whether this solution would have been achieved had it not been for the careful experimental work of Whitworth and Stopes-Roe [301 which forced a reconsideration of received ideas. A number of factors have hindered the satisfactory solution of this problem. The first is the inclination of physicists to believe that the general solution should be simple. Lowes’ quite complicated results for a much simpler problem show this expectation to be ill-founded. A second reason, and one which goes back to the turn of the century is the belief that the full solution could be immediately deduced from a microscopic or atomic interpretation of electromagnetic phenomena in matter. This expectation, as we shall see, is unlikely to be fulfilled in the near future. Another reason is an equally unfounded hope that the solution could be found by a straightforward application of the principles of special relativity and mechanics, without invoking the detailed structure of the electromagnetic equations. We shall find that this approach does yield one significant result but it is, alas, more of philosophical interest than an empirically verifiable physical result. As a final reason we may also mention the belief, or rather the suspicion, that the problem might not be soluble in purely macroscopic terms. We shall see that this suspicion is unjustified but it has diverted effort from a frontal attack on the problem, towards a study of simplified microscopic models. Because of these factors, operating separately or in conjunction, the literature of the subject contains many treatments which either fall short of a full solution, or fall short of that scrupulous attention to logical detail which should characterize a physical argument. These papers, though often both plausible and illuminating have tended to obscure the essential nature of the problem. This is that it is a purely macroscopic problem, and must be entirely soluble in terms of the laws of macroscopic electromagnetism, properly understood. In fact the problem was almost completely solved by Helmholtz many years ago. Helmholtz (1 882) used the principle of virtual work, in conjunction with the macroscopic electromagnetic equations to derive an expression for the force density acting in a medium subject to static fields. An account of his treatment may be found in Abraham and Becker [31 or Robinson [251. There is nothing in Helmholtz’s treatment that would have precluded him from generalizing it to time dependent phenomena although, in the absence of the theory of relativity, his calculation would have been wrong in some details. Of modern authors, only Penfield and Haus [23] have had the ,

316

F.N.H. Robinson, Electromagnetic stress and momentum in matter

stamina to produce a complete generalization of the Helmholtz treatment. It is on their work, which provides a complete solution to the problem, that we shall base our final conclusions. If a complete solution already exists it may properly be asked why we engage in yet further discussion. To this there are several answers. Firstly, attention has been distracted from the significance of Penfield and Haus’ work by the numerous papers in which authors purport to “pull results out of the hat” without engaging in the tedium of attempting an explicitly derived solution. Secondly, Penfield and Haus presented their results in the context of a book entitled “Electrodynamics of Moving Media” and their conclusions relevant to this problem only emerge gradually, and somewhat indistinctly, in the course of some 250 pages of closely argued text. Thirdly, they also chose to employ, as their main theoretical tool a formulation of electrodynamics due to L.J. Chu [91 which, whatever its pedagogical and practical advantages, is unfamiliar to the general reader. Finally, the relevant results are nowhere presented as a coherent whole and, as Penfield and Haus use a notation somewhat different from that used by many other authors, the reader has to do a rather considerable amount of work bt:fore he can either appreciate their significance, or compare them with the results of other authors. My aim therefore, in this article, is ultimately to present the salient features of Penfield and Haus’s argument and results, in a form which facilitates comparison with other results obtained by less substantial methods. I do not propose to set out the details of their calculation for these may be found in their book, which, once one has grasped their notation, is most cogently argued. Section 2 consists of a brief statement of the laws of macroscopic electromagnetism. In approaching a controversial subject it is necessary for the reader to understand the author’s point of departure, and I hope that the reader will find this statement of the laws acceptable, if perhaps unconventional. Although our topic is exclusively macroscopic, atomic (or microscopic) interpretations of macroscopic laws are often illuminating and so section 3 deals with this interpretation. The Helmholtz calculation is presented in section 4 and in section S we discuss what we can learn by simply manipulating Maxwell’s equations. Section 6 deals with the consequences of general conservation laws and sections 7 and 8 with two approaches based on microscopic models due to Gordon [141 and Shockley [27]. Gordon’s model has the virtue of being rigorous within its limited range of applicability. Shockley’s ingenious model is of particular interest because it not only illuminates the connection between the macroscopic laws and the structure of a medium, but also has been quite erroneously thought to show the impossibility of a fully macroscopic treatment. The core of the article is contained in section 9 which presents the method of virtual power introduced by Penfield and Haus [23]. Here at last we encounter unambiguous and general results. The final section is devoted to a general discussion. A note about references is in order. In the text we only quote authors whose work is immediately germane to the text. In order to avoid adding more fuel to the fires of this controversy we do not draw attention to authors whose work we believe to be incorrect or inadequate. However, at the end of the numbered list of references we append a brief list of further references. The content of many of these references is quite unexceptionable and readers are left to guess or discover which these are. Taken together, all the references, together with the references they contain, should give the reader access to the whole literature of this antique but perplexing controversy.

F.N.H. Robinson, Electromagnetic stress and momentum in matter

317

2. The macroscopic electrodynamic equations Although, in teaching, it is often useful to introduce these. equations by, either following a more or less historical course, or by (see e.g., Rosser [26]) basing them on a combination of special relativity and the electrostatic inverse square law, for our purposes it will be more useful to state them outright. The form that I adopt, though not perhaps conventional, will I hope appeal to the reader as expressing his own understanding of how he uses, and interprets, the laws. I. The electromagnetic properties of matter can be specified in terms of four densities; one, p is a scalar, the others J, P and M, are vectors. II. J is the flux vector for p. The rate at which the property p crosses a fixed element of area dA isJ~dA. III. The densities vanish in empty space. IV. The densities act as sources for two fields E and B according to the scheme: V~B0,

(Ia)

VAE+B=0;

(ib)

V~E=p— V’P,

(lc)

VAB—E=J-I-P+VAM.

(id)

V. Special material media exist in which P and M can be neglected. VI. The forces per unit volume acting in a stationary, special medium is f=pE+JAB.

(2)

This completes the laws, though perhaps we should also add that the properties p, J, P and M are “carried” by matter in the sense that the force f given by (2) acts on the matter that “carries” p and J. We also note that we have chosen rationalized units and set the velocity of light c = 1. Compared with S.I. units we have set ~ = 1, 0 = I. When, later on, we introduce a dielectric constant e and a permeability /1, they will be dimensionless. The refractive index is n = (jj~)1~’2. In our final formulae we can always retrieve S.I. units by inserting factors of j.t~and ~ to get the dimensions right. The most important feature of these laws is that, taken in conjunction with macroscopic mechanics, they are complete. We can, if sufficient quantities of the special media are available, devise experiments, using only mechanical measurements, which allow us to deduce all the electromagnetic variables. In general, a long series of measurements or a few measurements combined with access to a table of electromagnetic constitutive relations, will be required to solve even the simlest case. As a further aspect of the completeness we consider a particular example. If, in an isolated body, the two non-vanishing densities are a steady current J and a magnetization M we can define the magnetic moment m of the body as m

f

(M+~rAJ)d~r.

body

It then follows from equations (1) and (2) that if this body is placed, in vacuum, in a uniform impressed magnetic field B*, due to distant sources, the couple on the body is F m A B*.

318

F~N.H.Robinson, Electromagnetic stress and momentum in matter

This is a deduction. not a definition of m. We must not expect, nor do we find, equally simple results to hold if the medium outside the body has magnetic properties. Indeed in other cases the couple depends on the shape of the body as well as its volume. This example is included as a warning. Subsidiary quantities such as m are of’ten useful and they are usually defined because, in some particular context, they lead to a simple result e.g., F = m A B*. However, once they have been defined, in terms of the basic variables which enter eq. (1), they must not be redefined in terms of this simple result. The property m* of a body that leads to a couple F = m* A B in all circumstances, though an interesting parameter, is not its magnetic moment. Nor indeed is it a property of the body alone, but rather of the body and its environment. Next we come to some further auxiliary definitions associated with the vectors D and H and with constitutive relations. In the majority of media the density vectors P and M are uniquely related to the field vectors E and B, for a given thermal and mechanical state of the medium. In some cases the relation is exceedingly simple: P = X~’Eand M = ~B where Xe and are constants depending on the thermal and mechanical states of the medium. We also note that the inhomogeneous pair of Maxwell equations can be written as ~,

V

(E+P)p,

VA(B~—M)--(E+P)=J.

It is therefore useful to introduce two new vectors D tions can be written as

=

E

+

VDp

P and H = B—M so that these equa-

(3)

and (4)

VAH—D=J.

If P is a function of E so is D, and in particular, if P XeE, then D = eE where e = 1 + Xe. Also if M is a function of B it is also a function of H B —M and, if M is proportional toB, it is also proportional to H. If M = XmH we have B = pH where p 1 + xm. Relations between the density vectors and the field vectors, which are characteristic of the medium, are known as constitutive relations. They need not be linear and they may depend on the mechanical and thermal state of the medium. Some important types of constitutive relation are: ~

I

/

i/

B 1 B P1

=

=

=

p11H1

=

~

/

linear but anisotropic.

(5)

(~+

pH + M0

permanent magnetization M0.

dIIkEIEk

non-linear.

We shall be primarily concerned with the linear isotropic case. The vectorJ can also in some cases (ohmic materials) be expressed as uE but, since J does not occur in a unique combination with E in the field equations it is not worthwhile, except in a few special cases, introducing any further new vectors such as I = J + F or I = J + D. If, in view of all the accumulated experimental evidence, we regard the field equations (1) and the force equation (2) as the valid laws of macroscopic electrodynamics, then any consequences

F.N.H. Robinson, Electromagnetic stress and momentum in matter

319

of these equations is also a valid law. For example, if we combine equations (1 c) and (1 d) we obtain VJ+p0

(6)

and so p is the density of a conserved quantity in view of law II. We may perhaps remark that, had this not been so, the introduction of the vector J would have been rather pointless. We have now collected together the macroscopic results needed for our later use and we close this section with a remark about the force eq. (2). This equation, to some extent,furnishes an operational definition of E and B, i.e., a definition related to another branch of physics (mechanics) which we believe to be more fundamental. It is, however, only valid in special media. We have no need, nor any grounds, to believe that forces in other media will involve E and B in simple ways. In particular just because —v P behaves in the same way (as a source of E) asp we have no reason to believe that a term such as —E(v P) will represent a force density in a polarizable medium. .

3. Microscopic and macroscopic electromagnetism In macroscopic electromagnetism, if we have an isolated body, or localized system of charges and currents, in which P = 0 and M = 0, we can describe its external electromagnetic effects in vacuum in terms of a series of multipole moments. The first two electric moments are the charge: q

I pd3r

(6a)

body

and the electric dipole moment: pfprd3r.

(6b)

If q = 0, p is unique, but in other cases we must specify the origin of r. The magnetic dipole moment: m

r A Jd3r,

(6c)

is always unique. In atomic physics we believe that matter consists of interacting massive point-like charged particles. To a first approximation these particles are characterized solely by their charge, though they also to some extent exhibit internal structure and possess multipole moments. In the case of nuclei these are certainly due to their composite structure, but in the case of the electron we can, if we wish, interpret the magnetic moment in terms of the charge alone and the quantum dynamics of the particle. In what follows we shall treat nuclei and electrons as structureless point changes and ignore their intrinsic moments. They can, without much difficulty, see e.g., Crowther and ter Haar [12], but at the expense of extra complication, be incorporated in the theory. It follows from the basic macroscopic electromagnetic laws that the force acting on a stationary isolated body in a uniform impressed field E* in vacuum is qE* = E*fpdV and that, if it moves with a velocity v in a uniform impressed field B*, there is an addition force qv A B*. As the size

320

F.N.H. Robinson, Electromagnetic stress and momentum in matter

of the body shrinks to a point we can relax the requirement that the field be uniform. If therefore, e’ and b’ are the fields in vacuum due to a set of moving or stationary point charges, the force F acting on an additional charge q~at r0 with a velocity v,~is F

q~e’(r~) + q~v~b’(r~).

=

(7)

Note that in evaluating e’ and b’ we omit the contribution from q~itself’. The field laws for a set of point particles in vacuum are obtained, by a similar process of reasoning, from the macroscopic equations with P = 0 and M = 0. They are V~bO,

(8a)

VAe+b=O.

(8b)

V

-

e(r)

(8c)

~q~6(r—r~)

=

and b(r)

V A

—

e(r)

=

~ q~v~6(r—r,5)

where there are charges q~with velocities

(8d) it0,

at r0 and 6 is the three dimensional Dirac delta-func-

tion. The force F can also be expressed as a force density f(r)

=

~ (q0e’(r~)+ q~v0A b’(r0))6(r—r0).

(9)

We may regard these equations as either a plausible extension of the macroscopic equations to the microscopic domain of atomic physics, or as a new set of basic laws, justified by the way in which, in conjunction with the laws of quantum mechanics they describe the varied phenomena of atomic physics. We are not concerned here with their logical status but solely with the interpretation that they afford of the macroscopic equations. A macroscopic investigation is primarily characterized by a certain degree of coarseness and a lack of discrimination. Events which succeed each other within some small time r and events which occur within some small distance X of each other, are not resolved. In other words if u is a physical variable and we expand it as a Fourier series, or Fourier integral, temporal and spatial frequencies above l/X can be disregarded. If the grain a of the medium involved is such that a ~\. the truncated Fourier expansion of u will not betray aiiy features associated with the grain, or atomic structure, of the medium. Because the velocity of light is finite a finite resolution in space implies a finite resolution in time. If u is known exactly, i.e., on a microscopic scale, we can obtain its macroscopic equivalent U, that is the low frequency part of ii, as ‘~

U(r)

=

fu(s)ffr—s)d3s,

(10)

where f is any suitably behaved function whose Fourier spectrum is constant up to I /X and then falls to zero. The nature of the function f need not concern us, it is enough to know that it exists. The importance of eq. (10) is solely that it shows that U can be derived from a by a process which is compatible with the subsequent formation of true temporal, spatial or statistical averages of U.

F.N.H. Robinson, Electromagnetic stress and momentum in matter

321

The field equations (8) are linear and, as a result, if E is the macroscopic part of e, etc., we have, as a consequence of these field equations, the macroscopic equations V’B=O,

VAE+B=0,

V~E=p’~,

VAB—E=J’~,

where pC and JC are the course, or truncated, expressions corresponding to the sums. If we are dealing solely with an assembly of structureless point charges, these expressions reduce naturally to the macroscopic charge and current densities. However, in matter, the charges usually occur in stable aggregates, or atoms, and even if these aggregates have no net charge, thôy still influence the macroscopic fields. To deal with this situation (see for example Robinson [25]) we consider each atom (or ion, or force electron) as a single particle located at its center of mass R~and assign to it a set of multipole moments. If the atom contains particles of charge q~at positions re with velocities Ve relative to R~the first few multipole moments are =

q~,

Pn

=

q~r~

and

mn

=

~

~

AVe~

We then find that we can express pC and JC as pCp

VP,

JC_J÷p+ V AM

where p(r) and J(r) are obtained by truncating the expressions ~ q06(r—R,~)

and

~Dq0V~6(r—R~),

in which V1,~is the velocity of the nth aggregate, or atom. If we omit terms involving the atomic velocities V,~,and also ignore the higher-order multipole moments of atoms, the other two quantities P and M are the electric and magnetic dipole moment densities obtained by truncating the Fourier spectra of ~ p,~(r—R,~)

and

~

m~6(r—R~),

but, in general both P and M will involve the atomic velocities V~and the moments p~and m~. Their dependence on the higher-order multipole moment densities is, however, always in the form of spatial derivatives. Thus, if Q is the electric quadrupole moment density obtained by truncating ~ Q~6(r—R~), the first additional term in P is div Q. Thus, the microscopic field equations lead naturally to the macroscopic equations, and provide a microscopic interpretation of the densities p and J, and P and M. The force equation (9) presents more serious problems for, as we see, it involves products such as e’(r~)6(r—r~) of two terms, both of which have high frequency Fourier components. If, however, we are dealing solely with an assembly of point-like structureless particles this can be evaded. We note that e’ and b’, which appear in (9) in conjunction with the position of the nth charge q~, do not include the contribution from q~.If the particles are well separated, compared with their size, which will certainly be the case both for point particles, and for nuclei and electrons in ordinary matter, we can regard e’ and b’ in (9) as already partly truncated. Thus neither e’ nor b’ varies appreciably over the region in which 6(r—r~)is non-zero, and there are no high frequency components in e’ and b’ to beat with the high frequency components of 6, and so produce —

322

F.N.H. Robinson, Electromagnetic stress and momentum in matter

macroscopic low frequency components. in this special case (9) leads directly to f

pE+J A B.

If however, the charges are aggregated in atoms we shall only be able to proceed in this way if the separation between the atoms is large compared to their dimensions, i.e. in a rarefied gas (see the discussion in section 7). In other cases we cannot derive a general force law from (9). This is the prime reason why attempts to derive macroscopic stress tensors from microscopic models are doomed to failure. We may note that it is also the reason why the application of local field corrections in theories of dielectric media is so uncertain. Fortunately the somewhat inscrutable consequences of the microscopic force equation do show up in a macroscopic property of a medium, and this is the dependence of its dielectric constant and magnetic permeability on its state of strain. As we shall see the coefficients which describe this dependence play an important role in the macroscopic theory.

4. The Helmholtz calculation The account that we shall give here is rather brief, and applied only to the electrostatic case. For further details the reader may consult Abraham and Becker [3] and Robinson [25] Our aim is merely to illustrate the principle of the method and to give the results. If each point of a medium is given a small arbitrary velocity v and the force density in the medium isf, the change in its total energy can be expressed as .

(Ha)

dU/dt_ffvdV.

If, at the same time, we can express this in terms of a function X of the electromagnetic variables as

(llb)

dU/dt=_fX.vdV

then, because v is arbitrary, we can identify f with X. From the basic macroscopic field laws we can calculate the energy required to assemble a dielectric system containing intrinsic charge p as Ue=f(f E~dD} dv, in addition, of course, to any purely mechanical or elastic energy. Because we want to regard D as the dependent variable rather than E we write this as U~=f~E.D_fD.dE}dV so that EaD

~=f(E.D÷E.D_D.E+f(_) dt

0

at

EaD E

.~)dvf~E.nf(~) at 0

.dE}dV

F.N.H. Robinson, Electromagnetic stress and momentum in matter

For an electrically isolated system in which J fE. D dV

_fJ.E dV- _fpv

=

323

pv we have

EdV

thus, one term in the force is, as we expect, pE. We now make the major assumption that the medium, though anisotropic, is linear, so that it is permissible to write the displacement as =

e~1E1where .~, though it may depend on position and the mechanical state of the system

does not depend on E. We thus obtain due

dv.

Changes in due to the impressed velocity occur partly from the motion of the medium, which brings matter with a new value of e~to a given point, partly from local rotation of the medium which mixes the components of and partly from local strain resulting from the motion. It is a matter of taste whether we incorporate the local rotation with the strain term, which is the procedure adopted by Penfield and Haus, or as we do here, keep them separate. To describe the effect of (symmetric) strain Uki on Eq we introduce a tensor 0~ijkl

=

aE,//aukl

(12)

which, because and Ukl are symmetric, is also symmetric in the first pair and the last pair of indices. Then, after some rather tedious manipulation in which we pay careful attention to the distinction between partial time derivatives and substantial time derivatives, we find that we can indeed express dU/dt in the form (11 b) and that the components of the force are (13) where the symmetric electrostrictive tensor ~ rstr

=

iS

1aEE

(14)

The first term in parentheses arises from changes in e~,due to matter transport. It can also be written as

~ aD. / ar, /

aE.~,

—

D/ ar,‘ 1

~—

a. 1EE jk_ 2 / k ar,

—

axe 1EE 2 / k ar,jk

15

The second term in parentheses arises from rotation. It can be incorporated in T 7str at the price of destroying the symmetry of this tensor. It can also be expressed as V A (E A ~) and it vanishes in isotropic media. Using the field equations we can express f, as the gradient of a totally symmetric stress tensor. Thus, —

tr)

f1=~(T17+

where

r,e;

(16)

324

F.NJI. Robinson, Electromagnetic stress and momentum in matter

T.~=‘{ED +E1D1

—

E

.

D611}.

(17)

Many texts, unfortunately, express f~as

~~EIDJ

=

—

~

D6~/+ T1str*

}

(18)

with the rotational 1(E.D. term incorporated ED.). in —

=

1/

T.~

—

2

i

/

/

T7t1’*,

so that

(19)

7

It is, as we shall see, much more convenient to segregate, in T 1~5tr,effects arising from the detailed structure of the medium, and manifest in the fourth ranl tensor coefficient a~/kl, which are not, like the rotational term, completely described by the anisotropic dielectric constant To obtain the total force density we must add to (13) or (1 6) the derivative of the ordinary elastic stress tensor Y,1. Even in isotropic solids both this, and the electrostrictive coefficient and tensor T~, are usually quite complicated, but in a fluid of density r in which is simply a hydrostatic pressure, and in which e = I + Xe 15 necessarily isotropic, we get 0k1i1

=

a —r ~6kl6li6ij

(20)

and so Testr

=

‘E2i- ar 6 1/

.

2

=

iE2r 2

ar~ 6.‘I

In a fluid, therefore, we see that, with Y

f.m = —~-E2 axe 2 ar 1

5

1/

=

—ir6~,where it is the pressure, and with p

=

0

(21) ~ +~ —E~r a ax ar, ar, ar 5/ar~= 0, and it is also in equilibrium, then the effects of the If the fluid is uniform, so that ax —

—~-

—.

electrostrictive item must be cancelled by the pressure gradient. Thus, it

=

where

ir~-1- ~E~rax~’/ar, it

0 is the pressure in zero field. The effective total stress tensor is then simply T1~JSgiven by (1 7), with the appropriate simplifications. In a rarified gas we have ~ axe/ar = Xe and now we obtain, again with p = 0, 2 air aE• air = —~E2——+~ —(x~E2) __= ~_pi axe a ait ~ aE

ar~

ar~

—

—

ar

1

ar~ ar.

/

ar1

ar~

Since we are dealing with electrostatics, V A E = 0 and aE1/ar1

f,. =(P V)E,

—

air/ar,.

=

aE~/a~~. and thus (22)

The force on an individual atomic dipole p is just (p~V )E and so we see that (as we indicated at the end of section 3) when the atoms are far apart, the macroscopic force is simply related to the

F.N.H. Robinson, Electromagnetic stress and momentum in matter

325

microscopic force. Note, however, that this equivalence arose because, in the macroscopic calculation, we retained the electrostrictive tensor which, in this case, had a particularly simple form. We emphasize this point. Because the electrostrictive stress tensor involves the relation between e and the strain, there is a tendency to feel that it is part of a mechanical problem, especially if we are dealing with high frequency phenomena, and that it can therefore be ignored as a needless complication. This is not so. Strain, like rotation, is a geometric parameter and the electrostrictive term does not describe the changes in e due to strain produced by electromagnetic stress but rather an aspect of the dependence of c on atomic positions which cannot be ignored. We also note that the electrostrictive term is not, in general, negligible compared with the other terms in

f1. In a rarified gas the electrostrictive coefficient has a particularly simple form because we have basically ignored interactions between the atoms, other than those described by the macroscopic fields. In other cases al/kl and Ttr depend on those interactions. In the most general case these interactions will be mediated by forces which cannot be described simply in terms of the atomic dipole moments, and then we are forced to regard a~/klas an empirical parameter to be determined by experiment. There is, however, an intermediate case, which is associated with idea of a local field and the Clausius—Mossotti formula, or one of its many variants. The general notion here is that, although e’ in eq. (9) cannot be replaced by the macroscopic field E(R~)at an atomic site. yet we can calculate a corrected field, Eb0c, which is the same at each atomic site. The result is that, for atoms of polarizability a with a density N, e_

Na

______

1—7Na where ‘y is a numerical factor near 4. This leads to r axe/ar

=

N axe/aN = xe(l

If this, with ‘y f,

=

4, is substituted in (21) we obtain

axe ar,

=

+ ,~e)

—

—

a ar, {ir —

2xe(I + ~Xe)}. —

(23)

~E

The derivation of the corresponding magnetic results (see e.g., Abraham and Becker [3] or Robinson [25]) is much more complicated, because we have to introduce currents and deal with internal sources of power and dissipation, the results, however, are similar:

aB. f,

=

(J A B) 1

—

~(n~—1

a —

B1

-~)

—

~

—(H1B1

a —

H1B1) + —T~.

(24)

The magnetostrictive stress tensor is 11k11j =

(25a)

2l~kli/

where I3ijki

=

apIJ/auk,.

(25b)

326

F.N.H. Robinson, Electromagnetic stress and momentum in matter

Again T~t1 is complicated, except for fluids, where

ap/ar

2T

T~tT = ~]1J

=

4H2r axtm/ar.

The corresponding total stress tensor, excluding purely elastic terms, is

T~ 7S+

T~tr,where

5= ~(H

T~

1BJ+ H,B,

—

(26)

H B61J).

We emphasize that these results are valid only for linear media, and only for quasi-static fields, i.e., in circumstances where the field equations reduce to V A E = 0, V D = p, V B = 0 and V A H = J. The power of Penfield and Haus’s method is such that it can deal with cases where neither of these restrictions is needed. For future reference we combine these electrostatic and magnetostatic results to give an expression for the static force density: = —

pE1+ (JAB)1 —~(E1 D1~J)_~ -~-(E~D1E1D1) (H. B. —(H.B. H.B.) +—(T~~ + T~tr). ~_!_

~

2

‘—

/

ar1

—~

/

ar~1

—

—

~

2

(27)

—

ar1

/

/

‘

ar1

“~

‘-

We can express this as the derivative of a tensor T11 and the result is

a

a

T.. =—(T~+j~’estr+ ar1 ar1 7/ “

‘/

Tm5tm~)

(28a)

1/

where ~

(28b)

5. Manipulating Maxwell’s equations If the field equations are written in the form VB=Q,

(29a)

VAE+B=0,

(29b)

VDp,

(29c)

VAH=D+J,

(29d)

then, by taking the scalar product of H with (29b) and of E with (29d), we obtain V~(EAH)+ED+EJ+HB=0.

When this is applied to a volume V within a closed surface 5, with a positive outward normal, there results

(30)

F.N.H. Robinson, Electromagnetic stress and momentum in matter

327

f(E.b+H.h +EJ)dV= —f(EA H)~dS. (31) In the event that the media within S are all of that special type for which P = 0 and M = 0 we can, using the basic force law (2), identify fE .J dV as the rate at which the fields do work on matter, increasing either its internal energy or its kinetic energy. In this case E~D = E~E and H B = H~H = Th B, and we can further identify E~ D + H B as the rate of increase of the field energy density and, at the same time, identify F A H as the energy flux vector. By virtue of the field equations the normal component of E A H is continuous across any surface and therefore, if S is the surface of a general medium, the rate of energy efflux from the medium can be calculated by evaluating f(E A H) dS over a surface either just inside, or just outside, the medium. It follows that E A H is the general energy flux vector, and from this it then follows that the left-hand side of (31) is always the sum of the rates at which energy associated with the fields is accumulating and being transferred to matter. If, in the expression pE +J A B, we replace p by V~D andJ by V A H— D we obtain an equation which can be written as .

a~1

aE1

~

a ——(E~D1—E~,D~)

—~

=

(H1

-~-~

—

B1

~-‘)

—

~

~-(H1B1

-~--~ (E1D~+ E~D1 (E D)611 —

+

—

H~B1)+ ~(D

H~B1+ H1B1

—

A B)1

(H~B)61~).

(32)

Apart from the omission of the electrostrictive and magnetostrictive terms and the inclusion of an extra term ~(D A B)/at on the left we see that this equation is just the static force equation (27). Thus, it is at least plausible that (32) is the general momentum balance equation, when of course the appropriate electrostrictive and magnetostrictive terms have been added to both sides. This is certainly the case in vacuum, or in media where P = 0 and M = 0, for then (32) reduces to

pE1

+

(J A B)1

+

~—(D A B)

=

(E1EJ

—

2o 4E 1~+ B1B1

—

~B2o11)

(33)

and, on the left-hand side, we recognize pE + J A B as the density of force acting on matter. Thus, A B = E A H must be the field momentum density and the right-hand sideof theunits, derivative of the 2, suppressed by our choice the momenstressdensity tensor.isInequal vacuum, apart fromflux a factor tum to the energy vector1/c and the whole collection of terms can be assembled to yield a symmetric 4-dimensional energy-momentum tensor. The distinction between (32) and (33) is important. Equation (32) is merely an identity derived from the four field equations (29) with no reference to the force equation. Equation (33) is a similar identity for vacuum and regions where P = 0 and M = 0 but, because in these circumstances the two terms pE and J A B on the left can be identified as forces, it can also be directly interpreted as a momentum balance equation. In (32) except in static fields, we do not know the significance of a single term on the left-hand side. The tensor on the right of eq. (32) is just the static tensor introduced in eq. (28b) and, if we

D

328

F.N.H. Robinson, Electromagnetic stress and momentum in matter

add to this tensor, ~II (32) as

a at

— (D A B),

‘

=

the electrostrictive and magnetostrictive tensors to form T.. 7/ we can write

aT/ar.. ~

(34)

/

It is natural, but incorrect as we shall see, to identify f~as the force density and D A B as the electromagnetic momentum density. At this stage this marks the limit of our possible progress. We shall in fact show, in section 9, that the field momentum density is not D A B but rather E A H, even in a region containing polarizable and magnetizable matter, thus, the force density is

a at

J~.=f~+—(DA B—EAH)=aT../ar.— 1/

/

a

---(E AH). at

(35)

We emphasize, however, that we have not yet deduced this result. If, for the moment we accept (35) then, for a plane wave propagating in a homogeneous isotropic medium, we obtain

a =

(D A B — E A H),

a +

—

(T~tr+ T~t~D.

(36)

It is tempting to assume that this describes material momentum of density D A B — F A H propagating along with the wave which also carries field momentum of density E A H. The remaining terms are thus relegated to a limbo of obscure and unimportant effects which probably vanish at high frequencies. Many authors have thus been led astray. The fallacy is easily demonstrated by considering a simple fluid whose susceptibilities —i and p— 1 are proportional to its density. We then have Ttr + T~tr= ~ {(e—1)E2 + (p—l)fP}611. (37) For a plane wave

2-~V~EAHI,

E

H~ 3/~-IEAHI p

and —-*_v~—-,

ar,

at

(38)

and so (36) becomes

f

{pe— 1 —4p(e— 1)—~e(p— 1)}-~EAH~(p

+e_2)±EAH.

(39)

We see that the electrostrictive and magnetostrictive terms are significant. They change the density of material momentum from (p—l)E A Hto ~(p+e—2)E A H. We emphasize, once again, the crucial importance of these terms. They express, in an admittedly obscure way, fundamental electromagnetic consequences of the structure of the medium which, though an essential part of its overall macroscopic constitutive relation, are not entirely described by the constants c and p alone. Several authors have by diverse, and sometimes devious, processes obtained results equivalent to (39) and surmised that it is a general result for linear media. We see, however, that it only arose because we took a particularly simple form for the electrostrictive and magnetostrictive tensors. It cannot therefore be a general result.

F.N.H. Robinson, Electromagnetic stress and momentum in matter

329

Equation (32) is by no means the only identity, of the correct dimensions, that we can deduce from the field equations. In choosing it for discussion we were guided by the known form of the static result. Various authors have been guided by other considerations and we now give two examples. The equation V E = p — V~P suggests that P is equivalent to a charge density, and similarly the equation V A B = J + P + V AM suggests that P + V AM is equivalent to a current density. Thus, it is plausible that the force density in matter is, apart from electrostrictive and magnetostrictive terms, (p—VS P)E + (J+ P + V A M) A B. On this basis we are led to consider the identity —

(~—VP)E+(J+P+

V AM)AB+~(EA

26 B)~~(E1E~ —4E 1~+B1B1_4B2611.

(40)

We should then interpret E A B as the total momentum density and, if part of this, E A H, is the field momentum the remainder (E A M) is associated with matter. Since a (E A M)/at = E A M — M A (V A B) we obtain two extra force terms. The first E A M, though not widely known, is indeed a plausible force term (see e.g., Costa de Beauregard [11], Shockley and James [28] and Coleman and Van Vleck [10]) but the origin of the second term is obscure. There are two reasons for distrusting (40). The first follows from the discussion at the end of section 3: the vectors P, M are introduced solely as sources of fields, we do not know their relation to forces. The second is more conclusive. Even if we add electrostrictive and magnetostrictive terms to (40) it yields the wrong static force. Thus, if a/at, p,J and B are all zero we get fS

=

—(V P)E

+

a ~~_rstr ar~

which cannot be reconciled with (13). Another equation which again is an identity, is 2ó (PE+JAH+I~AH+EAI~I+(P. V)E+(M V)H+~(EAR))

=

~{E,D

~4E

26 11+H1B ~4H

11}. (41)

Since the force acting on a single dipole p is (p• V)E this looks plausible but we note that although E A M is reasonable the three terms J A H, P A H and (M~V)H contain H, where we might have expected B. In fact, in a simple fluid medium, (41) without the addition of electrostrictive and magnetostrictive terms, is a correct momentum balance equation but it has no more general significance. By the exercise of further ingenuity we can construct other identities but the exercise is pointless. Unless these identities can be related to the one fundamental force law,f pE +J A B in vacuum, they are little more than idle curiosities. They also yield no information about the electrostrictive and magnetostrictive terms.

6. Mechanical conservation laws Since we have failed to extract any ambiguous results from Maxwell’s equation we now see what we can extract from the various general conservation laws, without paying more than passing

330

F.N.H. Robinson, Electromagnetic stress and momentum in matter

attention to the field equations. This is rather like going to the well to draw water without taking the bucket, but we shall be in good company and, though we may not actually extract water, we shall at least see that it is there. We have an umabiguous solution for particles moving in vacuum and this solution can be used in a microscopic formulation. There it leads to a number of conservation laws, i.e., those for angular momentum and energy. In addition, in conjunction with special relativity, it leads to the constancy of the velocity of the center of mass of a system provided that, in vacuum we attribute to a wave of energy u a mass u/c2 or, in our units, a (Taylor [29]). The relation between macroscopic and microscopic theory is such that any microscopic conservation laws can be directly transcribed to macroscopic laws, provided that they are only used in conjunction with a system whose boundary surface lies in vacuum, and at least a macroscopically significant distance from the nearest material particle. Arguments based on these ideas have been presented by Balazs [7] and in a somewhat simpler form by Shockley [27] and Arnaud [4] We shall see that they are insufficient for a complete solution but lead to one important partial result. This is that, when a wave propagates in an isotropic medium, a part E A H of its total momentum density is not associated with the motion of matter. We shall give a rather more general treatment than these authors since it is rather easy to suppose that the limited results derived from the approach occur because only the simplest cases have been investigated. The system that we shall discuss consists of a pulse, or wave-packet, of approximately monochromatic radiation incident on the surface of a plane slab, or lamina, of refracting material characterized by isotropic constants and p and a refractive index n = (pc)112. We shall also assume that the pulse, though of finite lateral extent is approximately a plane wave and, for simplicity, that it is polarized with its electric vector in the plane of incidence. In fig. 1 we indicate the lines along which the center of gravity of the pulse travels after the various partial reflections that will occur and, on each line, we indicate, in terms of the reflection coefficient i~,the fraction of the initial incident energy in the pulse as it traverses this line. The figure also indicates a set of axes whose origin is at 0 the point of first incidence. The thickness of the lamina is 1. For our choice of the polarization the power reflection coefficient is -

~1= (~ cosO—P~cosø \ cos 0 + p”2 cos ~

)2

(42)

L~

The initial momentum of the incident wave is p 0=f(E AH)OdV

(43a)

and the magnitude of the momentum is Po

(43b)

where ur, is the pulse energy. If a is the energy of the pulse in the medium then, because it travels with a velocity 1/n we have fIE AHIdJ7u/n.

(44)

In fig. 1 we have tacitly assumed that the wave energy is conserved, which implies that the mass

F.N.H. Robinson, Electromagnetic stress and momentum in matter

%k-1 2% r~1 (1-17)~ --

k+2

k

%\

\8I8

2%

~ 7I1~~~L\

4_

_~z

‘i\i/

2

0

:3h1?,

~~{1-i~)

~

——

k+1

VACUUM MEDIUM

(1-,~)

MEDIUM

_________________

k+3

331

3 .,72(l_,1)2

i

VACUUM

(1_~I2

Fig. 1. A pulse of radiation incident on a plane refracting lamina, showing the intensities and directions of the various transmitted and reflected pulses relative to an incident pulse of unit intensity.

density of the medium is so large that any kinetic energy associated with momentum transferred to the medium is negligible. This is almost certainly justified. At each partial reflection momentum is shared between the reflected wave, the transmitted wave and a localized momentum S imparted to the surface. If S propagates away from the surface it does so at the velocity of sound, rather than of light. As a first step we consider the angular momentum of the system about 0, the point of first incidence. Before the wave arrives the angular momentum is zero. Since neither the reflected wave nor the surface momentum S0 lead to angular momentum about 0 we conclude that the transmitted wave also has no angular momentum about 0 and thus that its linear momentum is parallel to its direction of propagation, and therefore to E A H. Two comments can be made on this result. First, if the wave did have transverse momentum it would, as it propagated, lead to an ever increasing angular momentum about 0 which could not be cancelled by a localized torque on the surface. Thus, the result does not depend on whether such a torque exists. Secondly the result applies only to the whole of a wave pulse of finite lateral extent. We shall see, in connection with Shockley’s model, that the results for a plane wave of finite lateral extent are rather less obvious. Because the momentum of the wave is parallel to E A H we can write it as PaJ’EAHdV

(45a)

where a depends only on p and e, or at least not on the angles 0 and ~ of incidence and refraction. We also have pau/n.

By balancing linear momentum at the first incidence at 0 we obtain

(45b)

332

F.N.H. Robinson, Electromagnetic stress and momentum in matter

(1

csp0 i~)pocos0= S0~+ (l—i~) —cos~ n

+

(46a)

and ap0

(l—r~)p0sin0 S0~+(1—~)-——sin~.

(46b)

n

Suppose that S0 is zero at all angles of incidence, then it is also zero at normal incidence and so from (46a) we obtain l+r~ -~(p+e). I —r~

an

(47)

2 = p and so we conAt any other angle of incidence (46b) yields the incompatible equation a = n clude that S is not zero, except perhaps at some one special angle, which may possibly be normal incidence. Note that ifS = Oat normal incidence a is given by (47), and this (see eq. (39)) is the result for a simple fluid medium in which —1 and p—l are proportional to density. In general, then, we must have

S

0~ Po{(l +~)cosO—~(l _n)cosø}

(48a)

S0~p0~(l—~)sin0—~(l_~)sin~). We can also calculate S, and S~for the (k

SkX

=

(~)k_

1(1

=

~k_1(1

—

— ~)

{( I

+ ~)

+

(48b) I) partial reflection when k ~ 1, and the results are

cosØ — (1 —

fl)cos0} Po,

(49a)

and Sky

~)2~

sin~— sinO) Po’

(49b)

The sum of all the surface momenta Sk gives the total momentum imparted to the lamina. The results are ~

10x

k0

Skr=~P0CO50

(50a)

1+?7

and =

0.

(SOb)

It is easy to check that these are just the differences between the initial and final momenta of the waves in vacuum.

F.N.H. Robinson, Electromagnetic stress and momentum in matter

333

Next.we calculate the position coordinates of the center of mass of the radiation at a time T, much later than the time of first incidence. The rather extensive sums over the partial waves, in vacuum, give 1—77 ~l_—77\2cos0 x =Tcos0 1—377 1+77 + l+~i—nil\l+flJ I cos~ and Yr

=

Tsin0 — (n2

—

1)1 tan~.

To calculate the position of the center of mass of the slab at the same time we assume that, as the pulse travels through the medium with a total momentum au/n, only a part /3u/n of this momentum is imparted to the matter. The changes Xm and Ym in the coordinates of the center of mass of the slab are then obtained from 2+etc.),

MXm=~iSkx(T_ knisec0)+13p 01(1 —77)(l —77+77

and MYm=~Sky(T~ knisec0)+flp

2+etc.), 0itan~(l —i~)(l+fl+77

which give

Mx

477

=

p~ 1+~TcosO

+

1 —77 (/3 — a)i 1+77

+

/1_—77\2cosO cos~

nil\l+rjj

and

MYm = Po~(/3— a)tan~+ n21 tan~}. To conserve the motion of the center of mass we must have P&~r+ MXm

=

p 0Tcos0,

PQYr + MYm

=

p0TsinO,

and we see that this requires a — /3 = 1. Thus, of the total propagating momentum afE A H dV a part (a — fJ)fE A H d V = fE A H d V is not associated with the motion of matter. It is natural to regard this as the wave momentum and we see that its density is E A H. This is the one substantial result to be obtained by this approach, but it is important to realize its limitations. We do not arrive at an expression for a and therefore we do not know how to partition the material momentum between propagating momentum described by a, and localized surface momentum described by S. If perchance we knew that S was zero at normal incidence then we should know that a ~( + p). In the absence of any evidence on this point several authors have assumed this to be the case. It is clearly not so in general for, as we were at pains to point out in section 5, this result depends on a particular form for the electrostrictive and magnetostrictive tensors.

334

F.N.H. Robinson, Electromagnetic stress and momentum in matter

The results of this approach are, as we might have anticipated, of rather more philosophical than physical significance. If we still do not know how to calculate the distribution of momentum and stress in the medium, it is of little interest to know that we must first substract E A H from the momentum density.

7. Gordon’s model We have, so far, failed to obtain an unambiguous and precise description of momentum and stress, either by manipulating Maxwell’s equations while ignoring the laws of mechanics or by using the mechanical laws but ignoring the field equations. A similarly discouraging experience has led a number of authors to attempt to deduce the desired results by consideration of microscopic models. Among these treatments we select two for discussion; Gordon’s [14] because though limited in its applicability, it is rigorous, and Shockley’s [27] because it is both exceedingly ingenious and slightly misleading. In both cases the essential features of the models are the way they avoid discussing all interactions between the microscopic units of the medium which are not mediated solely by macroscopic fields. Gordon’s model is essentially a rarified gas of atoms whose electromagnetic properties are completely described by their linear electric polarizability a. The atomic density N is so small that the separation between atoms N”3 is large compared with the extent of the atoms. Thus, the effects of distant sources and neighboring atoms on a particular atom can be entirely described in terms of the macroscopic fields F and H = B. Because a is, in general, of the same order as the volume of an atom we see that this model is restricted to the case where c 1 = Na is small. The force acting on a single atom with a dipole moment p = ctE is -—

Fa=(p~V)E+pAH=a(E~V)E+aEAH.

(51)

When a plane wave propagates through the medium the first term is zero and the force density due to N atoms in unit volume is therefore

a

a

fNaEAHzz~Na_(EAH)4(e_l)__EAH.

(52)

If we keep in mind that we have set p = 1 we see that this corresponds to our earlier expression + c — 2)E A H for the material momentum density. The considerations of the last section can be very easily applied to this model system to show that the additional non-material momentum density is E A H. If we express (51) as

a

a

.

f+—(EAH)=(P~V)E+PAH÷—--(EAH)

and note that

~ /

aE. ar 1

aE. /

ar,

,~aE. aE. /

\ ar1

ar,

F.N.H. Robinson, Electromagnetic stress and momentum in matter

335

we obtain

a aE1 a f1+—(EAH)~=P1—+—(DAH),, at ar1 at

(53)

or

a

a

f+_(EAH)=4(e_I).VE2+_(DAJJ).

(54)

We can also express this as

a

ai~ a

f1+—(E ~

(55)

Because we have both the total effective stress tensor and the separation between field and matter momentum this model had led to26,~ a complete solution. We note that for this medium the electro= 4E. P6, strictive stress tensor is ~(e — l)E 1 and so the expression on the right of (55) can be written as

aT1,

a

—=——(E1D1+H,H1 ~ ar, ar~

a ar, l~

(56)

which is completely compatible with all our earlier results. We note a typical feature of model calculations which, if they yield anything at all, always yield the total stress tensor including the electrostrictive term. We expect this for, as we have remarked earlier, the electrostrictive term reflects, in macroscopic terms, effects due to the structure of the medium, other than those described by e and p. Gordon’s model is inherently confined to media in which e — 1 is small and attempts to enlarge its scope, by allowing for nonmacroscopic atomic interactions by using a Lorentz local field correction, are of doubtful validity. Its main virtue is its rigor, and the fact that within its limited context the solution is complete. Because in (55), we have a complete momentum balance equation we can use it to discuss not only plane waves, but also waves of finite lateral extent. Gordon has used this, as follows, to discuss the interpretation of experiments described by Ashkin and Dziedzic [5, 6]. In these experiments a short intense pulse of radiation of small transverse dimensions enters a fluid medium, from vacuum, and fluid is observed to be drawn out of the medium within the beam of light, both where the beam enters the medium and where it leaves the medium. The pulse, though short, is yet so long that there is enough time for mechanical equilibrium to be established in the fluid in the region of the light beam. We therefore, need only consider steady-state conditions in which the time average of ~(D A H)/at vanishes. There 2 tending to attract fluid into the beam, whereinE (54) is most intense. Thisremains force aacting forceinwards 4(e — l)VE over the periphery of the beam raises the pressure within the beam to it = iT 2, and this increased pressure forces fluid out through the surface, both where 0 + 4(e — l)E the beam enters the medium and where it leaves the medium. The situation on entry is shown in fig. 2. The total momentum transferred to the surface, for a pulse of duration r is —r f iT dS, where the integral is over the cross section of the beam. We can express this as —~(e— 1 )rfE2 dS.

336

F.N.H. Robinson, Electromagnetic stress and momentum in matter

1~

BEAM

VACUUM FLUID

-~

I—

FORCE

FORCE

Fig. 2. A narrow beam of light entering a fluid medium.

Since, in the medium the refractive index n = and we have E2 = iF A HI/n we can also express the momentum transfer in terms of the pulse energy a as —4( l)u/n. Because t’ie driving force is transverse to the beam an equal, but opposite, momentum must be transferred to the fluid below the surface. Thus, the total momentum in the region of the pulse is fE A H dV due to the field momentum together with 112,

—

—

l)J’E A HdV+4(

—

l)u/n

(e

-~l)f(E

A H) dv.

The sum of all these terms is e f E A H d V. In this sense the total momentum associated with the wave is fD A H dV, but we must remember that part of this has come from the surface reaction. If, however, we were to take fDA HdVas the total momentum in the medium we could calculate the surface reaction from the difference between this expression and the momentum of the incident wave. Since this is just the observable mechanical effect, there is something to be said for this identification of the total momentum with fD A B dv. There is a discussion of the significance of this expression in Gordon’s paper and we shall return to this question later. The extreme simplicity of Gordon’s treatment is both a virtue and a fault. On the one hand it yields a complete solution for his model, on the other it gives no indication of how we might generalize it to more complex media. Suppose for example that we try to modify it by a local field correction Eb0c = E + 4P. This is equivalent to adding 4P26 1, to the electrostrictive stress tensor in (56). This we know to be satisfactory in the static case but, if we add a similar term to (55), we have no grounds for believing that it will apply to the time-dependent case. It adds a force 4p1 aP1./ar, to the left-hand side of our basic equation (51) whereas we might have expected 4p1 aP1/ar,. The difference between these two terms is —~-(e I )p A H, whose significance is obscure. It does not appear that any easy generalization of this model is possible. —

F.N.H. Robinson, Electromagnetic stress and momentum in matter 8.

337

Shockley’s model

Shockley [27], to avoid having to discuss the force acting on atoms, considers a medium, which he calls “cube-stuff” built out of an array of small cubes and packed rather like a child’s building blocks, but with a minute space, or crack, between each block. The blocks themselves are characterized by constants e and p and each block, though large on an atomic scale, is nevertheless small on a macroscopic scale. Thus, on the one hand we avoid discussing atoms and on the other we can treat “cube-stuff” as a continuous medium in macroscopic applications. If F and H are the fields within the cubes we can calculate E” and H” the fields within the vacuum cracks, using the ordinary macroscopic equations. The force acting on a single cube can then be calculated by integrating the vacuum stress tensor over a surface lying entirely within the cracks surrounding the cube. With axes chosen in fig. 3 and taking account of the fact that a tensor element T11 only has to be evaluated over a surface normal to j, we find that the effective stress tensor for cube-stuff is (57)

When a plane wave propagates through the medium, in a direction making an angle 0 with the x1 axis and in the x1x2 plane, and has its magnetic vector normal to this plane, we find that the

momentum entering a cube (of side i) is =

[(~_t~— p(c —

1)2 sin 20) E A H — p(e

3.

—

1)2 sin40 k A (E A H)) i

(58)

Part of this momentum is associated with the fields, we may assume it is i3E A H, and the rest is material momentum. There are two puzzling features about this formula. The first is that it has a component of momentum transverse to the direction of propagation. The second is that, although from a macroscopic point of view, the medium is isotropic, it contains the angle 0. If the result (58) is applied over the whole of a beam of finite width it would, because of the transverse term, lead to violation of angular momentum conservation. In fact (58) is only valid for cubes in regions

/VE

Fig. 3. Cube-stuff showing axes.

338

F~N.H.Robinson, Electromagnetic stress and momentum in matter

where the wave is exactly plane. Near the edges of a finite beam the results are much more complicated and, at the edges there are shear stresses which leave behind a localized, non-propagating, wake of momentum. Taken altogether momentum for the system as a whole is conserved. Clearly the distribution of momentum in this medium is far from simple. The appearance of the angular terms in (58) is more puzzling. As Shockley correctly states “this example demonstrates the futility of attempts to establish energy-momentum tensors based solely on considerations of e and p”. We may not, however, infer from this that a completely macroscopic treatment is impossible. As we have continually emphasized a macroscopic treatment must also include electrostrictive and magnetostrictive terms, whereas in a model treatment these terms appear automatically. In cube-stuff a shear strain (relative to the cube axes) does not alter either e or p, but extensive strain along the axes does, by altering the width of the cracks. A short calculation yields the dcctrostrictive coefficient 6if6k1( I + (~ I )6~)~ (59) al/kl = dcIJ/dukl = —( 1~ —

—

where no sum over i is implied on the right-hand side. There is a similar magnetostrictive coefficient. The nonvanishing components of ailki are a, 111 = a2222 = a3333 = --e(e 1) and a1122 = a2211 = a1133 = a331, = a2233 = a3322 = -—(e 1). Although a1fkl has f’ull cubic symmetry it 6ll. The resulting contribution to the stress is not isotropic; we cannot express dc,1 as a1u11 + a2ukk tensor is: —

1)6i16k1(’ ~

—

-- l)öjk)EJEI+4(p

—-

‘)61j6k1(1

+(p

—

l)6jk)J~JjHl

(60)

and, when this is added to the field tensor 1/

1

/

I)

7/

2

(61)

the sum yields the effective cube-stuff tensor. Thus a macroscopic treatment of a medium with the same, rather bizarre, macroscopic properties as cube-stuff would also lead to the same results. Cube-stuff far from being a simple medium is in fact quite complicated. A full macroscopic description of the medium includes, in addition to the isotropic constants c and p, a specification of the electrostrictive and magnetostrictive coefficients which are fourth rank tensors.

9. The method of virtual power Of all the methods so far discussed only the Helmholtz approach, applied to quasi-static systems, has yielded general and unambiguous results. We recall that in this method the virtual work done when each point of the medium was given an arbitrary small impressed velocity v was calculated and expressed as f systems v X dV, and X could then be identified with the force density. The method of virtual power to Penfield and Haus [231 is essentially similar but, instead of restricting v to be a small time-independent velocity, they consider an arbitrary impressed velocity. As a consequence all the mechanical aspects of the system require a relativistic treatment. The results that they obtain are quite general and apply to moving media, dispersive media and dissipative media but, for the sake of clarity, we shall confine our attention to stationary, nondispersive systems. Wheras in the Helmholtz treatment we considered the total energy of the system, Penfielci and Haus find it more convenient to deal with energy densities and differential relations and, as a result, their calculation takes a rather different form. —

-

F.N.H. Robinson, Electromagnetic stress and momentum in matter

339

There are essentially two basic steps in their argument. The first is concerned with the division of a complete system, in which energy and momentum are conserved, into two, or more sybsystems of which one, labelled by a superscript k, stores energy and momentum only through the motion of the medium. The other subsystem or subsystems are associated with energy and momentum stored, either by fields, or by internal mechanical and electromagnetic stresses in the medium. The second step consists of an analysis of the rates at which momentum and energy are transferred between these subsystems. Each subsystem is characterized by a momentum-flow, (or stress) tensor T11, a momentum density G,. an energy flux vector 5, and an energy density W. If the subsystems were isolated, or closed, these quantities would obey conservation laws of the form

aT11/ar, +

0

=

aS,/ar,

and

+

h/ = 0,

but, because the subsystems are not closed, these equations have to be modified by the addition of a force density f1 and a power conversion density so that we have ~,

aT1,/ar1

+ d~,=f1

(62a)

and aS,/ar,

+

hi

=

(62b)

0.

Because the system as a whole is closed the force densities fk densities ~1k ~m etc., satisfy fk +frn+ etc. = 0 =

0.

fm

etc., and the power conversion (63a) (63b)

If there are only two subsystems and we can find T~and G,~in terms of the electromagnetic and strain variables then we can obtain fk as m= —{aTtm/ar. + ~ (64) fk= i

~f i

ii

/

I

To get to this result we begin by considering the transformation law which leads from subsystem quantities T,~,G~,S°~ and W°in the reference frame, in which the material at r is at rest, to the laboratory frame in which it is moving with the impressed velocity v. The relativistic transformation laws are rather complicated and so for clarity we give only the nonrelativistic transformations. These are

T., = T,’ =

+

v,G,°,

G,

=

G~,

S~+ v.W°+ T

11t’1 + v.v1G?, 1, W—W°+v~G~, fl=f;: When these are substituted in the energy eq. (62b) the resulting expression can be rearranged as v

1W°+ T1,v1)

ar,

+±(W0 +

at

v1G,) =

0° +

v,f1.

340

F.N.H. Robinson, Electromagnetic stress and momentum in matter

The momentum eq. (62a) can then be used to eliminateJ1 and, with a little further rearrangement, we obtain

a

aw°

au

av “ar1

—(S?+v.W°)+-___Ø°=_T._~.L_G._-L.

ar,

/

at

/

(65)

‘at

On the left appear only quantities relating to the energy balance equation which we assume to be known. When these quantities, which are expressed in terms of the rest frame variables, have been re-expressed in terms of the laboratory frame variables then, since av1/ar1 and av~/atare arbitrary, we can equate their coefficients on the left to —T11 and —G,. Equation (65) is in fact the generalization of eq. (1 lb) used in the Helmholz calculation. The relativistic equivalent of(65) is ta aw° a av 2/c2)~2}+~-~ v2/c2)”2 = --T a 1 ~ v,Si)j +-~- +~(v~W°)—Ø°(1 1 av1 1 —v 11 -i-— G, (66) ‘~

~-~(

~

where S~and S°1 are the components of S°parallel and perpendicular to v. To use (65) or (66) we apply it to the non-kinetic part of the system and thus finally, after a long series of manipulations in which very careful account is taken of the distinction between total and partial time derivatives, and between derivatives in the rest frame and in the laboratory frame, there eventually emerges an expression for TT and G~in the laboratory frame. Equation (64) then yields the force f~acting on the material medium and the problem is solved. Partly because Penfield and Haus present most of their working in terms of the Chu formulation of electrodynamics, and partly because they do not separate the symmetric part of the strain tensor from its anti-symmetric or pure rotational part, their results need some further manipulation before we can the compare them with the neglecting results in our sections. Their general for m and momentum Gm are, any earlier field-independent elastic termresults in T~, the tensor T E

H

—T~=D~~+B

1H,_E2+~÷fP.~+fM-

f1aM

E~P

~

(67)

and m=FAH.

(68)

G

In (67)

w~,

1= au1/ar1 is the unsymmetrized strain. These results refer to both linear and nonlinear media but, for the purpose of comparing them with our earlier results, we shall confine our attention to linear media. Then, since P1 = x~kEk= x~lEkand ax~1/aW11= ackl/aWEI we obtain =

D,E,+B1H1_~.(F.D+H~B)6,1 —4EkElak//aW1I—+HkH! apkl/aWlJ

If we now express W11 as a11 + w,~, where ~ =-~(au,/ar1+ au1/a,~)is the symmetrized strain and = ~(au1/ar~-- au./a,~) is the anti-symmetric or pure rotation part of w1,~,we find, after some manipulation, that

EkE! aek,/aW~I= EkEI aekl/aulI + E1D,

—

E,D1 = aklI,EkE1 + E,D,

—

E1D1,

where ak,,, is the electrostrictive coefficient introduced in section 4. With a similar rearrangement of the magnetic term we finally obtain

F.N.H. Robinson, Electromagnetic stress and momentum in matter

~

341

(69)

where T7t1 and T~5tr are the electrostrictive and magnetostrictive tensors introduced in section 4 as equations (14jmand (25). We can write this as + + T~tr (70) —

T~=

where

T,1 = T,~

T,~m= 4{EID,

+

E 1D, — E D6,~+ H,B1 + H,B,

—

(71)

H~B6,~}

is the tensor introduced in eq. (28b) of section 4. The force density is then, from (64) and (68), A II)~. (72) a a m + T~tr+ T~tI~)——(E a ara f, =—T,1 ——(E AR)1 =—(T’ 1 at ar1 / / / at mis the tensor appearing on the right of eq. (32) we can use this equation (an identity Because T~ Maxwell’s equations) to write down f derived from 1 directly. It is

a at

~ aB.

—4 (i~_~ —

/

aH’ B. —_‘)

aD. ar,

a —

~(fJB

—

H1B1)

+

aE.\ a ar, —~—(E1D~--E,D~) ar,

a t~ + ~ —(T~

(73)

Equations (71), (72) and (73) with the identification (68) of the field momentum as E A H are, at least as far as stationary, linear media are concerned, the complete solution. We have the general stress tensor and the separation of field and material momentum. This separation means only that the rate of change of the material momentum density, taken alone, cannot be expressed as the derivative of a stress tensor. The missing term, ~(E A 1-1)/at is, however, most naturally interpreted as the rate of change of a field momentum density. Finally, we note that, although we have only given the results for stationary, linear media Penfield and Haus’s principle of virtual power leads to equally unambiguous, though much more complicated, results for both moving and non-linear media.

10. Discussion Equations (68), (71), (72) and (73) are a complete description of electromagnetic stress and momentum in stationary, linear non-dispersive media. By the same methods equivalent, but more complicated, results can be obtained for both moving and more general media. The formulae have been obtained entirely by macroscopic arguments and involve only macroscopic variables. We note, however, that in addition to the macroscopic fields and the parameters c,, and p, 1 they 13 These additional contain the describe electrostrictive and magnetostrictive coefficients al/k, andstructure of the medium parameters macroscopic features resulting from the internal which are not fully exposed to view by the constants and p, 1. They are macroscopic parameters and can be obtained by macroscopic measurements. Their appearance in no way commits us to a microscopic study of the structure of the medium. We remember, however, that except in the very simplest medium, a fluid, the constants ai/k! and /3.~ki need not be isotropic even if and

342

F.N.H. Robinson, Electromagnetic stress and momentum in matter

p11 are isotropic. A crystalline medium of cubic symmetry is not, in respect of either its electrodynamic stress or its elastic stress, a completely isotropic system. The general formulae though compatible with a microscopic theory are not dependent on microscopic models. This is just as well, as we have seen in section 3 that the passage from the microscopic force law to the macroscopic force law is fraught with difficulty. The situation is in many ways analogous to that obtaining in the atomic theory of dielectric behavior, where a rather inscrutable local field connection intervenes between the atomic interpretation and the empirical behavior. This problem is common to static and wave fields and we have already encountered it in connection with Gordon’s model. Attempts to interpret the explicitly time-dependent term a(D A B -- E A H)/at in microscopic terms by, for example, writing it as PA H+ F AM—f AM — PA (V A E) — M A (V A H) are particularly unrewarding. We now make a macroscopic examination of the nature of the individual terms in (73), ignoring the first two terms pF and J A B, whose significance may be regarded as already sufficiently obvious. The next term ~(D A B — E A 1-1)/at clearly vanishes for static fields, but it also has a vanishing time average for fields with a steady sinusoidal variation. It is, therefore, only important in transient phenomena, such as the passage of a pulse of radiation through a system. Since force is rate of change of momentum, it leads to a material momentum density D A B — F A H which propagates, or travels, with the pulse. The sum of this term and the field momentum density E A H is not, however, the only propagating momentum density for, as we shall see, contributions arise from other in terms in (73). In the remainder of the equation we need only discuss the electric terms, since the magnetic terms are similar. The fourth term can be expressed as —-4E,Ek aeIk/ar~,and we see that it only leads to forces where the medium is inhomogeneous. In particular it acts at boundaries between two media, where there is a discontinuity 6 in the dielectric constant across the boundary surface S. The resulting force on an element dS is —~-6eJkEJEkdS,and it is always normal to the surface even if e is anisotropic. We note that this term is non-zero for static fields, and also has a non-zero time average for steady sinusoidal fields. When a pulse traverses a boundary, the localized momentum imparted to a surface layer of the medium during the transit of the pulse does not travel with the pulse; rather, it propagates away from the surface with the velocity of sound. It is convenient to take the electrostrictive term next and to express it in terms of the electrostrictive coefficient a. Differentiated by parts it yields —~EkElaakllJ/ar, 2aklIIaEkEl/arl.. and we see that the first term leads to boundary forces at discontinuities in a. The force acting on a surface element is —~-6akllJEkEldSwhich will only be normal to the surface if a depends on the indices i and! as 6,~.This is usually the case only for fluid media. This part of the electrostrictive term imparts a localized momentum to the surface of a medium, and this momentum does not propagate with a pulse crossing the boundary surface. The other part of the term leads to forces even within homogeneous media, and it is non-zero for static fields, as well as having a non-zero time average for steady sinusoidal fields. In the special case of a plane wave, of indefinite lateral extent, propagating with a velocity c’ = I/n along the r~axis, we can replace a/ar1 by — n61~a/at. and then the force density becomes~nakl1PaEkEl/at,which has a zero time average for steady sinusoidal fields. On the other hand a pulse of radiation has an additional material momentum density 2nakIIPEkEl, which travels with the pulse. In6~a truly isotropic medium, as glass, denthe 6 6k 6 and so thesuch momentum electrostrictive coefficient is of the general form a~ 1 1~ + a2 1 sity is }n(a 26 1E1E~+ a2E 1~).Since, in an isotropic medium, a plane wave has no component E~ —

1~,

F.N.H. Robinson, Electromagnetic stress and momentum in matter

343

in the direction of propagation, the momentum 4a2E261p is in the direction of propagation. This is not the case (see section 8) for a crystalline medium, even if it has full cubic symmetry, and so the total momentum, (E A H) + (D A B — F A H), + 4nakllPEkEl, propagating with an indefinitely extended plane wave, need not be in the direction of propagation, even if c and p are isotropic. This does not, of course, violate conservation of angular momentum which only applies to the entire wave, including edge effects, when the wave has a finite lateral extent. The remaining electric term —4a(E1D~— E~D1)/ar1vanishes in isotropic media, and it also vanishes for plane, transverse waves in anisotropic media. It therefore vanishes for ordinary waves in birefringent media. It is, however, non-zero for extraordinary waves in which, although D is transverse, the electric field E has a component in the direction of propagation. If we express D1 as e,kEk, we see that, on partial differentiation, this term will yield both forces localized at surfaces, where e is discontinuous, and a material momentum which propagates with a pulse. In general, the direction of this propagating momentum coincides neither with the normals to the wave fronts, nor with the direction of the energy flux vector, F A H. We see then that the term —~(E,aD,/ar1 — D1aE1/ar,) leads to localized momentum (at surfaces) which does not propagate, except with the velocity of sound, while the term ~(D A B — E A H)/at leads to a momentum density which propagates with a pulse of radiation. The electrostrictive and anisotropy terms (and their magnetic analogs) yield contributions of both types, and so greatly complicate the picture. In particular, the total propagating momentum density is not D A B = peE A H. Indeed, in the simplest case of all (Gordon’s model), with p = I, we have seen that the propagating density is 4(e + may l)E Afurther H. Theverify readerthat may this,the byforce the substitution 1)6k16 ,~6 j~.He in verify this case at the interface, where ak!I! = —(e — a pulse enters a medium from vacuum, is always normal to the interface and also that it vanishes at normal incidence. We have, so far, ignored the force density due to elastic stress. If the elastic stress tensor is Y~ 1 we must add aY11/ar1 to the force density given by (72) or (73) to obtain the total force density (acting on matter) ~ The rate of change of material momentum density per unit volume is then ~m_frn_f+aya(Ty)a(ER)

(74)

In equilibrium, in static fields, this must vanish and equilibrium will be achieved by strain in the medium which generates the appropriate elastic stress. In wave fields the displacement of matter required to achieve a balancing elastic stress will not usually be able to follow the rapidly changing electromagnetic stress; however, in a steady state, with steady sinusoidal fields, the timeaverage electromagnetic force will be cancelled by a time-average elastic force. In a solid, in which elastic shear stress may be present, the consequences of eq. (74) will be far from obvious, even in equilibrium. In a fluid medium, with no elastic shear stress, things are, fortunately, much simpler but, before we confine our attention to fluid media, there is one important general result that we can deduce. We consider an indefinitely extended medium which contains all the field, and whose external surface is free of applied mechanical stress. Then, from eq. (74), we obtain f

(f~+±(E A H)1)

dV =f(T11

+

Y1~)dS~t = 0,

(75)

344

F.N.H. Robinson, Electromagnetic stress and momentum in matter

which is, of course, just a statement of momentum conservation. Now let S, a closed surface within the medium, divide the total volume into an inside region and an outside region. We can then write (75) as r J inside

a dV= fm+—(EAH) ‘ at

a Jr j~°+—(FAH)dv.

—

at

outside

Suppose next that there is initially a single pulse of radiation in the outside region, and that it enters the inside region where it is either partially adsorbed, or scattered, and then re-emerges. Initially, and finally, F A H is zero in the inside region and so the total impulse imparted to

matter in the inside region is

I~=fdt

fmdv=fdt

f inside

+~(E A

I

dV= —fdt

at

inside

(im +~(E A

f

at

outside

H)

1}dV.

If, in addition, the medium outside the surface is isotropic and homogeneous this can be expressed as

I,

r

=

—Jdt

C

i

outside

a at

—(D A B)

a

(Y.. ar~ /

+—

+ T~~.5tr +

T~tT) dV

~‘

and, in terms of the initial and final values of D A B, as = I {(D A B)~~ —(D A B)~2)}dV_fdt f ~(Y1. outside

outside

ar1

+

~

+ Tmstr)dV 11

(76)

1,’

or we can express the last integral in terms of an integral over the dividing surface 5, and obtain

=ft~A B)~’~—(D AB)c2)}dV+fdtf(Y..

i~

+ T~stT+

T~tr)dS1,

(77)

where the positive normal to the surface is outward. The importance of this result is that in certain circumstances, which we shall shortly describe, the integrals involving the tensors in (76) and (77) are zero, and so the total impulse given to the matter within S is simply equal to the change in the volume integral of D A B. In these cases the momentum density associated with the pulse in the outside medium can, for the purpose of scattering calculations, be taken to be D A B. In the homogeneous, isotropic, external medium the force density is tr+Tt1)

a a J=.(DAB_EAJJ)+__(y+Tes

and the first term oscillates with twice the frequency w of the wave. At the same time its spatial variation, either along the pulse or across its transverse dimensions, is only appreciable over distances of the order of c/w. Because the velocity of sound s is much less than the velocity of light c, the strain and elastic stress in the medium cannot follow the rapid variations of this term. The approach of the medium to a steady-state, or state of quasi-static equilibrium, can only involve the approach of the elastic force, aY../ar~,to a value where it cancels the time average value of a(T~tr+ Tmstr)/arr If this state of quasi-static equilibrium is reached, which will take a time 0. where 0 = d/s and d is the least of the linear dimensions of the pulse, the second integral in (76)

F.N.H. Robinson, Electromagnetic stress and momentum in matter

345

is proportional to 0 whereas the first integral increases indefinitely as the pulse duration ‘r is increased. Thus, for pulses of narrow transverse dimensions and, long durations r, the second integral in (76) can be ignored and the momentum transferred to the matter in the scattering region within S is simply 1=

I outside

~(D A

B)(1) —

(D A B)(2)}dV.

(78)

When, therefore, we have to.deal with pulses whose transverses width w and duration r satisfy w ~ sr, we can calculate the momentum transfer in a scattering process by ascribing a total momentum density D A B to the pulse. A second case in which (78) is valid occurs when, perhaps because the pulse is wide and its duration short, the elastic force aY,1/ar, does not have time to change before the pulse is past, so that we can ignore Y,1 in both (76) and (77). If, at the same time both T,5t1 and T,~tIare zero on 5, eq. (77) again yields (78). This can, for example, happen because the magnetostrictive coefficient is negligible and we are discussing the reflection of a plane wave, at normal incidence, from a conducting surface 5, so that F and T,5tI~are both zero on S. There are, therefore, important cases where the complications due to the electrostrictive absent.need We note 3m sec’, theterms pulseare duration only that, exsince the velocity of sound in most media is about 1 0 ceed 1 psec for a beam 1 mm wide, for eq. (78) to apply. We now specialize our results to fluid media, in which the elastic stress tensor is simply = —H6,~, where H is the pressure. At the same time the electrostrictive and magnetostrictive tensors are: =

T~~stI = 1/

4aE26jj

_!/3J.126.. 2 i~

~

(79a)

.!-IPp--~~6.. 2 ap ‘~

(79b)

where p is the fluid density. The force equation is now

ftm =—a-(D

A B —F AR) _4E2V6

—4H2S7p — V(.~.aE2+-~-/3fP+ II).

(80)

With static fields in a homogeneous fluid only the last term does not vanish. Thus, in equilibrium, the fluid must be strained (have moved) so that the electrostrictive and magnetostrictive forces are cancelled by a pressure gradient, and the required pressure is fl

=

11o

_~aE2

_4j3H2

=

Ho+4E2p~-+4H2p~-,

(81)

where H 0 is a constant pressure, which may be ignored. Since the elastic stress tensor now exactly cancels the electrostrictive and magnetostrictive tensors the total effective stress is simply T~m=+(E,D1+E1D,_E.D6~,÷H1B~+H1B1_H.B611).

(82)

F~N.H.Robinson,

346

Electromagnetic stress and momentum in matter

This can be used to express the force acting on the matter within any region of the fluid in terms of a surface integral. If the matter within the surface merely consists of the same homogeneous fluid, the result will be zero, but, if the surface encloses a foreign body, there will be a net force corresponding to discontinuities in and p at the surface of the body, and to any charge or current associated with the body. The reader who finds these remarks either totally obscure or totally transparent, is invited to show that the force, per unit length, acting on a round, enameled, copper wire, carrying a current i in a direction normal to a uniform impressed field B = pH, within a magnetically-permeable, fluid medium, isIB; despite the fact that the impressed field within the conductor is 2B/(l + p). He might also show that the result is associated with mechanical stress in the enamel.

In dealing with time-dependent fields we must distinguish between transient phenomena. in which there is no time for the pressure to reach equilibrium, and steady-state phenomena. in which a slight distortion of the fluid has brought the pressure into equilibrium with the electromagnetic stress.

Clearly, if a homogeneous fluid medium contains only steady beams of radiation, the time average (or time integral) effect of the electrostrictive and magnetostrictive forces will be cancelled by a pressure gradient. The time average force acting across a surface can therefore be calculated from T~malone. This result will also apply to beams of finite width w and duration r if w ~ sr, for then the pressure will have time to equilibrate, by the flow of fluid across the sides of the beam. We have already discussed the consequences of this, in connection with Gordon’s interpretation of Ashkin and Dziedzie’s experiment, but we now present the calculation in a slightly different form. Since their beam-width was about 1 pm and their pulse duration 50 nanosec, we have sr 50 w and we can assume that steady state conditions are applicable. We consider, as in fig. 2, a finite beam entering a fluid medium from vacuum at normal incidence. Since within the medium the effects of pressure just cancel the electrostrictive and magnetostrictive stresses, the total effective stress tensor everywhere is just T. We take the x 3 axis in the direction of gives propamwhich gation, to H. TheE only term in T~ rise to xthe x1 axis parallel to E and the .v2 axis2+parallel pH2). Because and H are continuous the force 3 directed forces is then T~ ---~(eE per unit area acting on a thin lamina, with one face at x 3 in vacuum and the other at x3 + ox3 in ‘~-

the medium, is

F3

T33(x3 + Ox3)

=

—

T33(x3) =

2 ~-~(e

—

—

~(p

—

I )H2.

I )E

Poynting’s vector N = F A H in the medium is related to the Poynting vector N

0 of the incident wave by N = (1

--

i7)N0, where

112 + ) p is the reflection coefficient. We also have E2 ~/~i/~3N and H2 mentum is brought up to unit area of the surface is (I + T7)N eh/2\2

h/2

=

~

0

transmitted momentum is (1 ~

~++(

-~

1)

+

77)N0

~/~+~(p

=

=

v7~N. The rate at which mo-

{(

I + 77)/( I

77)

}

N and so the

F3, which is

—

I)

V~)N=V~N=D A B/~.

(83)

F.N.H. Robinson, Electromagnetic stress and momentum in matter

347

As long as we are not interested in the actual distribution of momentum and stress in the medium we could ascribe to the wave a momentum density D A B traveling with the velocity 1 ~ Notice that in this calculation the electrostrictive and magnetostrictive terms cancelled exactly. The result is therefore valid whatever the nature of these terms. It is only when we want to describe the state of affairs in the medium in more detail that we require these terms. Jones and Richards [18] showed experimentally that, when light falls on a metallic reflector immersed in a liquid, the force exerted on the reflector is, for a given intensity, accurately proportional to the refractive index n = (pc)”2 of the liquid. Their result is most easily explained if we assume that the momentum density of the radiation is D A B = peE A H, and that this is brought to the mirror, and reflected back, with a velocity 1/n = I/(pe)”2, so that the force is 2(pe)”2E A H = 2n(E A H’), per unit area. Since their pulses of radiation had a duration exceeding 10 millisec we can be sure that this was effectively a steady-state situation, and therefore their result is consistent with eq. (78). Also, because it was a steady-state situation and the media were fluids, we can find the force by integrating T,7°over a surface just outside the mirror, in the fluid. If this surface is normal to the r 2 + pH2) = 2~/~IEA HI where, again. 3 axis, T~,which is ~(eE E A H is the energy fluxwe in require the incident beam. This also agrees with Jones and Richards’s result. As Gordon [141 remarks, Jones and Richards could have found the same result had they used very short pulses, so that the pressure would not have had time to equilibrate with the electrostrictive force (we can ignore the miniscule magnetostrictive term), for they used a metallic reflector and so E and Ttttr would have been zero at the mirror surface. This, as we have seen, also leads to eq. (78). We have now described two steady state experiments where the results are consistent with the notion that the momentum density of a wave is D A B, and we have indicated that the same result might have been obtained in a transient version of Jones and Richards’s experiment. It will be instructive to try to relate these results to our earlier assertion that the momentum density of a plane wave is not D A B but rather D A B + 4(pe)1~’2(aE2+ /3JP)k, where a and /3 are the electroand magnetostrictive coefficients and k is a unit vector parallel to D A B or E A H. The difference between the two expressions, which can be written as epE A H and ep(l + a/e + j3/p)E A H, is clearly connected with the experimental conditions which, in the steady state experiments, allow the pressure to reach equilibrium within a pulse of radiation. In the hypothetical transient experiment the difference is associated with the vanishing of E at the mirror surface, and it will also be instructive to see whether we could devise even a transient experiment that might lead to a different result. We begin by considering a pulse, of transverse width w, propagating through an extended, homogeneous, fluid medium and, for simplicity, we assume that w is sufficiently larger than the wavelength for us to treat the wave as approximately plane. Figure 4 shows the pulse moving to the left, and the plain arrows indicate pressure gradients, while the wavy arrows indicate electrostrictive forces. At the leading edge of the pulse there is a region, of length roughly equal to wc’/s (where c’ and s are the velocities of light and sound in the medium), in which the pressure has not yet equalized. There is therefore a net, inward, electrostrictive force and this, as shown in the upper graph (i), imparts an additional negative momentum to this part of the medium. However, as the central part of the pulse, in which the pressure is equalized, arrives, the pressure gradient destroys this momentum. No further momentum changes occur until the tail of the pulse arrives. Beyond this tail there is a wake of increased pressure extending backwards for a distance wc’/s. At the tail of the pulse itself there is an unbalanced inward electrostrictive force, so that

\

F.N.H. Robinson, Electromagnetic stress and momentum in matter

348

4

MOVING PULSE

~t ~vt*

~t

p~O *~— +p

+~

<

~t

~t

+p

~

-i-p

WAKE

(~t~

+p

—,

~ 4~”

ELECTROSTRICTIVE FORCES

4—

PRESSURE GRADIENT FORCES

4 MOMENTUM

I®

(ij N~J MOMENTUM

MOMENTUM

o

a

a

U® ®

Q~~J

Fig. 4. Electrostrictive and pressure gradient forces acting on a pulse and the resulting momentum distribution for (i) a long, thin pulse, (ii) a short, wide pulse, (iii) an intermediate case.

the material in the wake acquires a forward momentum, as shown in graph (i). This momentum is finally destroyed by the pressure gradient at the trailing edge of the wake. The resulting momentum distribution may be compared with that in the middle graph (ii), which is what would happen if the pressure should not have time to develop during the passage of the entire pulse. The bottom graph (iii) shows the momentum distribution for an intermediate case. Here the length of the wake is approximately equal to the length of the pulse. We see that, in addition to whatever other momentum it possesses, the pulse contains negative and positive contributions to its momentum arising from the combined effects of electrostrictive and pressure forces. We also see that there is no net exchange of momentum between the pulse and the medium. We may therefore conclude that both the total momentum, and its distribution within the pulse, are determined at the point where the pulse is generated or enters the medium. It will be convenient to discuss this in terms of a pulse entering the medium at normal incidence. from vacuum. From the instant the beam enters the medium, up to a time w/s, when a length wc’/s of the beam is within the medium, the pressure in the medium will be virtually unchanged, and the momentum density in the beam will be D A B + aE2/2c’ (for brevity we ignore magnetostriction and the finer points of vector notation, but we note that a is usually negative). This establishes conditions in the leading edge of the pulse. At the same time there will be a net outward pressure on the surface, due to the discontinuities in and a and, because F is continuous, this is 4 (e — 1 )E2 + aE2. Since usually a — (e — 1) this force will be small. After a time w/s the pressure under the surface will have built up to —~aE2and the net outward pressure on the surface, which might, for example, be restrained by surface tension, will become 4(e l)E2. From now on, until the tail of the pulse arrives, the rate at which momentum is transported away from

4

‘~

—

F.N.H. Robinson,

Electromagnetic stress and momentum in matter

349

unit area of the surface is reduced by 4aE2, corresponding toa reduction aE2/2c’ in the momentum density (actually an increase because a is negative). This establishes conditions in the body of the pulse and gives a momentum density D A B. During this time surface tension forces exert an outward traction on the liquid outside the boundary of the beam. Finally the tail of the pulse arrives and the electromagnetic forces collapse, but a wake, of length approximately wc’/s, at an excess pressure _~4aE2 trails after the pulse. In this wake the medium has a net forward momentum, which is generated by surface tension forces, as the outward bulge at the surface collapses, over a time of the order of w/s. We have thus arrived at a description of how the momentum in the pulse originates and why, if the pulse duration is long and its width small, the average momentum density is very nearly D A B, for, under these conditions, the negative and positive momenta at the two ends of the pulse are both a small part of the total momentum, and also very nearly exactly compensate each other. There are two further points to be made in connection with this discussion. The first is that, although in our earlier discussion of plane waves we were at pains to distinguish between forces which propagate with the pulse and forces which are generated at surfaces and propagate away from the surface with the velocity of sound, here we appear to have effects which originate at a surface and yet propagate with the pulse. The explanation is as follows. Normally, if a mechanical impulse is generated at a surface it propagates away from the surface as a pressure wave associated with a mechanical compression involving the movement of matter. In the present case the excess pressure in the beam is generated by electrostrictive forces acting across the periphery of the beam. To maintain the required pressure in the beam these forces have only to propagate, and move matter, a fixed distance w (the beam width), not a distance c’t, which is increasing with time, as the pulse travels away from the surface with a velocity c’. The second point concerns the distribution of momentum in the fluid when a pulse enters the fluid, and is subsequently reflected or absorbed by a body immersed in the fluid. When a pulse enters a fluid at normal incidence, the total momentum within the fluid can be calculated from the change in momentum between the incident and reflected waves. In terms of the pulse duration r, the pulse cross-section A and the fields in the medium, it is (e + p)(E A I-1)Ar/2(cj4”2. If the pulse is wide and its duration short, the impulse imparted to the surface of the fluid is —~-{(e — 1 + a)E2 + (p — 1 + /3)H2 }Ar, which can also be expressed as —(2p — p — + ap + j3e)(E A H)Ar/2(ep)”2. The difference between these two expressions (pe)”2(l + a/2c + /3/2p)(E A H)Ar, is the momentum propagating with the pulse in the liquid. If, however, the pulse width is small and its duration long, the impulse imparted to the surface within the cross section of the beam is —4 {( — 1)E2 + (p — 1 )H2 }Ar, and the momentum propagating with the pulse is (pe)”2E A HAr = fD A B dv. In both cases the surface momentum will eventually be dispersed throughout the fluid as a whole, and we see that the distribution of momentum within the fluid is different in the two cases. The total momentum is the same, but for a short, wide pulse the momentum propagating within the pulse is (1 + a/2 + /3/2p)(pe)”2E A HAr = (1 + a/2e + /3/2p)fD A B dV while for a long thin pulse it is simply fD A B dv. We now consider what happens when the pulse is absorbed or reflected by a solid body immersed in the fluid. In the case of a long, thin pulse nothing unexpected happens. The impulse given to the body is, in this quasi-static case, always equal to the change in the pulse momentum fD A B dv. In the case of a short, wide pulse the situation is quite different. We have already seen that, when the magnetostrictive coefficient /3 is negligible and the pulse is a reflected at a

350

I”.N.H.

Robinson, Electromagnetic stress and momentum in matter

conducting surface (where E vanishes), the impulse given to the reflector is also simply the change in JD A B dv. This leaves the change in the part, (a/2e)fD A B dv, of the pulse momentum to

be accounted for by a net impulse given to the fluid near the reflector. To investigate this we consider a pulse of duration r, in which the electric field is E = E0sin w(t--x/c’), approaching a plane obstacle at x0. We suppose that at the surface of the obstacle the amplitude of the reflected wave, Er, is related to the incident amplitude E~by a real coefficient, so that Er = ~E1(for a conductor = -I) and that the pulse length r satisfies wr = 2Nir. At a point x for which x0 - x> 4-c’r, there will only be progressive waves present but if x0 x < }c”~there will, for part of the time, be a standing wave.on The contribution theregion. time integral the the force per unit volume,to the 2/ax, acting theonly fluid, comes fromtothis We findofthat impulse imparted 4aaEwithin a distance 40c’r of the surface of the body, is —~a~E~Ar fluid, { I (I -0 )cos 0 wr Since wr is necessarily large we see that for small values of 0, i.e., near the surface this oscillates between zero and —a~E~Ar, and that the total impulse imparted to the fluid out to the edge of the standing wave region is —-~a~E~Ar. If we remember that E 0 is 1’I)Ar the peak field, not the r.m.s. 2(EA = --(~a/)j’jncidentD A B dv. field, the incident wavetowe can express this asthis —a~(pft)” If ~ = in —1, corresponding a metallic reflector, is (a/)J’incidentD A B dV which is equal to —

--

the difference between (a/2)JD

A

B dJ/ in the incident and reflected waves. If, on the other

hand, the body is a perfect absorber so that ~ = 0 there is no net impulse given to the fluid. The impulse given to the body is now no longer the change in ,fD A B dV but, instead, the change in (I + a/2e)fD A B dv, i.e., the value of this expression for the incident wave. We can confirm this directly by using eq. (77) and retaining the non-vanishing surface integral which, in this case, is J’dtf(—~aE2)dS.An even more interesting case occurs if ~ +1, which we could achieve by coating the surface of a metallic reflector with a solid dielectric layer. one quarter-wavelength thick. The net impulse given to the fluid is now —(a/)fincjdentD A B dV and the impulse given to the reflector becomes (1 + a/e)j’{(D A B)~’~ — (D A B)~2~}dV where (I) and (2) refer to the incident and reflected waves. Since we expect a to be approximately --(c 1) we see that this impulse is approximately (I/)f{(D A B)~0— (D A B)(2)}dv and, since we have tactily ignored the difference between p and unity, this is equal to the change in f(E A H)dV, the field momentum. We thus see that, for short, wide pulses, the fluid is left with a net momentum in the vicinity -—

of a solid body, scattering or reflecting radiation. Since a is likely to be negative the net momentum for a metallic reflector is away from the body in the direction of the reflected pulse, zero

for a perfect absorber, and towards the body for a high-impedance, coated reflector. Whatever the magnitude or direction of this momentum it will (like the surface momentum set up where a pulse enters the medium) eventually be dispersed throughout the entire volume of the fluid. It does not propagate away from the scatterer, with the pulse. Perhaps more significantly we also

see that the momentum transfer when a short, wide pulse is scattered depends not only on the initial and final states of the pulse, but also on the nature of the scatterer. The nature of the scatterer determines the fraction of the change in the pulse momentum which is transferred to the scatterer and the fraction which is dispersed throughout the fluid. Only for a long, thin pulse, when the pressure within the pulse has reached its equilibrium value, can we say that, in all circumstances, the momentum transferred to a scatterer is equal to the change in fD A B dv. The behavior of a pulse depends on the ratio of its length 1 = cr to its width w. If this aspect-ratio. l/w, exceeds the ratio c’/s of the velocities of light and sound in the medium, the effective momentum of the pulse is always given by I’D A B, but, if the aspect ratio is less than c’/s, there is

F.N.H. Robinson, Electromagnetic stress and momentum in matter

351

no unique effective momentum and, indeed, if a’ < ( — I), the effective momentum may in some circumstances be less than fE A H d17. It has been remarked by Blount [81 that the connection between the momentum density D A B and the momentum density of the wave E A H is analogous to the connection between the pseudomomentum Ilk of an electron, or a photon, in a crystal, and its true momentum. This analogy has been further discussed by Peierls [221 who points out that although Ilk may not be the true momentum it nevertheless accounts for the momentum transferred in absorption or scattering processes. The balance of true momentum is maintained by a reaction on the medium which supports the wave and whose own motion is generally ignored. In view of the dependence of the momentum transfer on the aspect-ratio of the pulse and the nature of the scattering process, it appears that the indiscriminate use of this analogy is restricted to pulses whose aspect-ratio exceeds cl/s.

We have now come close to a discussion of the Abraham — Minkowski controversy and we see that Abraham’s contention that the true electromagnetic momentum density is E A H (or E A H/c2), is correct. When a wave propagates through matter a part E A H of the propagating momentum density is not associated with the motion of matter, although this part of the momentum plays a role in determining the force exerted on obstacles in the path of the wave. Minkowski’s momentum density D A B in certain circumstances (amongst which is the quasi-static situation associated with pulses of large aspect-ratio) will often determine the total momentum transferred to an obstacle and so, in a sense, is the analog of pseudomomentum. It is not however, in the case of an indefinitely extended plane wave, the total momentum propagating with the wave. This propagating material momentum is not simply D A B — F A H but contains in addition components of electrostrictive, and possibly magnetostrictive origin. In a simple fluid, whose susceptibilities are proportional to its density these additional components reduce the material momentum density from (pe — 1) E A H to 4(p + — 2)E A H. If p = 1 the total propagating momentum is not cE A H, but only 4(e + 1)E A H. We have not paid any attention to the construction of a symmetric, relativistic energy-momentum tensor, but we may remark that in constructing such a tensor it is necessary to include, in the energy flux tensor, terms describing the transport of rest-mass energy. Even if the non-relativistic ‘kinetic energy of the medium, 4pu2, is negligible, the rest-mass energy pc2, and its rate of transport pc2v, cannot be ignored. These considerations have been discussed in some detail by Haus [161 but the reader who consults this paper should be aware that Haus regards a medium whose susceptibilities are proportional to its density, as free of electrostrictive and magnetostriction, whereas we would regard this as a simple, but typical, electrostrictive, or magnetostrictive, medium. In our treatment a non-electrostrictive medium is a medium whose susceptibility does not depend on its state of strain. This is an important point for, as we have seen, the electrostrictive and magnetostrictive terms in the stress not only have significant magnitudes, but also play a crucial role in reconciling macroscopic theory with results based on microscopic considerations. It must be admitted that our preference for Abraham’s rather than Minkowski’s, contention is based almost entirely on theoretical grounds including Penfield and Haus’s virtual power argument. While there is (see e.g., Hakim and Higham [15]) reasonable experimental evidence for the static results, there is no direct evidence for Abraham’s formula, although James [171 has presented results which are not compatible with the Minkowski formula. On the other hand, because Gordon’s rigorous treatment of a simple system, which since it is realized in nature by a gas is more than just a model, also leads to Abraham’s result, there can be little doubt that this, rather than

352

F.N.H. Robinson, Electromagnetic stress and momentum in matter

Minkowski’s, result is correct. In view of Gordon’s analysis of the problems involved, an experiment to decide between these two hypotheses appears likely to be difficult to devise. The whole subject may therefore be regarded as little more than an academic curiosity. There is, however.

one reason why it is important, and this is because the notions of radiation pressure, electromagnetic momentum and electromagnetic stress play a fundamental role in other branches of physical

theory such as, for example, the theory of black-body radiation. When we make use of these concepts it is important that they should be based on a theory that is not only self-consistent, but

also consistent with the laws of mechanics and electromagnetism. The great virtue of Penfield and Haus’s results is that they are firmly based on the accepted laws of macroscopic electromagnetism, special relativity and mechanics. We may also remark that, because they involve a symmetric field stress tensor and identify the electromagnetic momentum density with the energy flux vector, they fit much more naturally into the general scheme of relativistic electrodynamics. We see then that it is possible to give a completely macroscopic derivation of the form of the stress tensor, and the magnitude and nature of electromagnetic momentum. Short cuts to the answers are not only misleading but also unneccessary. In the final result only macroscopic variables appear, although in addition to those aspects of the constitutive relation described by c~ and p~,we also need the electrostrictive and magnetostrictive coefficients. These coefficients appear in terms which are by no means negligible in their final result, even at the high frequencies associated with optical phenomena. They describe in macroscopic terms, microscopic aspects of the structure of matter which do not appear in Eq and p11 alone, and as we have seen in connection with Shockley’s cube-stuff, they need not be isotropic even if and p themselves are isotropic. In view of the complicated form of the final results it appears unlikely that any direct general derivation can be based on microscopic models and, in section 3, we have indicated why

this should be so. Finally we remark that the electrostrictive coefficients are closely related to the photo-elastic coefficients. A useful discussion of the experimental values of these coefficients and their relation to the atomic structure of a medium has been given by Pinnow [241. We note that although in gases and liquids the dominant influence on the change in the electric susceptibility with strain is due to the changing number of atoms per unit volume together with the resulting change in the Lorentz local-field, in solids, and especially covalent solids, an equally important influence, usually of the opposite sign, is the change in the atomic polarizability as the space available to each atom is altered. This affords a particularly striking example of the way in which the forces between individual charged particles in matter, though primarily electrostatic, are not adequately described by macroscopic fields. Nevertheless their macroscopic consequences are completely described when we add to the two usual macroscopic constants and p the macroscopic electrostrictive and magnetostrictive coefficients. The reader still interested in these problems will find further references below. They have been roughly classified in terms of their content.

Acknowledgment Throughout the writing of this paper I have had the benefit of many discussions with J.P. Gordon, and it is with considerable pleasure that I acknowledge both his encouragement and his criticism.

F.N.H. Robinson, Electromagnetic stress and momentum in matter

References [1] M. Abraham, Rc. Circ. Mat. Palermo 28 (1909) 1. [2] M. Abraham, Rc. Circ. Mat. Palermo 30(1910) 33. [31M. Abraham and R. Becker, Classical Electricity and Magnetism (Blackie, 1937). [4] J.A. Arnaud, Am. J. Phys. 42(1974) 71. [5] A. Ashkin and J.M. Dziedzic, Phys. Rev. Lett. 24 (1970) 156. [6] A. Ashkin and J.M. Dziedzic, Phys. Rev. Lett. 30(1973)139. [71N.L. Balazs, Phys. Rev. 91(1953) 408. [8] E.l. Blount, 1972, unpublished. [91L.J. Chu, see Fano et al. [131. [101 S. Coleman and J.H. Van Vleck, Phys. Rev. 171 (1968) 1370. [11] 0. Costa de Beauregard, Phys. Lett. 24A (1967) 177. [121 J.M. Crowther and D. ter Haar, Proc. Kon. Ned. Acad. Wet. 51(1971) 793. [131R.M. Fano, L.J. Chu and R.B. Adler, Electromagnetic Fields, Energy and Forces (Wiley, 1960). [141J.P. Gordon, Phys. Rev. A8 (1973) 14. [151S.S. Hakim and J.B. Higham, Proc. Phys. Soc. 80(1962)190. [161 H.A. Haus,Physica 43 (1969) 77. [17] R.P. James, Proc. Nat. Acad. Sci. 61(1968)1149. [18) R.V. Jones and J.C.S. Richards, Proc. Roy. Soc. A 221 (1954) 480. [191F.J. Lowes, Proc. Roy. Soc. A 337 (1974) 555. [20] H. Minkowski, Nacht. Ges. Wiss. Gottingen (1908) p. 53. [21] H. Minkowski, Math. Annalen 68(1910)472. [22] R. Peierls, 1974, to be published. [23] P. Penfield and H.A. Haus, Electrodynamics of Moving Media (M.I.T. Press, 1967). [24] D.A. Pinnow, I.E.E.E. J. Quant Electronics QE 6 (1970) 223. [25] F.N.H. Robinson, Macroscopic Electromagnetism (Pergamon, 1973). [26] W.G.V. Rosser, Classical Electromagnetism via Relativity (Plenum, 1968). [27] W. Shockley, Proc. Nat. Acad. Sci 60 (1968) 807. [28] W. Shockley and R.P. James, Phys. Rev. Lett. 18 (1967) 876. [29] T.T. Taylor, Phys. Rev. 137B (1965) 467. [30] R.W. Whitworth and H.V. Stopes-Roe, Nature 234 (1971) 31.

Additional references The origins of the controversy Helmholtz, H. von, Wied. Ann. 13(1882)798. Laue, M. von, Z. Physik 128 (1950) 387. Livens, H.G., Phil. Mag. Ser. 6, 32 (1916) 162. Pauli, W., Theory of Relativity (Pergamon, 1958).

General de Groot, S.R. and L.G. Suttorp, Foundations of Electrodynamics (North-Holland, 1972). Landau, L.D. and EM. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960). Page, C.H., Am. J. Phys. 42 (1974) 490. Podolsky, B. and K.S. Kunz, Fundamentals of Electrodynamics (Decker, 1969). Stratton, J.A., Electromagnetic Theory (McGraw-Hill, 1940).

Macroscopic and microscopic theory de Groot, S.R., The Maxwell Equations (North-Holland, 1969). de Groot, SR. and J. Vlieger, Physica 31(1965) 254. Irving, J.H. and J.G. Kirkwood, J. Chem. Phys. 18(1950) 817. Ma, S-k., Rev. Mod. Phys. 45 (1973) 589. Mazur, P. and B.R.A. Nijboer, Physica (1953) 971. Robinson, F.N.H., Physica 54 (1971) 329.

353

354

F.N.H. Robinson, Electromagnetic stress and momentum in matter

Macroscopic stress Lax, M. and D.F. Nelson, Phys. Rev. B4 (1971) 3694. Maradudin, A.A. and E. Burstein, Phys. Rev. 164 (1967) 1081. Mazur, P. and S.R. de Groot, Physica 22 (1956) 657. Tiersten, H.F., Intl. J. Engng. Sci. 9 (1971) 587. Wong, H.C and J. Grindlay, Adv. in Phys. 23, No. 2 (1974) 261.

Wave momentum Arnaud, J.A., Electronics Lett. 8 (1972) 541. Arnaud, J.A., Optics Comm. 7 (1973) 313. Arnaud, J.A., U.R.S.I. Symp. on Electromagnetic Theory, London, 1974. Burt, M.G. and R.E. Peierls, Proc. Roy. Soc. A 333 (1973) 149. Costa de Beauregard, 0., C.R. Acad. Sd. Paris 274 (1972) 164. Greenberg, J.M. and J.L. Greenberg, Am. J. Phys. 36 (1968) 274. Gyorgi, G., Am. J. Phys. 28 (1960) 85. Joyce, W.B., Phys. Rev. D9 (1974) 3234. Ratcliff, K.F. and D. Peak, Am. J. Phys. 40 (1972) 1044. Thornber, K.K, Phys. Lett. 43A (1973) 501.