A Revisiting of Scientific and Philosophical Perspectives ...

A Revisiting of Scientific and PhilosophicalPerspectives on

Maxwell's Displacement Current

Krishnasamy T. Selvan

Applied Electromagnetic Research Group, Department of Electrical and Electronic EngineeringThe University of Nottingham Malaysia Campus

Jalan Broga, Semenyih 43500, Selangor Darul Ehsan, MalaysiaE-mail: [email protected]

Abstract

The displacement-current concept introduced by James Clerk Maxwell is generally acknowledged as one of the mostinnovative concepts ever introduced in the development of physical science. It was through this that he was led to thediscovery of his electromagnetic theory of light. While this concept and its development have received much admiration in theliterature from the viewpoints of both scientific content and philosophical methodology, there interestingly has been criticism,as well. This article presents an overview of these perspectives. With the distinctive creative quality of the concept emergingon balance, it is suggested that effective use of it can be made to help students to contemplate innovation and creativity,among other factors.

Keywords: Displacement current; education; electromagnetic engineering education; electromagnetic theory; electromagneticwaves; Maxwell equations; Maxwell; scientific approach

1. Introduction

Maxwell's electromagnetic theory is one of the founding theorieson which modem electrical science is based. It is therefore anessential segment of the electrical-engineering curriculum in mostuniversities. While undergraduate teaching of these equationsacross universities in general probably follows the approach inwidely used electromagnetics textbooks [1-3] or a variationthereof, a presentation of Maxwell's equations in historical contextis recommended, for example, in [4]. In the present article, theconcept of the displacement-current concept that led Maxwell tohis electromagnetic theory of light is considered.

Although it was through the concept of his displacement cur-rent, introduced in 1862, that Maxwell was able to come up withhis great discovery of the electromagnetic nature of light, this con-cept "has been the subject of both admiration and controversy formore than a century" [5]. The questions that continue to receiveparticularly significant attention in the literature are the following:

Is the displacement current equivalent to an electric cur-rent?

Does the displacement current produce a magneticfield?

Several authors have addressed the above questions. Therehave also been articles that offer either admiring or skeptical cri-tiques' from the viewpoint of the philosophy of science. Not with-

standing this, although to the undergraduate student the concept ofdisplacement current is presented as one of the most revolutionaryideas conceptualized by Maxwell, it is generally done without ref-erence to the arguments that led him to its introduction and the richscholarship on the subject. Justification is often given for its needbased on the well-known capacitor example and the equation ofcontinuity. However, considering the topic in historical and scien-tific perspectives can possibly better enrich students' understandingof the ways of innovators. This belief is a chief motivation for thisarticle, the other motivation being the presentation of an overviewof the different perspectives.

The paper proceeds as follows. Section 2 presents thephilosophy and context of Maxwell's displacement current. Sec-tion 3 presents the typical textbook presentation of the topic. Sec-tion 4 reviews the scientific perspectives, while Section 5 reviewsthe philosophical perspectives on displacement current. Section 6suggests that the context and development of and perspectives ofdisplacement current can be used to stir contemplation of creativityand innovation in students. Section 7 concludes the paper.

2. Maxwell's Displacement Current

2.1 The Underlying Philosophy

Consistent with the philosophical notion formulated by theseventeenth-century mechanical philosophers that any acceptable

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understanding of nature called for mechanical explanations [6],Maxwell, who initiated his research in electromagnetic theory inearly 1854 (when, incidentally, he was 22 years of age), believed inthe tradition according to which the physical sciences, in order tobe physically understood , would warrant a reduction to a "changein the arrangement and motion of a material system." The attitudehe displayed towards the mechanical world-view however changedthroughout the 25-year-period (mid-1850s to 1879) of his activityin the area.

In the early stages, Maxwell favored the use of physicalanalogies in preference to both purely mathematical and physicaltheories . He said that while a purely mathematical theory wouldresult in loosing "sight of the phenomena to be explained," aphysical hypothesis could just be a "partial explanation," and thusbe a threat to our discovering facts [7]. He therefore employedphysical analogies as heuristic aids to "obtain physical ideas with-out adopting a physical theory." In this way, he set out to translatethe ideas of Faraday into a mathematical form. He took care toemphasize that the theory did not profess "to explain the cause ofthe phenomena" [7]. At the same time, he had hope for the future:" ...a mature theory, in which physical facts will be physicallyexplained, will be formed by those who by interrogating Natureherself can obtain the only true solution of the questions which themathematical theory suggests" [7].

In his later work [8], Maxwell sought to explain the origin ofelectromagnetic effects "in the medium surrounding the electric ormagnetic bodies," assuming the existence of the medium as prob-able. Maxwell had the clue for this approach in WilliamThomson 's showing that magnetic action on light "cannot beexplained without admitting that there is motion of the luminifer-ous medium in the neighbourhood of magnets and currents" [8],and his related work on molecular vortices [6]. In his Treatise [9],Maxwell gave justification for the introduction of a medium onseveral occasions . One such justification was the following:

We may conceive the physical relation between theelectrified bodies, either as the result of the state of theintervening medium, or as the result of a direct actionbetween the electrified bodies at a distance. If we adoptthe latter convention, we may determine the law ofaction, but we can go no further in speculating on itscause.

Maxwell thus believed that the introduction of a medium was nec-essary to physically account for the electrical forces. He tended toreject the possibility of proceeding in a strict mathematical waywithout reference to any mechanical representation [10]. In thissense, he rejected the action-at-a-distance theory. However, headmitted that the two approaches were mathematically equivalent[9]. Although the mechanical model was thus intellectually close tohim, Maxwell was open to believing that "it is a good thing to havetwo ways of looking at a subject, and to admit that there are twoways of looking at it" [6]. In this light, he presented his approachessentially as an alternative to the theory of action-at-a-distance forexplaining electromagnetic phenomena.

In building his mechanical model, Maxwell employedThomson's hypothesis of molecular vortices, introduced in 1856[8], and successfully proposed a coherent and comprehensive the-ory of electricity and magnetism from the field-primacy viewpoint.Although he tended to take a skeptical stance towards the mechani-cal representation towards the end of his career, in the middleperiod, which was also "the period of intensive innovation" in his

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electromagnetic theory, his work displayed a strong commitment tothe mechanical approach. It must be mentioned that the "two majorinnovations in Maxwell's electromagnetic theory - the displace-ment current and the electromagnetic theory of light - receivedtheir initial formulations in the context of the molecular-vortexmodel" [10].

2.2 The Molecular Vortex Model

One of the goals of Maxwell appears to have been to con-struct a general mechanical illustration of the properties of anydielectric [9]. In this endeavor, during 1855-56 Maxwell "...startedout using an analogical approach to mechanical representa-tion...(and) presented the mechanical images...as purely illustrative,with no claim whatever to realistic status." During 1861-62, hedeveloped a molecular-vortex representation for the electromag-netic field that he believed to be more realistic. While the physicalvalidity of this model could not be guaranteed , Maxwell opinedthat by virtue of offering a comprehensive picture of the phenom-ena, it was a serious candidate for reality. He said:

If, by the hypothesis, we can connect the phenomena ofmagnetic attraction with electromagnetic phenomenaand with those of induced currents, we shall have founda theory which, if not true, can only be proved to beerroneous by experiments which will greatly enlargeour knowledge of this part ofphysics [9].

According to Maxwell's molecular-vortices hypothesis , origi-nally proposed by Thomson, the medium can be considered to bemade of a large number of very small portions , or molecular vor-tices, as shown in Figure 1. Under the influence of a magneticfield, each of these molecular vortices rotates on its own axis.During the propagation of an electromagnetic wave, the vorticesmay be so disturbed as to in tum affect the propagation of the wave11].

In the molecular vortex model, there are innumerable tinyether cells that are separated from each other by small particles,acting as idle wheels, which enable the adjacent vortices to rotatein the same sense. The axes of the vortices represent lines of mag-netic field. In a conducting material, the idle wheels could move

Figure 1. Maxwell's vortices and idle-wheel particles (redrawnfrom 16]).

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from vortex to vortex, constituting a conduction current. In insula-tors, they could not leave their place, with the implication that anymovement on their part would result in a distortion of the corre-sponding cells. These distortions represented an electric field. Byusing this difference between conductors and insulators, Maxwellwas able to account for the accumulation of charges at the bounda-ries between insulators and conductors. "The details of Maxwell'smodel were able to accommodate the major electromagnetic phe-nomena known at the time, the magnetic field accompanying con-duction currents, the interaction of currents and magnets, electro-magnetic induction and electrostatics" [6].

2.3 Displacement Current

3. A Typical Textbook Presentation

A typical textbook presentation of Maxwell's modification ofAmpere's law (and thus his introduction of the displacement cur-rent) proceeds by first showing that Ampere's law does not satisfythe equation of continuity, and is thus not valid for time-varyingfields. The displacement-current term is then added, and it isshown that the modified equation obeys the continuity equation. Itis also shown to remove an inconsistency when only the conduc-tion current is used in the Ampere's-law-based analysis of a circuitcontaining an ac source and a capacitor [1-3].

Since current is charge in motion, and since charge is con-served (i.e., never created or destroyed), the charge density, p, andthe current density, J, satisfy the continuity equation:

Maxwell's first mention of the term "electric displacement"appears to have been in [8]. His immediate context for the intro-duction of the displacement current was the molecular vortexmodel [10]. He "had long been aware that. ..Ampere's circuital lawin differential form had restricted applicability" - that is, only toclosed circuits- "and that was a matter of concern to him."Towards rectifying this law, in January 1862 he proposed and usedthe concept of displacement current: one of the chief innovations inelectromagnetic theory, the other being the electromagnetic theoryof light [10].

8pV·J=--.at

Ampere's original law is

VxH=J

Taking the divergence on either side,

V·VxH=V·J=O.

(1)

(2)

(3)

In developing the modified Ampere's law, Maxwell intro-duced the concept of displacement thus: "Electromotive force act-ing on a dielectric produces a state of polarization of its parts ....The effect of this action on the whole dielectric mass is to producea general displacement of the electricity in a certain direction" [10].It is the variations in electric displacement that constitute dis-placement currents. As for the justification of introducing theconcept of displacement current, Maxwell had this to say [9]:

Ampere's original law can thus hold only for stationary currentdistributions for which J is constant in time. On the other hand, ifthe charge distribution is such that aplat * 0 (a discharging orcharging capacitor is an example where this situation holds), thislaw cannot be correct.

This was one of the motivations for Maxwell to introduce thedisplacement-current term to modify Ampere's law:

The introduction of the new term is shown to remove theconsistency problem. It is also shown to ensure that the modifiedAmpere's law is valid for any loop, including when the loopencloses a surface between the capacitor plates and perpendicularto its axis. Two important implications of this law are that all cur-rents are closed (Figure 2), and that a changing electric field pro-duces a magnetic field.

+ c:::> +c::::>I + c::::> + I.. t c::::> +

-+ -+ --. -. c::::> c::::> -+ -+ --.. -.+ +.. ++c::::> +

We have very little experimental evidence to the directelectromagnetic action of currents due to the variationof electric displacement in dielectrics, but the extremedifficulty of reconciling the laws of electromagnetismwith the existence of electric currents which are notclosed is one reason among many why we must admitthe existence of transient currents due to the variation ofdisplacement.

With this background, although Maxwell suggested that the"electric displacement. ..is a movement of electricity in the samesense as" the normal "movement of electricity," he continued toadd that only in a conductor "a current of true conduction is set up"[9, p.69]. He also noted that "any increase of displacement isequivalent to a current of positive electricity ...and any diminutionof displacement is equivalent to a current in the opposite direction"[9, p. 166]. When defining variables in a later section [9, p. 328],he distinguished currents due to conduction and variation of dis-placement. Considering these in conjunction with Maxwell's viewof electricity in his Treatise as "an abstract, mobile, incompressiblefluid that did not consist of electric charges," we are able to under-stand how Maxwell could accommodate current in a dielectric.Thus, "in the charging capacitor circuit. .., Maxwell's...total current(would comprise) the conduction current in the wire and the dis-placement current in the dielectric," [10] as in Figure 2.

anVxH=J+-.at

Maxwell called the new term the displacement current.

(4)

With the concept of displacement current introduced,Maxwell deduced that every electric current (conduction plus dis-placement) must form a closed circuit [9].

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Figure 2. The conduction current (single arrows) and the dis-placement current (double arrows).

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Ramo et al. [1] additionally noted that the displacement cur-rent "contributes to the curl ofmagnetic field in the same way as anactual conduction current density....(While) there is an actual time-varying displacement of bound charges in a material dielec-tric (the) displacement current can be nonzero even in a vac-uum .It is essentiaL..to the understanding of all electromagneticwave phenomena" [1].

4. Scientific Perspectives on theDisplacement Current

4.1 Possible Motivation for theDisplacement Current

It has been suggested [12] that Maxwell's treating the dis-placement current as being equivalent to current was attributable tohis "constant choice of the Coulomb gauge for the potentials."

Maxwell's equations in vacuum are

...preferred to have scalar and vector potentials satisfyPoisson-like equations with source terms, the chargedensity as source for the equation for the scalar poten-tial and the total current for the vector potential. It neverbothered him that his total current was not really asource term, but contained an initially unknown dis-placement current.

Jackson opined that "it is in this sense, and within the frameworkof the Coulomb gauge, that Maxwell could insist on the reality ofthe displacement current as a contribution to the total current" [12].

However, from a modem viewpoint the following flaws areidentified with such an interpretation [12]:

an/at cannot be a source term, as it involves fields;

p and J are the true external sources; and

the Coulomb gauge has an instantaneous scalar poten-tial.

Making use of the scalar potential, <1>, and the vector potential, A,

the inhomogeneous Maxwell's equations, explicitly showing thedisplacement current, can be written as

Choosing the Coulomb gauge, V· A =0, the equations for thepotentials become

aDVxE+-=Oat '

anVxH=J+-,at

V.B=O,

aAE=-V<I>--at '

B=VxA,

2 a p-v <I>--(V.A)==-,at EO

-V2A+V(VoA) =.uo [J+

2 [an]-\7 A=f.Jo J+at .

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(However, although the Coulomb gauge is unorthodox for mostpeople these days, there is mathematically nothing wrong with it[15]. Besides this, the fields derived from the Coulomb gaugetravel with finite speed [14]. This being so, Bartlett [17] wondered"why labor the Coulomb gauge when Maxwell's theory is gaugeinvariant and the Lorentz gauge is generally much easier to use?"Of course, this kind of variation in perspectives is widely prevail-ing (and highly desirable as well) in research approaches.)

In addition to the above, the questioning of Maxwell's dis-placement current as a true current appears to be essentially basedon the following counts:

1. The displacement current does not produce magneticfield for slowly varying fields;

(Jackson has noted as follows: "This blanket statementis false. There are electromagnetic waves of frequenciesas low as 8 Hz (ELF waves) around the Earth, excitedby thunderbolts. Clearly, the displacement current isproducing a magnetic field" in this case [14]. It may beadditionally noted that these waves are slowly varyingbut not quasistatic.)

2. The Biot-Savart law using the conduction currentsalone can be used to estimate the magnetic field inquasi-state conditions for both closed and open circuits.

4.2 Displacement Current andMagnetic Field

Purcell [3], Bartlett [16, 17] and Rosser [18] showed that forslowly changing fields, one cannot observe the magnetic fieldcaused by the displacement current.

While Equation (13) is Poisson's equation, Equation (14)"has the appearance of a Poisson's equation for the vector poten-tial, with a source term that is the sum of the conduction currentdensity and the displacement current" [12]. Maxwell thus [13, 14]


This was shown by Purcell [3] by considering the case of aslowly discharging capacitor. The electric field is slowly dimin-ishing, and it can therefore be practically considered to be an elec-trostatic field. Its curl is thus practically zero. By implication, thecurl of the displacement-current density must also be zero.

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Starting with

J =aD=c aEd at at '

(15)

in this equation, "it seems pointless to make statements such as achanging electric field produces a magnetic field" [18]. It shouldbe noted that in this approach, the displacement-current term ismathematically eliminated from the wave equation by using thescalar potential.

and taking the curl of either side,

(16)

French and Tessman [21] also showed that in all cases ofquasistatic situations, whether they are closed or open, the Biot-Savart law with conduction currents alone can be employed todetermine the magnetic field.

Bartlett [16, 17] showed that the magnetic field for slowlyvarying fields can be estimated without including displacementcurrent when the Biot-Savart law is employed. He showed that thislaw, for the magnetic field close to the axis of a capacitor, can bewritten in the form

With his coworkers, Bartlett also carried out careful experi-ments to measure the displacement current in capacitors [17, 19,20]. Although their initial finding appeared to show a displacementcurrent in the azimuthal direction, this finding turned out to actu-ally be a "confusion caused by a poorly aligned detector."

where J is the total current density. He then deduced that sinceV x Jd = 0 for slowly varying fields, as noted earlier, the qua-sistatic displacement current cannot produce a magnetic field eitherin vacuum or in a homogeneous dielectric. In this case, it is onlythe conduction currents on the capacitor plates that would contrib-ute to the magnetic field.

where we have made use of Equation (5). The above will be negli-gible for sufficiently slowly varying, or quasistatic, fields. Theimplication of the displacement-current density having no curl isthat "it can be made up...by superposing radial currents flowingoutward from point sources or in toward point sinks." The mag-netic field due to the displacement current should therefore be zero,as "the magnetic field of any radial, symmetrical current distribu-tion, however calculated" must be zero by symmetry. It followsthat conduction currents alone account for the magnetic field in thequasistatic field.

The magnetic field inside a slowly charging capacitor may becalculated by using the Biot-Savart law, in which case it is only theconduction currents that contribute. Alternatively, when Ampere'scircuital law is used, it requires only the real currents for certaincontours, while the inclusion of displacement current is warrantedto obtain the correct result for arbitrary choices of contours [12, 16,22]. The displacement current thus makes an alternative approachfor the estimation of the magnetic field possible.

Jackson [12] demonstrated this approach by employing thelinear superposition approach to calculate the magnetic field, con-sidering Ampere's law around two loops: loop A and loop B, as inFigure 3. While the application on loop A leads to a result thatrelates the magnetic field only to the conduction current, the appli-cation on loop B requires the inclusion of the displacement currentso that the result can be consistent with that of loop A.

4.3 The Displacement Current as anAlternative Approach

Jackson also employed a "perturbation approach" - whichappears to be consistent with Maxwell's own interpretation - toshow that the displacement current is the only source of magneticfield within a charging capacitor. In this approach [12], a current,I (t), flows along the negative z direction, as shown in Figure 3. It

brings a total charge of Q(t) to the plate at z = 0, and removes an

equal amount of charge from the plate at z = d . Assuming thestatic limit, and neglecting fringing (as a» d), Q(t) is uniformly

distributed over the inner side of the positive plate. The surface

(17)dt'

cB(r}=- fVxJ !r-r'I'

The right-hand side of Equation (18) can be written as

Equation (18) is obtained by using aE/at, found from Equa-tion (9), in Equation (14).

Figure 3. A charging capacitor. Each circular plate is of radiusa. The plates are separated by a distance d, such that d « a .

;r..x

(19)

(18)

where Jt , called the transverse current density, can be calculated

from the conduction-current density, J, alone and the scalar poten-tial. Since the displacement current in empty space does not appear

Rosser [18] showed that the displacement current does notproduce magnetic field in empty space by starting with the inho-mogeneous wave equation for the vector potential and by using theCoulomb gauge:

40 IEEEAntennas and Propagation MagaZine, Vol. 51, No.3, June 2009


Since there is no conduction current between the plates, theAmpere-Maxwell law for the situation is

Making use of the expression for the displacement current, Equa-tion (20), and recognizing that the displacement current causes anazimuthal magnetic field, the first-order magnetic field foro< p < a and 0 < z < d as given by Jackson is

charge density there is thus u(t) = Q(t . The electric dis-

placement is uniform over the entire volume between the platesand is directed along negative z. The first-order displacement cur-rent is therefore given by It is worth noting here what Maxwell said of his electromag-

netic field equations, the concepts expressed by which alone,according to Hertz, represent Maxwell's theory [24; 11, p. 254]:

These equations may be regarded as the principal rela-tions among the quantities we have been considering.They may be combined so as to eliminate some of thesequantities, but our object...is not to obtain compactnessin the mathematical formulae, but to express everyrelation of which we have any knowledge. To eliminatea quantity which expresses a useful idea would berather a loss than a gain....

4.4 The Electromagnetic Equivalence ofaD/at to Current

current. His equation is in fact equivalent to Equation (27), whichinvolves aD/at.

(20)

(21)

(22)

B.dl =JaD .ftdA.'fc s at

Substituting into the wave equation, Equation (23),

Let us again start with the inhomogeneous wave equation forthe vector potential [13]:

Closely agreeing with the above views, Roche [5] stated thatwhile the term aD/at is "enormously important," it can in nosense be "described as an electric current" in the case of a uni-formly charging capacitor. He also proceeded to suggest that thisterm should not even be mentioned to undergraduates.

The presence of the term aD/at in the right-hand side ofAmpere's law literally means that a changing electric field causes amagnetic field, even when no conduction current exists. In thiscontext, and essentially in the light of the observations noted inSection 4.2, Maxwell's calling the term aD/at has received mixedopinions in scholarly literature. We consider these observations inthis section.

Around the turn of the twentieth century, the displacement-current concept had been "completely emancipated from its origi-nal context in the theory of molecular vortices" [10]. For example,Warbarton [25], while agreeing that the displacement current was areasonable concept when it was introduced, felt that the name hadlost its relevance as the "ether is gone and we do not visualize D invacuum as a physical displacement of charge." He also suggestedthat this term might mislead students. Rosser [18] suggested that "alot of confusion about the role of the displacement current in emptyspace might be avoided, if it were called something else that didnot include the term current. If a name is needed, it could be calledthe Maxwell term in honour of the man who first introduced it."

Interestingly, Roche [5] proceeded to appreciate that "in theCoulomb gauge," which is frequently used in advanced electro-magnetism, "the displacement current can, in general, be consid-ered to be equivalent to an electric current," but added that the"Coulomb gauge is not a physical gauge and the equivalence ofaD/at to a current is always purely fictional" [26]. Responding tothis observation, Jackson [27] observed that [26]

(25)

(23)

(24)aA

Vt/J=---E.at

The first-order result for the magnetic field in the region betweenthe capacitor plates thus depends only on the existence ofMaxwell's displacement current [12].

Zapolsky [23] argued that Rosser's argument [18] that no partof the displacement current gives rise to a magnetic field (as theterm does not appear in the wave equation for vector potential) isessentially a semantic argument. This can be explicitly shown asfollows [13, 14].

Differentiating either side of Equation (24) and multiplying byPOGo,

The scalar potential, Equation (9), can be written as

It can thus be seen that Rosser's Equation (19) in terms of trans-verse current density has actually hidden away the displacement

Thus,

2 (aD)v A=-po J+at . (27)

...the choice of gauge is purely a matter of convenience.The Coulomb gauge is no more or less physical thanany other. It is convenient for some problems, incon-venient for others. The fields are the reality. TheCoulomb gauge has potentials with peculiar ("unphysi-cal") relativistic properties, but the fields derived fromthem are the same as the fields derived from the poten-

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tials in any other gauge, causal and with finite speed ofpropagation.

While Jackson [27] agreed with the "fact that...the vastmajority of physicists and engineers living today do not think thatthe unfortunately named 'displacement current' is a true cur-rent....The external charge and current densities are the true sourcesfor the fields," he also noted that "Maxwell is wrong (only) if heasserts that the displacement current is a real external current den-sity on a par with the conduction current density, but he is right ifhe says that it is electromagnetically equivalent" [12].

According to Zapolsky [23], the Lorentz vector-potential cal-culation not needing the displacement current does not mean thatthe displacement current does not exist. He rather suggested thatboth approaches could be useful.

According to Roche [5], the investigations by Purcell,Zapolsky, and Bartlett demonstrated that "the displacement currentof a rapidly changing induced electric field will generate a signifi-cant magnetic field." When one has rapidly changing fields, theinduction effect of changing electric field producing magnetic fieldis thus observable. This is the reason the demonstration of dis-placement current needed Hertz's experiments, conducted manyyears after the law had been developed by Maxwell [3]. It is ofinterest to note here that Hertz, in a significant work done in 1884,derived Maxwell's equations in a way that dispensed with therequirement for both the mechanical models and the displacementcurrent. In this work, his fundamental contribution was to developa theory of source-field relation [28].

In 1922, Max Planck, while agreeing that an/at produces amagnetic field, "demonstrated that the symmetries of this functionensure that no magnetic field is actually produced when thechanging electric field is a conservative field" [5].

The above studies appear to explain why the displacementcurrent remains "deeply rooted in electromagnetic intuition," andalso why the denial of its existence "has not been generally wel-comed" [5]. The textbook tradition of equating the term to currentalso probably stems from this viewpoint.

4.4 Additional Salient Features of the Term

French and Tessman [21] acknowledged that without theintroduction of the displacement current, "the treatment of elec-tromagnetic waves would be absurdly complicated if the fieldswere always referred back to the motions of real charges." Theyfurther suggested that "even in many circuit problems it is muchsimpler to compute magnetic fields from the circuital theorem thanfrom the Biot-Savart law."

Maxwell's consideration of the full displacement currentreceived much admiration from Zapolsky [23]. Although Maxwellwould have been aware that the inclusion of the longitudinal (curl-free) part of the displacement current would be sufficient to makeAmpere's law consistent with the continuity equation, in his defi-nition of the displacement current, he included both the longitudi-nal and transverse (divergence-free) components of the displace-ment current. According to Zapolsky [23], "This was his truestroke of genius, since it is the transverse component of the dis-placement current, coupled with Faraday's law, which gives rise toelectromagnetic radiation." Zapolsky [23] also considered the sug-

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gestion that it was Maxwell's theory that "forced on us" the specialtheory of relativity [13]. Stating that the decomposition of E intotransverse and longitudinal components is not relativisticallycovariant, and that the full displacement current is required toensure both charge conservation and relativistic covariance simul-taneously, Zapolsky wondered if it could be that Maxwell "alreadyhad an early glimmering of the revolution which was to come?"[23].

5. Philosophical Perspectives

Maxwell's coming up with a successful theory unifying elec-tricity and magnetism could be chiefly attributed to his use of ana-logical reasoning, or model building [29]. It was consistent withthe philosophical notion prevailing in the nineteenth century thatMaxwell "sought to explain physical phenomena mechanically." Itwas his view that "when a physical phenomenon can be completelydescribed as a change in the configuration and motion of a materialsystem, the dynamical explanation of that phenomenon is said tobe complete" [9]. This thinking also led him to introduce amedium, and "by directing his attention to the medium surroundingelectrified bodies, Maxwell was certainly led to innovations ofmajor importance" [30].

In constructing a physical picture, Maxwell carefully distin-guished between nature and our abstraction of it. It is illustrative tonote here what he wrote [24]:

...molecules have laws of their own, some of which weselect as most intelligible to us and most amenable toour calculation. We form a theory from these partialdata, and we ascribe any deviation of the actual phe-nomena from this theory to disturbing causes. At thesame time we confess that what we call disturbingcauses are simply those parts of the true circumstanceswhich we do not know or have neglected, and weendeavour in future to take account of them. We thusacknowledged that the so called disturbance is a merefigment of mind, not a fact of nature, and that in naturalaction there is no disturbance.

To Maxwell, the model thus did not necessarily represent reality. Infact, he took care to "stress that no physical explanation could be inperfect correspondence with reality" [31]. However, the usefulnessof the model lay in simplifying reality and in possibly leading to asatisfactory theory for explaining and predicting observations.

Maxwell's generalization of Ampere's law for open circuits,originally done "in an effort to get at a numerical value for theelasticity of the electromagnetic ether" [32, 33], was clearly themost important contribution in the sense that the other major pre-dictions followed this generalization. Since Maxwell did not havemeasurement data on currents flowing in open circuits, his intro-duction of displacement current was fully a theoretical venture[34]. "Maxwell's great successes in electromagnetism stemmedfrom his introduction of a displacement current into the theory.Once the appropriate form of that current had been introduced,dramatic consequences, such as the propagation of electromagneticeffects in time through empty space, and an electromagnetic theoryof light followed from it" [35].

According to Siegel [34], Maxwell's work on electromag-netic theory was oriented "toward the goal of theoretical complete-



ness, and this orientation would have been a significant factor inhis establishment of a complete and enduring foundation for elec-tromagnetic theory. In this respect, then, Maxwell's electromag-netic theory would emerge as a shining example of systematic andgoal-oriented theoretical endeavor rewarded." This statement, in away, summarizes the philosophical perspective on Maxwell's the-ory.

6. Teaching Displacement Current

In general, scientific methodology has model building as anintegral part. While a mathematical approach without physicalabstraction may be possible, many well-known scientists have pre-ferred analogy. Maxwell was an example; Neils Bohr also held that"mathematical clarity had in itself no virtue...." He feared that theformal mathematical structure would obscure the physical core ofthe problem, and in any case, he was convinced that a completephysical explanation should absolutely precede the mathematicalformulation" [36]. How straightforward and simple could thismodel-building process be? According to Albert Einstein [37],

...the external conditions which are set for [the scientist]by the facts of experience do not permit him to let him-self be too much restricted, in the construction of hisconceptual world, by the adherence to an epistemologi-cal system. He, therefore, must appear to the systematicepistemologist as a type of unscrupulous opportunist....

This would mean that science is essentially a "complex and hetero-geneous historical process which contains vague and incoherentanticipations of future ideologies side by side with highly sophisti-cated theoretical systems..." [37]. Therefore, as Einstein noted,only an incomplete picture (model) of the physical universe isobtainable in the pursuit of science [38]. It follows that any modelcould in principle be subject to criticism.

Notwithstanding the criticism against Maxwell's mechanicalapproach, it was the mechanical model building that was an"engine of discovery" for Maxwell. It was through this approachthat he was able to introduce the displacement-current concept, andhence to unify electromagnetism and optics [14]. More importantly[14],

...the displacement current and the electromagnetic the-ory of light were not independent constructs looselygrafted to the molecular-vortex model, but ratherorganic parts of that model, growing naturally out ofit....The new term in Ampere's law was introduced inorder to allow for the accumulation of electric charge inthe model, and the formulation of a complete and con-sistent set of electromagnetic equations, incorporatingthe displacement current, followed directly from that.

Considering the immense scientific and philosophical depthof Maxwell's displacement-current concept, it appears that itwould be desirable to make particular use of this concept toenhance teaching and learning. Not mentioning the term displace-ment current - as has been suggested in [5], for example - maydeprive us of a very good opportunity to enhance the student'slearning. This is because while the primary objective in teaching isto give students a good grounding in the understanding of funda-mental principles and concepts, an important additional objectiveof most teachers is that they should try to give them "an apprecia-


tion of the significance of the scientific approach (and) the variousrevolutions in man's understanding of nature" [39]. Consideringthat development of science does not often occur in a simplisticway, this would often require that scientific theories be presentedalong with the context in which they were developed, the method-ology adapted, and an appropriate historical account. It has beensuggested that such a presentation may entail certain additionalbenefits, such as acting as a motivator, facilitating intellectualopenness, and making the lecture interesting [4, 37, 40, 41].

In view of the above discussion and in the context of electri-cal-engineering teaching, in the opinion of this author Maxwell'stheory is a particularly creative edifice that could be profitablyemployed in teaching. (Where they are accommodated in the cur-ricula, the theory of relativity [38] and the atomic theory [45] areamong the other theories that could also provide such a possibil-ity.) It may be noted here - particularly in the context of the generalimpression that the modem generation of students may find his-torical perspective boring - that the student participants in a recentsurvey have indicated that in their perception, the integration ofhistorical content into teaching would have positive effect [42, 43].However, these observations notwithstanding, in general there is"complete lack of historical perspective on the part of the youngergeneration of today" [40].

As this article has attempted to trace, and as a reviewer haspointed out, "a variety of positive, negative, and sometimes mis-leading discussion" has surrounded the development of the dis-placement-current concept. Given this, what version of the storycan best represent the creative process? While there can be differ-ent perspectives on this question, this author would suggest that thefollowing be discussed, perhaps in an hour-long lecture:

1. Discuss the context of development of displacementcurrent by Maxwell;

2. Emphasize "the most consequential innovations in thehistory ofphysics" [10] that flowed from it;

3. Present both the positive and negative criticism of theterm and its development;

4. Take care not to hint at any particular judgment;

5. Point out that different perceptions of the history of sci-entific development may be possible.

In this way, this author believes, the students' "image of the cer-tainty of scientific knowledge" could be challenged, and they couldbe encouraged to try to find "place for their own interpretation"[45].

Of course, such an approach cannot be without its limitations,as modem requirements force the accommodation of "new or aug-mented subjects" and soft skills in engineering curricula, leading tothe shrinking of such "strongly mathematical subjects like electro-magnetic theory" [44]. However, considering that "students oughtto experience the how of scientific enquiry, rather than merelybeing exposed to what is known about and by science," and that"the study of scientific ideas in their original context of discoverywill help to develop students' conceptual understanding" [45], inthe opinion of this author, the approach of teaching in context mustbe seriously considered.

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7. Conclusion

Maxwell's electromagnetic theory evolved as a dynamicscientific and philosophical process of immense importance over aperiod from 1856 to 1879. During the course of this, Maxwellemployed two different physical models for representing nature:the lines-of- force approach, and the theory of particulate polariza-tion [24]. (It is of interest to mention here that Maxwell's contri-butions were not limited to electromagnetics. For an overview ofhis other achievements, which were in topics as varied as digitalphotography and Saturn's rings, the reader may refer to [28].)

As has been noted by J. W. Arthur [46] in a recent article inthe Magazine, "Maxwell's equations have been the basis for thedescription and analysis of all electromagnetic phenomena to date."The displacement-current term introduced by Maxwell is at thevery heart of Maxwell's theory, in the sense that it led to predic-tions of fundamental importance. Although it does not appear thatMaxwell ascertained the displacement current to be similar to con-duction current, he considered it as being electromagneticallyequivalent to conduction current. This consideration led him to usethe term as a source term in formulating his equations. This aspect,and the question of if displacement current would cause a magneticfield, have been the subject of many interesting scholarly papers. Aprimary objective of this article has been to present an overview ofthese papers. Although there has been both scientific and meth-odological criticism of Maxwell's approach to the an/at term, ona balance of these views, it appears that the term, for the path-breaking implications it led to, has received very respectful consid-eration - and for the right reasons.

The term "displacement current" has with it a wealth ofscientific and philosophical significance. As pointed out byEinstein, this concept is "closest to...a genuine, useful, profoundtheory ...built purely speculatively" [38]. As such, this theoryappears to be a fertile pedagogical ground for possibly inspiringstudents to innovative thinking. Of particular value is the theorylending itself to philosophical discussions, the importance ofwhichin any intellectual pursuit need not be overemphasized. For exam-ple, as has been suggested it was the "philosophical approach toanalysis of electromagnetic phenomena (that provided a) necessarylink in the series of steps leading to discovery of the theory of rela-tivity" [38]. When an effort is made to present to the students theterm along with its context and development, it will have thepotential to have positive effects in the long run, consistent withthe currently emerging ideas of education that lay importance oncreativity and innovation and higher level skills [47, 48]. As thisapproach gives a good idea to students concerning the process ofscientific enquiry, it can perhaps also facilitate "successful learningof science" [45]. Highlighting the desirability of this approach ofcontextual teaching in view of these potential benefits has been anadditional significant object of this paper.

8. Acknowledgement

The author gratefully acknowledges helpful discussions withDavid F. Bartlet during the preparation of the paper, and usefulobservations on the manuscript from Tapan K. Sarkar. The thor-ough review and comments of an anonymous reviewer were par-ticularly helpful in improving discussions related to pedagogicalaspects in the paper.

44

9. References

1. S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves inCommunication Electronics, Third Edition, New York, JohnWiley, 1994.

2. F. T. Ulaby, Fundamentals of Applied Electromagnetics, FifthEdition, Upper Saddle River NJ, Prentice-Hall, 2007.

3. E. M. Purcell, Electricity and Magnetism, New York, McGraw-Hill, 1985, pp. 324-330.

4. K. T. Selvan, "Engineering Education: Presentation ofMaxwell's Equations in Historical Perspective and the LikelyDesirable Implications," IEEE Antennas and Propagation Maga-zine, 49, 5, October 2007, pp. 155-160.

5. J. Roche, "The Present Status of Maxwell's Displacement Cur-rent," European Journal ofPhysics, 19, 1998, pp. 155-166.

6. A. F. Chalmers, "Maxwell, Mechanism, and the Nature of Elec-tricity," Physics in Perspective, 3, 2001, pp. 425-438.

7.1. C. Maxwell, "On Faraday's Lines of Force," Transactions ofthe Cambridge Philosophical Society, 10 Part I, 1864, pp. 27-65(read December 10, 1855, and February 11, 1856).

8. J. C. Maxwell, "A Dynamical Theory of Electromagnetic Field,"Proceedings ofthe Royal Society ofLondon, 1864, pp. 531-536.9. J. C. Maxwell, A Treatise on Electricity and Magnetism, Volume1, Third Edition, Mineola, NY, Dover, 1954 (originally publishedby Clarendon Press in 1891).

10. D. M. Siegel, Innovation in Maxwell's Electromagnetic The-ory, New York, Cambridge University Press, 1991.

11. J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol-ume 2, Third Edition, Mineola, NY, Dover, 1954 (originally pub-lished by Clarendon Press in 1891).

12. J. D. Jackson, "Maxwell's Displacement Current Revisited,"European Journal ofPhysics, 20, 1999, pp. 495-499.

13. J. D. Jackson, Classical Electrodynamics, Third Edition, NewYork, John Wiley & Sons, 1999.

14. J. D. Jackson, private communication.

15. A. D. Yaghjian, private communication.

16. D. F. Bartlett, "Conduction Current and the Magnetic Field in aCircular Capacitor," American Journal ofPhysics, 58, 12, Decem-ber 1990, pp. 1168-1172.

17. D. F. Bartlett, private communication.

18. W. G. V. Rosser, "Does the Displacement Current in EmptySpace Produce a Magnetic Field?," American Journal of Physics,44, 12, December 1976, p. 1221-1223.

19. D. F. Bartlett and T. R. Corle, "Measuring Maxwell's Dis-placement Current Inside a Capacitor," Physical Review Letters,55, 1, July 1985, pp. 59-62.

IEEEAntennasand Propagation MagaZine, Vol. 51, No.3, June 2009


20. D. F. Bartlett and G. Gengel, "Measurement of QuasistaticMaxwell's Displacement Current," Physical Review A, 39, 3, Feb-ruary 1989, pp. 938-945.

21. A. P. French and J. R. Tessman, "Displacement Currents andMagnetic Fields," American Journal of Physics, 31, 3, March1963, pp. 201-204 .

22. A. J. Dahm, "Calculation of the Displacement Current Usingthe Integral Form ofAmpere's Law," American Journal ofPhysics,46, 12, December 1978, p. 1227.

23. H. S. Zapolsky, "Does Charge Conservation Imply the Dis-placement Current?" American Journal ofPhysics, 55, 12, Decem-ber 1987, p. 1140.

24. P. M. Heimann, "Maxwell, Hertz, and the Nature of Electric-ity," Isis, 62, 2, Summer 1970, pp. 149-157.

25. F. W. Warburton, "Displacement Current , a Useless Concept,"American Journal ofPhysics, 22, 1954, pp. 299-305 .

26. J. Roche, "Reply to J.D. Jackson's Maxwell's DisplacementCurrent Revisited," European Journal of Physics, 21, 2000, pp.L27-L28 .

27. J. D. Jackson, "Reply to Comment by J Roche on 'Maxwell'sDisplacement Current Revisited'," European Journal of Physics,21,2000, pp. L29-L30.

28. T. K. Sarkar, R. J. Mailloux, A. A. Oliner, M. Salazar-Palmaand D. L. Sengupta, History ofWireless, New York, Wiley, 2006 .

29. N. J. Nersessian, "Reasoning from Imagery and Analogy inScientific Concept Formation," Proceedings of the Biennial Meet-ing of the Philosophy of Science Association, Volume One: Con-tributed Papers (1988),1988" pp. 41-47.

30. A. F. Chalmers, "The Limitations of Maxwell's Electromag-netic Theory," Isis, 64, 4, December 1973, pp. 469-483 .

31. P. M. Heimann, "Maxwell and the Modes of Consistent Repre-sentation, " Archives for History of Exact Sciences, 6, 3, January1970, pp. 171-213.

32. J. Bromberg, "Maxwell' s Displacement Current and His The-ory of Light," Archives for History of Exact Sciences, 4, 1967-8,pp.218-234.

33. J. Bromberg, "Maxwell's Electrostatics," American Journal ofPhysics, 36, 2, February 1968, pp. 142-151.

34. D. M. Siegel, "Completeness as a Goal in Maxwell's Electro-magnetic Theory," Isis, 66, 3, September 1975, pp. 361-368 .

35. A. F. Chalmers, "Maxwell and the Displacement Current, "Physics Education, January 1975, pp. 45-49.

36. Paul K. Feyerabend, "On a Recent Critique of Complementar-ity: Part II," Philosophy ofScience, 36, I, March 1969, pp. 82-105.

37. P. Feyerabend, Against Method, London : Verso, 1988.

38. D. P. Gribanov, Albert Einstein 's Philosophical Views and theTheory ofRelativity, Moscow, Progress, 1987.

IEEE Antennasand Propagation Magazine, Vol. 51, No.3, June 2009

39. M. A. B. Whitaker, "History and Quasi-History in PhysicsEducation - Part 1," Physics Education, 14, 1979, pp. 108-112.

40. K. T. Selvan, "An Approach for Harmonizing Engineering andScience Education with Humaneness," Science and EngineeringEthics, 10,3,2004, pp. 573-577.

41. K. T. Selvan, "Comments on 'On "Bettering Humanity'" inScience and Engineering Education," Science and EngineeringEthics, 14,2,2008, pp. 291-293 .

42. K. T. Selvan and L. Ellison, "Incorporation of Historical Con-text into Teaching: Student Perception at The University ofNottingham Malaysia Campus," IEEE Antennas and PropagationMagazine, 49, 5, October 2007, pp. 161-162.

43. K. T. Selvan, "Integration of Historical Content into Engineer-ing Teaching: Possible Benefits and Perceptions," Proceedings ofthe IEEE Applied Electromagnetics Conference, Calcutta, India,December 2007.

44. J. B. Temes, "Teaching Electromagnetic Waves to ElectricalEngineering Students: An Abridged Approach," IEEE Transac-tions on Education, 46, 2, May 2003, pp. 283-288 .

45. A. R. Irwin, "Historical Case Studies: Teaching the Nature ofScience in Context," Science Education, 84, I, pp. 5-26, 2000 .

46. J. W. Arthur, "The Fundamentals of Electromagnetic TheoryRevisited," IEEE Antennas and Propagation Magazine, 50, 1, Feb-ruary 2008, pp. 19-65.

A7. L. Jamieson, "Engineering Education in a Changing World,"lEe DesignCon Conference, January 2007.

48. V. K. Arora and L. Faraone, "21st Century Engineer-Entrepre-neur, " IEEE Antennas and Propagation Magazine, 45, 5, October2003,pp.l06-114.

Introducing the Feature Article Author

Krishnasamy Selvan was born in Kumbakonam, India, in1966. He obtained his BE (Honors) degree from the MaduraiKamaraj University, Madurai, India in 1987; the MS degree fromthe Birla Institute of Technology and Science, Pilani, India, in1996; and the PhD (Engineering) degree from the Jadavpur Uni-versity, Calcutta, India, in 2002.

From January 1988 to February 2005, Dr. Selvan was withthe Society for Applied Microwave Electronic Engineering andResearch (SAMEER) - Centre for Electromagnetics, Madras,India . There, he was essentially involved in antenna analysis ,design, and testing. During 1994-1997 , he was the Principal Inves-

45


tigator of a collaborative research program that SAMEER had withNIST, USA. He was later the Project Manager/Leader of some suc-cessfully completed antenna-development projects. In early 1994,he held a two-month UNDP Fellowship at RFI Industries, Austra-lia. Since February 2005, he has been an Associate Professor in theDepartment of Electrical and Electronic Engineering, the Univer-sity of Nottingham Malaysia Campus. There, he leads the researchactivities of the Applied Electromagnetic Research Group. He isalso with the George Green Institute for Electromagnetic Research,University ofNottingham, UK, as an associate member.

Dr. Selvan's original contributions have been chiefly in theareas of hom antennas, antenna metrology, and engineering educa-tion. In these areas he has authored or coauthored a number ofjournal and conference papers. He is on the editorial boards of theInternational Journal on Antennas and Propagation and the Inter-national Journal of RF and Microwave Computer-Aided Engi-neering. He has been a reviewer for the IEEE Transactions onAntennas and Propagation, the IEEE Antennas and PropagationMagazine, lET Microwaves Antennas and Propagation, and theJournal ofElectromagnetic Waves and Applications. He is a mem-ber of the Education Committee of the IEEE Antennas and Propa-gation Society. He was the Secretary of the Organizing Committeeof the International Conference on Electromagnetic Interferenceand Compatibility 2003, held in cooperation with the IEEE EMCSociety. Dr. Selvan is a Senior Member of the IEEE and a Fellowof the Higher Education Academy (UK). His interests include thephilosophy of science and Indian carnatic music. @)

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