Bowdery, G. J. - The Concept of 'Field' in Electrical Theory (1946)

7/28/2019 Bowdery, G. J. - The Concept of 'Field' in Electrical Theory (1946)

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The Concept of 'Field' in Electrical TheoryAuthor(s): George J. Bowdery

Source: Philosophy of Science, Vol. 13, No. 4 (Oct., 1946), pp. 307-324Published by: The University of Chicago Press on behalf of the Philosophy of Science AssociationStable URL: http://www.jstor.org/stable/185211 .

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THE CONCEPT OF 'FIELD' IN ELECTRICAL THEORYGEORGE J. BOWDERY*In this paper we shall consider the circumstances under which the concept of'field' was introduced into electrical theory, the traditional use of the notion offield with particular reference to electrical theory, and sketch three charactersof a field in this context. These are its pervasiveness, its independent existence,and its status as an elastic body. In each case we will briefly bring to bear moremodern comment on these three facets of the traditional conception, attemptingto salvage the meaning for the term field that is currently accepted. Following

this, 'field intensity' will be compared with other terms such as 'displacementcurrent' and the fictional character of terms and the conventional character ofthe equations in which 'field intensitv' appears will be discussed. The papercloses with a summary of the points made.

IA

The idea of 'field' as employed by Maxwell and his contemporaries had itsroots in hydrodynamics and the mathematical theory of elasticity. Thesesubjects employed the notion of a space-filling fluid with certain well-definedproperties which included its incompressibility, i.e., in a region which is com-pletely filled, just as much fluid on the whole must enter any closed surfaceas leaves it, and its irrotationality, i.e., a fluid such that it is impossible to obtainan unlimited amount of work by guiding a particle of the fluid indefinitely arounda closed path.The hydrodynamical analogy was first suggested by Thomson' and was em-ployed to the fullest extent by Maxwell who used it in his earliest papers, e.g.,"On Faraday's Lines of Force"2merely to guide him in the construction of themathematical outlines of a theory and not to formulate the properties of anyactual fluids.It is insufficient for the purposes of electrical theory to consider fluids withonly the above properties. Some provision must be made for the continualgeneration of fluid at certain points (sources) and its continual destruction atothers. This need arises from the existence of positive and negative electrodes.By assuming a suitable system of sources any known electrostatic phenomenacould be formulated on analogy with a steady motion of an incompressible fluid.The transition from the hydrostatic analogy to electrical phenomena wasmade by interpreting electric charges, not merely as centers of force on the modelof Newton's mass particles but as sources of flux of force, where the latter means

* An obituary follows this article, which is being published posthumously.1 Thomson, W. 18422 Maxwell, J. C. Scientific Papers. Vol. I

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308 GEORGE T. BOWDERYthe product of the normal3 component of a force across a surface element by thearea of surface element through which the force acts. By so doing Gauss'stheorem which relates the total flux of force through a closed surface to the totalcharge within that surface became applicable. In hydrodynamics the 'totalcharge' is replaced by the term 'mass of fluid' within the surface.With Gauss's theorem a whole series of theorems became available includingthe theorems connecting volume and surface integrals (Green's theorem) andline and surface integrals (Stokes' theorem). All these theorems were in factemployed in hydrodynamics.Much earlier than the use of the hydrodynamical analogy was the applicationto electrical problems by Poisson4 and George Green5of the potential functionsatisfying LaPlaces equation. This function, defined everywhere, is closelyrelated to the force function, and depends upon the position of the chargedbodies. This notion was also employed by Maxwell in his construction ofelectrical theory. He found the notion of potential, which corresponded tothat of 'pressure' in hydrodynamics, extremely useful in rendering Faraday'sideas mathematically.This discussion suggests that two circumstances which led Maxwell and hiscontemporaries to utilize the idea of field were 1) relatively full mathematicaldevelopment of hydrostatics, and 2) the analogy of hydrostatics and Fara-day's electrical ideas which seemed so fruitful in their continual suggestion ofnew directions for investigation.

A third source which stimulated the development of a field theory was thetheory of wave propagation which suggested that some sort of continuous mediumwas necessary for the passage of light through space. The connections betweenlight and electricity suggested by Faraday made some such medium desirable.Finally, as early as 1820, Oersted appears to have used the notion of fieldas an explanatory tool for electromagnetic phenomena. He invoked an 'elec-tric conflict' acting in circles around a current-bearing wire to account for thedeflection of a magnetic needle in the neighborhood. In this context 'field'meant a spatial region possessing a physical quality that varies in intensity anddirection throughout this region. The fact that no deflection appeared ifcurrent did not flow through the wire was taken as evidence that the spacesurrounding the wire had acquired as a whole different properties when the cur-rent was on.We have thus far indicated the background for the use of the notion of field.We turn now to three general characters of the traditional usage of the termas well as a more detailed consideration of the work it did for Maxwell and classi-cal theorists of electricity.

BIn electrostatics a kind of qualitative alteration in the space near a charged

body was imagined by Faraday. The quality was conceived to be distributed3That is, perpendicular to the element of surface.4 Poisson. 1812r Green, George. "Essay on the Application of Mathematical Analysis to TheoriesElectricity and Magnetism", 1828



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THE CONCEPT OF 'FIELD' IN ELECTRICAL THEORY 309throughout this space. Evidence for this distribution was provided by the factthat an indefinitely small test body would detect this quality everywhere in thespace near the charged body. The pervasiveness of this quality was expressedin mathematical terms by the use of mathematical functions defined for everyreal value of space coordinates.A second feature of this usage of the term 'field' lies in the tendency to treat afield as existing independently of any testing body; in particular, persistingwith or without the presence of matter or electric charges. For Maxwell, "theElectric Field is the portion of space in the neighborhood of electrified bodies,considered with reference to electric phenomena. It may be occupied by air orother bodies, or it may be a so-called vacuum, from which we have withdrawnevery substance which we can act upon with the means at our disposal."6

Thirdly, the electrostatic field, which is neither a property of matter nor pro-vides the metric of the physical geometry of the space in question, is held to be amedium called the ether. It and not matter became the seat of electrical action.In order to explicate these three characters of fields, namely, their pervasive-ness, their alleged independence of test bodies, and their existence as media wewill discuss briefly the traditional use of the term 'field' in electrostatics."If a stick of sealing wax is rubbed with a piece of catskin, these bodies andthe space round about them are thrown into a peculiar condition, as is revealed

by the fact that light particles in the neighborhood are set in motion. We saythat the rubbed bodies are 'electrified', and that the space surrounding them isan 'electric field'."7"Let an electrified piece of metal be at rest in air. The electric field in itsneighbourhood is investigated with the help of a proof body, which may be, forexample, a small ball of elder pith covered with gold leaf, and electrified bycontact with the rubbed sealing wax or the catskin.8 . . . Suppose this force Fto be measured; both its magnitude and direction will be different at differentpoints of the field; even at a fixed point in the field they will vary according tothe way in which the pith ball is electrified. With regard to the latter type ofvariation, however, a very simple law governs the result; if the proof body wasmade to touch the sealing wax, then the direction and sense of the force IF,which acts on it at a given point of the field, are perfectly definite, and onlyits magnitude depends on the details of the process; but if it was the catskinwhich was touched, then the direction of the force is reversed, its magnitudedepending, as before, on the nature of the preliminary process. This suggeststhat we should put, for the force which acts on the proof body in the electric field,

F=e- E (1)where the scalar e depends on the electrical state of the proof body, while thevector F is independent of that state, but varies in magnitude and in direction

6 Maxwell. Treatise. 1:477 Abraham and Becker. The Classical Theoryof Electricity and Magnetism. p. 538 It is observed that the pith ball moves in different directions and different distanceswhen in the presence of the electrified body. Since there is motion it is assumedthat theremust be a force of some sort operating.



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310 GEORGE T. BOWDERYat the various points of the field. Experiment shows, in fact, that if two proofbodies which have been treated in different ways are brought in succession tothe same point in the field, the forces upon them are in a definite ratio,Fi: F2= el:e2 (la)and that this ratio remains the same when the point varies. Experiment showsalso that the magnitudes of the forces which act on one and the same proof bodyat different points P and P' of the electric field are in a ratio independent of theprevious treatment of the proof body, or

1FI: IF' I =El: E' (lb)The statements (la), (lb) are both included in (1). If el is given for the firstbody, e2 is defined for any other body by (la); E can then be found for individualpoints of the field by means of any proof body."The scalar factor e in the expression (1) is called the 'electric charge' of theproof body or the quantity of electricity upon it; the vectorial factor E is calledthe 'electric intensity'."9We note that the force acting on the test body is definedto be the product ofa factor which depends upon the electrical charge of the test body and one whichdoes not. The latter term is taken to have a referent which is independent ofthe test body.'0

"It is a characteristic feature of Maxwell's theory that it associates a magni-tude and intensity E with every point in space... ." "The physical significanceof E consists to begin with only in the relation expressed in equation (1), whichstates: if a small charge e were brought up to the definite point of space in ques-tion, then the force IF = e* E would act upon it there. Maxwell's theory thengoes on to ascribe to this vector E a self-existent reality independent of thepresence of a testing body. Although no observable force appears unless atleast two charged bodies are present (for instance the charged piece of metaland the proof body), nevertheless we assert with Maxwell that the charged pieceof metal by itself produces in the surrounding space a change of physical condi-tions which the field of the vector E is exactly fitted to describe. The primarycause of the action on a testing body is considered to be just this vector field

9 Ibid. p. 53-410An alternative procedure for the definition of E was suggested by Prof. Black. Con-sider these statements:At a place P in the space surrounding a body B, test body b1 moves in direction diAt a place P in the space surrounding a body B, test body b2 moves in direction d2

At a place P in the space surrounding a body B, test body bi moves in direction diSuppose that as the charges on the bodies (bi) decrease monotonically the directions

move mionotonically. And that if ei is the charge on bi and 0 the angle between di and acertain direction d, we can establish empirically a relation 0 f(ei) such that lim f(x) = 0.x oWe may then define d as the direction of the field at P. Obviously we cannot performthe passage to the limit empirically. The definition extrapolates beyond experimentalverification.



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THE CONCEPT OF 'FIELD' IN ELECTRICAL THEORY 311at theplace where thetestingbodyis situated. As for the piece of metal, its part inthe matter is merely to maintain this field.""

IIIn the traditional method of defining electrostatic intensity described aboveit is assumed that a unit charge can be placed at an empty point of space with-out disturbing the position of any of the existing charges, that is, without dis-turbing the electrification of the other bodies present. This is in principle im-

possible since, in general, the whole configuration of charges will be altered if aunit charge be introduced. As Maxwell puts it, "If an electrified body be placedat any part of the electric field it will, in general, produce a sensible disturbancein the electrification of the other bodies." He meets this objection by pointingout that "if the body is very small, and its charge also very small, the electrifica-tion of the other bodies will not be sensibly disturbed, and we may consider theposition of the body as determined by its centre of mass."'12 Clearly a bodyof finite size with finite charge, no matter how small, is not satisfactory. Infact, Maxwell is later led to remark that "in order to simplify the mathematicalprocess, it is convenient to consider the action of an electrified body, not onanother body of any form, but on an indefinitely small body, charged with anindefinitely small amount of electricity, and placed at any point of the space towhich the electrical action extends. By making the charge of this body in-definitely small we render insensible its disturbing action on the charge of thefirst body.""3 This amounts to defining the intensity as the limiting valueof the force per unit charge which would act on a charge were it located there,where the limit is taken as the magnitude of the charge approaches zero. As adevice to simplify calculation there is no objection to this definition at leastwithin the context of Maxwell's Treatise. But more modern conceptions of aminimum indivisible charge-the electron, are incompatible with such a limitingprocess as is involved in this definition of electrostatic intensity.The existence of a smallest unit of charge was established in part by Millikan'srepeated observations on a single oil drop carrying charges and moving in elec-trostatic and gravitational fields. However, the additional statement is alsomade that the unit charges are not only discrete electrically but also are discretespatially; more exactly that they are localizable by means of space (and/ortime) coordinates just as unit particles of matter are. This of course does notnecessarily mean that a single electron can be located precisely or even that itwould be significant in terms of the procedures used to assign a numerical valueof position an isolated electron, but that some very small, but macroscopic, bodycharged with one electron can be so located. Of course, it is said for the purposesof the construction of a theory that charges (which may be unit) are so localiz-able. In Mason and Weaver's exposition, for example, charges are conceivedas spatially distributed and an expression for the potential of a single charge"Abraham and Becker. op. cit. p. 55

12Maxwell. op. cit. 1:4813 Ibid. p. 75



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312 GEORGE T. BOWDERYrelative to a group of charges is obtained. Thus quantities like potential andelectric intensity are defined only at places where there are charges. The pro-cedure used is worthy of more detailed description since it is typical of the ap-proach to field of those who start with particles of some sort as fundamental.Consider a spatial configuration of charges made up of a charge e' and n othercharges el, e2, e3, ... , en. The total mutual electrostatic energy of this con-figuration may be written when we are interested in the force on the charge e'due to other charges el, e2, , en as a sum

i'e' + 0(other charges)e' [el e2 en 1 1 [ele2 ele3=- + 1+ + _+47rLre' re" reL 4r L rl2 r13

+ ...+ e2e3 + e2e4 + + +en-l Xenr23 r24 rn-I, n

where re,,is the distance from the charge e' to the charge el, etc. and r12is thedistance from the charge el to the charge e2 , etc. The first term in this expres-sion obviously depends on the presence and location of the charge e' sincethe positions of the charges e 1, e2 , - - , en are, in general, dependent uponthe location of the charge e'. The coefficient of e', namelyei . ...... +ren

is called the electrostatic potential at the position of the charge e' due to thecharges e1, - - *, en. The force acting on e' may be written as the product of thecharge e' and a certain function of the electrostatic potential (which we shall call4b). This function E is a vector function which gives the direction for the forceon e' as follows:F = e'E

where E is from its use in this expression the force per unit charge acting one' (not force on a unit charge conceived to be where e' is). E is called the elec-trostatic intensity due to the charges el , - - *, en . The electrostatic potential4 and the electrostatic intensity E, considered from a purely analytical pointof view as the scalar and vector point functions defined by the foregoing equa-tions, can obviously be calculated at any point of space. From this accountof the potential and intensity however, it is clear that they are defined as coeffi-cients of the particular charge e' selected for attention, and are not defined ir-respective of such a charge. It is true that it is possible to measure a distanceri from any point to a charge ei whether or not there is a charge at this point;if this is done, the quantity -[ el . ..... Jen can in principle becom-puted (though difficulties arising from the very large number of charges willoccur). Thus 4 as a scalar point function is continuous and is defined for everyvalue of r.,i . But to do this is to lose sight of the role that it plays in the ex-



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THE CONCEPT OF 'FIELD ' IN ELECTRICAL THEORY 313pression for the energy (1), where it is the coefficient of the magnitude of a chargee' placed at that point from which re,i is measured. Unless we are willing totake a term out of its context we are forced to conclude that this mode of formu-lating electrostatics does not imply thatthe electrostatic field is defined every-where. Thus if we assume the existence of indivisible unit charges and there-fore their spatial localizability and if we accept the above definitions theelectrostatic field loses its pervasive character.

IIIWe turn now to the second point, namely, the independent character of thefield in the traditional use of the term. "Maxwell's theory goes on to ascribe

to this vector E a self-existent reality independent of the presence of a testingbody."'4 In this remark lies the paradoxical character of the traditional con-ception of field. For a field is asserted to exist even though no testing body isused, although the field is only experimentally observable if in fact such a bodyis used. It was asserted that the term did in fact refer to something-but thisreferent could only be observed if a test particle (or charge) were used. Thus afield was supposed to exist without a test body and yet could not even be de-tected in principle without one.A way out of this difficulty may be suggested if we consider how E was de-fined in the Maxwell theory. We recall the equation F = eE defined 1F,and Ewas defined as a factor of the force which did not depend upon the electricalstate of the body. It was however, a quantity correlated with a certain ob-servable occurrence, as was shown by the fact that we invoked a unit charge.Now the fundamental equations of electrostatics assert that certain relationshold between the symbols E, p, pv, b. It can be shown that they give uniquesolutions (i.e., values of E.) if either p or 4' (not necessarily both) be given.16This is in fact the general mathematical problem of electrostatics. Becausethese simultaneous equations have E as a unique solution for given values of por 4)we can take them as defining a quantity E which has the same propertiesas the E previously defined.In the general electromagnetic field the same procedure may be followed. Amore general set of equations must be given-the so-called Maxwell equations ofthe electromagnetic field. In these equations E is interpreted as electric in-tensity and H as magnetic intensity. According to Lindsay and Margenau wemay "treat the equations themselves as defining E and H. This means thatthe only real importance of the quantities is involved in the fact that they satisfythe field equations. As a matter of fact, it has been pointed out (notably byRitz and Swann) that the only use which we make of E and H in atomic problemsis their calculation frormthe equations, assuming assigned values of p and pv."'In atomic problems the definitions in terms of unit charge and pole are whollyunnecessary. It seems most logical to go the whole way and treat E and HEEs

14 Abrahamand Becker. op. cit. p. 5515Here p is the charge density and pv the current density, v being the velocity of electriccharges considered as discrete particles.



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314 GEORGE T. BOWDERYdefined by the field equations in all cases. The commoner definitions can thenbe looked upon as mere picturizations. When we use E and IHI, ll we are reallyinterested in is a description in simplest terms of certain phenomena." "Undervarious conditions values can be assigned to these quantities, and from thesevalues other things such as the motion of circuits or charged particles can becomputed. If we stop to think of how we use the field vectors even in practicalproblems it is clear that the idea of them in connection with unit charge and unitpole never enters into our measurements. When we talk about the electricfield existing between the plates of a condenser we think of it in terms of thegradient of the potential rather than in terms of force per unit charge. Insimilar fashion the magnetic field which we calculate at the center of a circularcurrent carrying coil, for example, is used in the calculation of the force action,between this coil and other current-carrying conductors or magnetic materialor perhaps in the calculation of the deviation of a stream of charged particlesin the vicinity of the coil. The unit pole rarely if at all enters into our calcula-tions."16There is a fundamental ambiguity in our present discussion between thephysical space in which observable changes take place and the set of values ofquantities characterizing these changes. In the first use of the term 'field'some test body is clearly needed to detect experimentally the charges and weare again in the difficulty suggested at the beginning of this section. But if'field' means simply a function having certain properties a different set of prob-lems arises. Lindsay and Margenau suggest that such quantities as 1E and 13[are in many problems determined from assumed values of the charge density.17In obtaining particular solutions to problems no definition of E or so n termsof a unit charge is necessary. E, and soare quantities invented to obtain spe-cific results in specific problems. There is no need to consider them as possessingextra-linguistic referents. Only the charge density on the body must somehowbe known. In these problems charge density of bodies is taken as the characterhaving direct reference to physically measurable characters of electrified bodies.In the general electrostatic problem either the total charge (and therefore the

16 Lindsay and Margenau. Foundations of Physics. p. 311-31217 In electrostatics, for example, there is the problem of determining the capacity of a

charged metal sphere of radius a. Let us see how Lindsay and Margenau's suggestionoperates here. We infer from the symmetry that the distribution of charge is uniform sothat the surface density of charge is e/4wra2. Let equations ffEnJOS= 4ire and 4irco = E.Od=- , (, the potential) define a certain vector E whose normal component has the aboveproperties and which we assume is so specified as to be unique by adding sufficient condi-etions. Then Er =- and o = e/r + k. On the sphere it has the constant value p = e/a + k.r2Consider a concentric sphere radius b > a with a surface density co = -e/47rb2 andpotential at its surface pib= e/b + k.

A relationship independent of the size of the charge may be found by forming the quotient1 1 b -a ___ ab_of the charge to the difference SCa- = e(- - - ) = e we get C = aba b ab ' a- b b-awhich is called the capacity of the spherical condenser.



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THE CONCEPT OF 'FIELD' IN ELECTRICAL THEORY 315charge density if the volume or surface be known) or the potential of the surfaceof the conductors may be given. In either case the electrostatic field intensityE will be uniquely defined if certain general equations defining the relation of? to soand the charge density are given. Thus in the general electrostatic prob-lem if either soor the total charge is given the other will be determined by a cer-tain equation whence the field E will be determined. E is always a calculatedquantity that is not physically measurable, though the potential and chargedensity may take either role according to the particular problem.In this sense, a 'field' as a mathematical function which is defined by a set ofequations 1Ecan be called fictional for we have here a quantity defined by certainrelationships, which aids in the calculation of specific results in specific problemsand whose function is not to express anything about the subject matter of elec-trical theory.

IVIn the traditional use of 'field' the medium to which the term 'field' refers was

taken to be a direct mechanical causative agent operating directly to produceactions observable in the space which the medium fills. Maxwell is very ex-plicit on this point. "In the theory of electricity and magnetism adopted inthis treatise, two forms of energy are recognized, the electrostatic and the elec-trokinetic, and these are supposed to have their seat, not merely in the electri-fied or magnetized bodies, but in every part of the surrounding space, whereelectric or magnetic force is observed to act. Hence our theory agrees with theundulatory theory in assuming the existence of a medium which is capable ofbecoming a receptacle of two forms of energy.""According to the theory of undulation, there is a material medium which fillsthe space between the two bodies, and it is by the action of contiguous parts ofthis medium that the energy is passed on, from one portion to the next, till itreaches the illuminated body."The luminiferous medium is, therefore, during the passage of light throughit, a receptacle of energy. In the undulatory theory, as developed by Huygens,Fresnel, Young, Green, etc., this energy is supposed to be partly potential andpartly kinetic. The potential energy is supposed to be due to the distortion ofthe elementary portions of the medium. We must therefore regard the mediumas elastic. The kinetic energy is supposed to be due to the vibratory motion ofthe medium. We must therefore regardthe medium as having a finite density.''18The medium is thus treated as an elastic body of finite density like other pon-derable bodies. It thus may act as a source of both potential and kinetic energyand expressions for these may be calculated just as in the dynamics of Newtonianparticles and bodies reducible to particles."But," as Maxwell says, "the properties of bodies are capable of quantitativemeasurement. We therefore obtain the numerical value of some property ofthe medium, such as the velocity with which a disturbance is propagated throughit, which can be calculated from electromagnetic experiments, and also observed

18 I have transposed some paragraphsin this quotation. II:432



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316 GEORGE T. BOWDERYdirectly in the case of light. If it should be found that the velocity of propaga-tion of electromagnetic disturbances is the same as the velocity of light, and thisnot only in air, but in other transparent media, we shall have strong reasons forbelieving that light is an electromagnetic phenomenon, and the combination ofthe optical with the electrical evidence will produce a conviction of the realityof the medium similar to that which we obtain, in the case of other kinds ofmatter, from the combined evidence of the senses."'9Despite the equality of the velocity of electrical disturbances and that oflight, this conception of the ether is no longer accepted. It is not germane toour inquiry to describe in detail why this conception was abandoned. Thefailure of its detection in the ether drag experiments contributed to its downfall.A few remarks as to its theoretical dispensability are in order. In the Maxwell-Lorentz theory the ether was considered to be an absolute frame of referencerelative to which charges moved. The discovery, however, that the electro-magnetic field equations are invariant under the so-called Lorentz transforma-tions discredits the absoluteness of this frame of reference. "A medium of sucha nature that you cannot tell whether you are moving with respect to it, or atleast not by means of any electromagnetic phenomena (including of course light)is not a very useful medium in the material sense in which classical physicistsviewed media. The feeling therefore arises that one might as well dispense withthe ether concept altogether. But the inquiry may then be made: how is onegoing to talk about the propagation of electromagnetic waves if there is nomedium? The answer is that after all in such propagation the medium is theleast important thing. From hydrodynamics, elasticity, etc., we have growntoo much accustomed to thinking that a wave is a moving disturbance in a me-dium. Actually we may as well admit that the only relevant feature aboutwave motion is that there exist physical quantities which are functions of spaceand time in the form f(x i vt).20 Working with this idea and its developmentsone can cast aside the medium notion as irrelevant embroidery which is of valueonly in so far as pictures are valuable and which ceases to be valuable when thepictures break down."2'These remarks are subject to some modification. Maxwell's theory was adynamical theory. It utilized the traditional Lagrange equations. In thisconnection, the function of the field as medium was to provide a reference-systemfor the velocities which occur in the fundamental force formula (shown to beequivalent to Maxwell-Lorentz equations). That is, "the measurable forcebetween two charged particles is taken to involve 'something more than theirrelative velocity; it depends on their velocities with respect to some medium,framework, or background.' 7122However, the so-called density or elasticity of the medium are irrelevant to

19 Maxwell.20 Where 'x' is the distance measured in the direction of the wave, 'v' is velocity, and't' the time.21 Lindsay and Margenau22 O'Rahilly, Alfred. Electromagnetics.chapter on the aether. p. 630 circa



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THE CONCEPTOF 'FIELD' IN ELECTRICAL HEORY 317electromagnetics. A medium is relevant only as "something which enters phys-ically into the equations, without which the v and v' in our force-formula cannotbe defined or measured."A field, then, even if considered as an absolute frame of reference, is not thesubstantial medium with a finite density of Maxwell's day. At most it is aframe of reference with respect to which velocities are measured.

vA

The character of 'field intensity' has been discussed in older and modernusages. We can better appreciate its function in use by discussing the stationarycharacter of the so-called ether and comparing 'field intensity' with 'displace-ment current'.Let us consider just how the stationary character of the so-called ether didin fact aid Maxwell in formulating his theory. Its use may be stated thus:By using the notion of ether as a stationary body the form of the Maxwell-Lorentz equations are simplified. For if you referred the coordinates used todescribe electromagnetic phenomena to axes moving with a uniform velocity u,i.e., when x is replaced by x-ut, you would get equations of a different form.Pragmatically speaking, therefore, the use of a stationary ether prevented onefrom getting more complicated equations that would result from replacing x byx-ut and hence the notion saved the form of the Maxwell-Lorentz equations.After it was shown that the Maxwell equations were invariant in form underthe more complicated Lorentz transformation it was clear that it was in prin-ciple impossible to detect the ether experimentally, at any rate by looking formotion thru an ether as had been done in the Michelson-Morley experiments.Before this discovery, however, the problem of invariance of the equations wasavoided in practice by the use of the notion of ether.In another domain, that of Newtonian dynamics, it had long been suspectedthat there was in fact no fixed or privileged set of axes in the universe, denotedby the term 'absolute space'. In fact the ordinary Newtonian equation ofdynamics m dtx = F is invariant in form when referred to axes moving with auniform velocity u, i.e., when x is replaced by x-ut. Hence absolute spacewas not a physical existent, detectable by experimental means. The fact washowever, that the notion was used. It remained for Poincar6 to point out thatits use was to facilitate calculation.

BSimilarly the notion of displacement current in Maxwell's theory serves onlyto transform an equation of Poisson's type to the general equation of wavepropagation. No electric stresses in free space are used any longer.Let us see just how Maxwell used these two notions-held as stationary and



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318 GEORGE T. BOWDERYdisplacement current jointly, in deriving "the general equations of electro-magnetic disturbances."23To obtain the general equations of electromagnetic disturbances, Maxwellexpresses the sum of the term for conduction current C1E(which is experimentallyobservable) and another term4 defined as equal to KIE where K is the so-calleddielectric capacity of a substance and can be found experimentally as functionsof a vector potential A and the scalar potential 4. The sum of the two terms iskCE + - E. Now ]E is the sum of three terms: that due to the motion of a4wparticle through a magnetic field, that due to the time variation of the magneticfield and that due to the space variation of AV. Since there is no motion of themedium, the first term drops out, thus simplifying calculations, and so we canwrite

-(+C + 4 K d-t (dtA \6Maxwell then utilizes a relationship which connects currents with the vectorpotential. This relationship was obtained by an ad hoc extension of the equa-tions of electric currents derived for closed circuits to open circuits. "The ex-treme difficulty of reconciling the laws of electromagnetism with the existence ofelectric currents which are not closed is one reason among many why we mustadmit the existence of transient currents due to the variation of displacement.Their importance will be seen when we come to the electromagnetic theory oflight."24

In fact it appears that use of the expression 1 k dE is to prevent the con-ditions as Maxwell formulates them from disappearing altogether as they wouldin cases where C = 0, i.e., for non-conducting media.Maxwell has his eye fixed on the propagation of light thru a non-conductoras well as upon light as propagated like other electrical disturbances throughspace. By utilizing a formulation which includes both expressions for the cur-rent of closed currents and for the dependence of propagation phenomena ontime he connects the optical and electrical properties of bodies. Propagationphenomena are simply studied by setting C = 0. He remarks "the conditionof things at any point 0 at any instant depends on the condition of things at adistance Vt and at an interval of time t previously, so that any disturbance ispropagated through the medium with the velocity V."25The use of the stationary medium constantly simplifies calculation as in thecase of the plane wave. The general equations of electric intensity are consider-ably simplified by its use. The stationary medium like displacement currentis a sufficient not necessary condition for the solution of electromagnetic prob-lems.Both these terms are of a different type than that of 'field'. They were both

23 Maxwell. Treatise. II:43324 Ibid. II:25225 Ibid. I1:435



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THE CONCEPT OF 'FIELD' IN ELECTRICAL THEORY 319concerned with the changes in the algebraic form of equations which formulateelectromagnetic properties. The use of the stationary ether was tantamountto the exclusion of moving axes and hence of any change in form of the Maxwellequations. The use of the Lorentz transformation showed that the form of theMaxwell equations could be preserved without the use of an ether. Similarly'displacement current' was a term added to an equation to allow its extensionto "open" circuits. By its addition a new vector was formed which had thesame property (of irrotationality) as the old. That is, the equation as extendedhad the same form. These are examples of that type of term in a theory whoseonly function is either to preserve the algebraic form of the equations charac-teristic of the theory or to change the form so that particular problems may bemore easily solved (as in Newtonian dynamics where second order rather thanthird order equations are obtained by the use of 'absolute space', or in the use of'stationary ether' in the wave equations). Briefly, the subject matter of suchterms is the algebraic form of the equations characteristic of the theory. (Suchterms include those usually referred to as having a heuristic functions in aidingcalculation.)Clearly E differs radically from such terms when a point charge type definitionis used. For 'field intensity' may be defined at and is in principle, at least,measurable at, every point where there is a unit charge or particle, or aggregateof these, though, of course, only at these points.But what if 'field intensity' is defined by a set of equations as we have sug-gested earlier?26 What then is the function of 'field intensity'? We are led toinquire just how 'field intensity' is defined by equations and how these are re-lated to empirical laws.

VIThe so-called Maxwell field equations for empty space consist of five equa-tions27the first of which is a formulation of Ampere's law as supplemented byMaxwell's displacement current. The second is a formulatln of Faraday'slaw of induction, the third relates the electric intensity to charge density, thefourth the magnetic intensity due to all types of causes, and finally an equation

which gives the force per unit charge in terms of the electric and magnetic in-tensities, which are to be determined (if necessary boundary conditions areadded) through the field equations from the supposed known values of the chargedensity and the convection current.The first two equations are obtained from experimental laws, by assumingthat the laws hold for small volume elements of the bodies concerned and thatthey have meaning at any point of such bodies. For example, Ohm's law whichin its experimental form asserts that the difference in potential (which may bemeasured) between two points of a wire resistance, is proportional to the currentflowing, becomes the statement that the volume density of current defined at apoint of the conducting material is proportional to a certain function of the po-

26 See p. 1127 This discussion follows Mason and Weaver. TheElectromagneticField. Ch. III, part1, and Ch. IV.



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320 GEORGE T. BOWDERYtential, the electric field intensity. We need not pause on Maxwell's contribu-tions to the first equation; this was discussed above.The third equation is a generalization of Coulomb's inverse square law for theforce between two charges. The fifth equation which is the crux of the mattergives the force per unit charge. To calculate the force in any specific case wemust, as we have indicated, know the charge density and also utilize the gen-eralized form of Ohm's law which we have mentioned above.The chief characteristic of the equations we have mentioned is their generality.It is very difficult to find experiments which confirm them except in very par-ticular cases. Consequently it would not be surprising that equations of a dif-ferent form might be found which so far as consistency (i.e., derivability) withexperimental laws is concerned, are equivalent.As employed in the set of five equations mentioned, 1Eand H1are chiefly auxil-iary vectors whose values are determined by equations (1) to (4) only when initialvalues of charge and current density are given. They are auxiliary in thesense that they are needed to determine the force as given in the fifth equation.The intensities play a different role in the experimental laws in which they firstappear in non-differential form. They were as we saw much earlier definablein terms of point charges. Thus we must specify the context to which we referwhen we ask as to the fictional character of 'field'.The particular form that the fundamental force equation takes is not uniquelydetermined by experimental laws. In particular the force between two currentelements may be represented by two different expressions.This is explicitly recognized when one adds, to one deduced form for a differen-tial law, further terms which enable one to shorten the expression for the lawby the use of a vector identity. Both those expressions lead to the same resultfor the force on a closed circuit. Hence it may be said that the expressionusually adopted is conventional with respect to the experimental laws of thesubject since every magnetostatic problem may be solved by either. We arenot able to specify in sufficient detail the conditions under which a unique ex-pression for the force may be derived. This circumstance suggests a definitionfor 'convention' more specifically applicable to scientific formulations than thatin Chapter I. A formulation is conventional whenever the conditions of itsformulation are insufficient to make it a unique derivative; or in general, when-ever we are given a set E of experimental laws and a set S of stipulations of thattheory such that p is derivable from E on the basis of S and also p' from E on Swhere p may be obtained from p' by a transformation T such that p D q andp' D q where q is an observational statement giving the numerical value of aspecific measure which is a function of some quantity occurring in p. In otherwords, if T leaves invariant the numerical value of some measure occurring ina proposition implied by p, then p (or the proposition obtained by transformingp) is conventional.Maxwell's equations are not to be considered as the only possible formula-tions of the experimental laws of electromagnetism. Variations which are con-sistent with the laws which are here taken to mean statements capable of veri-fication by standard experimental means are available.



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THE CONCEPT OF 'FIELDI IN ELECTRICAL THEORY 321Because of this conventionality of Maxwell's equations, the definition of fieldintensity will vary with the particular set of equations adopted. Once, how-ever, a set has been chosen, the problem of the uniqueness of definition is a

mathematical one which has been solved. That is, it has been proven that thereis only one solution of these equations if the charge density and current densityare given.As employed in Maxwell's equations the field intensities are on the one handdependent upon experiment for the value of p and i and on the other, are auxiliaryto the calculation of the force which is the only directly measurable quantitybesides p and i. The E and HIhave no specific referent taken alone but only inrelation to the other quantities in the equations.

ViIIn the preceding section of this paper it was noted that the unique character

of a theory lies not so much in the experimental laws reformulated but in thestipulations involved in the reformulation of those laws into the fundamentalequations of the theory. We shall find that these stipulations are conventionsof a different type than those of which the force equation is an example. Brieflythe first are norms or criteria and the latter are not. In the second place con-ventions of the second type are connected with fictions. We turn first to a dis-cussion of the distinction between empirical laws and stipulations. For illus-tration we again return to magnetostatics.Ampere's first law which will serve as an example of a statement of an ex-perimental result is that "the action of a current on another current (or currentelement) is unchanged in magnitude but reversed in direction when the directionof the current is reversed." This statement is established by noting that theforce exerted by two straight parallel currents of equal magnitude but oppositedirection is zero when the wires are very close to each other. The total force isalso zero.28 Experimental apparatus can easily be arranged to test this law,though it is difficult to test some of Ampere's other laws-for example, the thirdlaw that the action of a closed circuit is always normal to the latter, involves a

suspension over mercury whose capillary action interferes with the accuracy ofthe test. A difficulty of a different sort occurs in the case of the fourth law:"In similar and similarly stituated circuits traversed by equal currents the forcesare equal." This is difficult to test in sufficient generality though Amperetested it by three circular closed circuits whose radii were in ratio 1:2:4, thedistance from the center to center of the first two and the distance from centerto center of the last two being in the ratio 1:2.These statements are capable of experimental verification despite difficultiesattendant upon this procedure. This is not true however of the statementswhich taken together comprise the distinguishing feature of a given theory.These statements have a legislative function and simply specify what require-ments are to be met by the statements to be deduced from the laws. For ex-ample, the principle of continuity provides for the introduction of continuousfunctions, by "assuming" that the above experimental laws hold for the action

28 Not clear, but left as author stated it.



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322 GEORGE T. BOWDERYof one current element on a second current element. This enables the vectorcalculus to be readily applied. This requirement is similar to that imposed uponthe experimental form of Ohm's law which was transformed into differentialform.

Since the only effect of this principle was to select which branch of mathematicswas to be used in formulating the laws of electromagnetics, it could not be con-firmed or disconfirmed by appeal to experiment with wires and electrical ap-paratus.Supplementing this first stipulation is the so-called principle of the super-position of effects namely that the total effect can be obtained by summation ofthe effects of the parts. In the present case, this amounts to saying that theforce is proportional to the magnitudes of the component arcs concerned and to

the current strengths. In ordinary mechanics there is a similar principle, theso-called parallelogram rule for the summation of vectors, in virtue of which,for example, two forces acting on the same point are equivalent to one forceacting along the diagonal of the parallelogram determined by the two vectorsrepresenting the forces, and equal in magnitude to the length of the diagonal.As in the case of the principle of continuity there is no empirical control over theprinciple of superposition. Finally it is stipulated that the forces between thecurrent elements mnustsatisfy the ordinary mechanical requirements for equi-librium.

The stipulations are conventions because with respect to the empirical lawsthey are arbitrary, but with respect to other theories of electromagnetism theyare not; for the totality of stipulations is one of the distinguishing characters ofone electromagnetic theory from another, e.g., Maxwell's from others. Theyare norms which are prescribed by the scientist that his theory must meet.Different scientists may prescribe different niorms. For example, not all othertheories of electromagnetism would require both the mechanical requirementsmentioned above. With respect to experimental laws in the sense we have usedthem however the stipulations are arbitrary.Besides these two sets of statements, 1) empirical laws and 2) stipulationswhose function is legislative and are conventional with respect to the first classof statements, there is a third class of statements-the consequences of thesetwo; these consequences amount to reformulations of the experimental laws,via the stipulations. A set of fundamental equations, e.g. Maxwell's, consti-tutes the third cluster of statements that are signalled for attention in the net-work of all the consequences of (1) and (2). These statements are not uniquefor their algebraic form may vary even if (1) and (2) are fixed. This was illus-trated in electromagnetic theory by the fact that no unique expression for themagnetic force exists. Hence these equations are conventional not merely withrespect to the experimental laws (1) but also with respect to the set of stipulations(2). They are thus derived conventions in that they depend on the conventionsof (2). But they are not norms nor do they have a legislative function in thesense that (2) do. Conventions of the first type are usually called principles.Instead of speaking of the conventionality of statements of (2) and (3) it



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THE CONCEPT OF 'FIELD' IN ELECTRICAL THEORY 323might be more appropriate to call the whole set of statements conventionalsince one or more might be missing and others added. This is only a matter ofterminology. However there must be some statements having the functionof (2) and (3); such statements are essential components of physical theories.Of course we do not mean to imply that the membership of any of the threeclasses is fixed. If the system is thrown into a deductive form, the equations ofset (3) will be the axioms and more empirical laws may be deduced besides thosealready in (1). Also stipulations may be dropped in some cases and reinstatedin others. For example equality of action and reaction (which is a consequenceof the mechanical requirements mentioned above) may not be demanded for theaction between current elements though required for closed circuits. Moreoverthis requirement may be enforced because it is analytically desirable at somepoints of an exposition and dropped at others because it is not.Fictional terms are often found in conventions of the second type. We haveseen how 'displacement current' enters into the first equation whose form mightbe altered by the addition to the displacement term of any function whose diver-gence is zero without altering the applicability of the equation, i.e. without beinginconsistent with any experimental laws. The displacement term is similar tothe term added to the expressions for the magnetostatic force in that the lattercan by a suitable transformation be eliminated. Hence conventions of thesecond type always contain fictional terms and the latter occur in such formu-lations.

VIIIWe conclude with a summary of the general points suggested by the pre-

ceding analysis. Some are obvious but bear restating; others require to be moreperfectly established than was possible in this paper.1) The role that a term plays depends upon the particular theoretical formu-lation in which it occurs; the term 'field' is defined differently in different formu-lations of the theory of electromagnetics.2) There are at least three types of definitions of terms as they appear intheoretical formulations of physics, corresponding to different stages in inquiry;a) the denotative type which serves to indicate the subject matter of the inquiry;for example, Oersted's use of 'field'; b) the use of definitions which serve to makethe resources of the differential and integral calculus applicable, e.g., definitionsinvolving passage to a limit; c) a definition of a term by a set of equations ofwhich it is the unique solution.2') Definition of a term by a set of equations also requires that some otherterm in that equation refer to some experimentally measurable aspect, e.g.,charge density or possibly the potential, in the case of the equations of the elec-trostatic field.

3) The term 'fictional' is used in two senses at least: a) as applied to some ele-ment of the extra-linguistic subject matter which is alleged to exist, but is notexperimentally detectable and b) as applied to a term which refers to no extra-linguistic object but has a heuristic function in aiding calculations; in other



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324 GEORGE T. BOWDERYwords the subject matter of such a term is the form of the equations in which itoccurs.4) A necessary condition for a term 'T' being fictional (in sense (3b) is thatthere exists a formulation 'B' transformable into the formulation in which 'T'actually appears, such that 'T' does not appear in 'B'. The transformationmust not alter the consistency of the two formulations with accepted experi-mental results.5) There are conventions of two types a) those which have a legislative func-tion as criteria or norms and differ in different theories but are alternatives withrespect to the experimental laws, b) those which exist in virtue of the incom-pleteness of formulation of the conditions under which theoretical statementsare made; neither experimental laws nor stipulations are sufficient to determineuniquely the form of the fundamental equations of a theory. In this sense al-ternative formulations exist with respect both to the experimental laws and thestipulations.6) The fictional character of a term is connected with the conventionality ofthe formulation 'F' in which it appears in this way: if 'T' is fictional, 'F' is con-ventional, but not conversely, unless 'F' is a convention of the second type. Inthis case fictions occur in the reformulation of experimental laws.7) The conventionality of a formulation appears only late in the history of atheory; the stage corresponding to the third type of definition is especiallyfruitful for here the equivalence or non-equivalence of formulations with respectto some transformation can be more easily ascertained.8) The study suggests that three essential components of a physical theoryare 1) the experimental laws, 2) stipulations which aid in the reformulation of thelaws into the 3) fundamental equations consistent with the experimental laws.Comparison with other theories, e.g., the kinetic theory of gases would be fruit-ful to see if this pattern is widespread. The conventional character of the stipu-lations with respect to experimental laws together with the conventional charac-ter of equations of a theory with respect both to the laws and the stipulationsactually made contrasts with the relative stability of the experimental laws.9) 'Field intensity' is not a fictional term; the equations together define it.It is contrasted with 'displacement current' and 'stationary medium' which arefictional. Some terms in the Maxwell equations must be given in order to makethe equations have a unique solution.10) Both extreme views of scientific theories, namely 1) that all the terms ina scientific theory are purely fictional (in sense 3a) and have no connection withthe subject matter of the theory, or at most that such connection is made onlyby the terms of some lower type of language, as in protocol statements, and 2)that every term of a theory has its correlate in the subject matter of the theory,are inadequate. 1) For though many terms satisfy the necessary condition ofa fictional term, namely that it be replaceable by others in an equivalent formu-lation, there are always other terms with reference to which the given term isdefined which have extra-linguistic reference and in every formulation there aresome such terms.

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Bowdery, G. J. - The Concept of 'Field' in Electrical Theory (1946)

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