The first half of the nineteenth century was marked by the discovery of a variety of new phenomena in electricity and magnetism, and the general task to which Maxwell then addressed himself, beginning in the 1850s, was the development of a unified electromagnetic theory that would incorporate all of these phenomena. Section 1 of this chapter explores the broad background in nineteenth-century electromagnetic experiment and theory for Maxwell's work. Section 2 deals with the particular scientific and mathematical education and training that Maxwell brought to his work in electromagnetic theory, including his educational experiences at the universities of Edinburgh and Cambridge, as well as the epistolary tutelage of William Thomson. Finally, Section 3 deals with the methodological approach – growing out of the combined Scottish and Cambridge backgrounds – that Maxwell brought to his work.

Electricity and magnetism from Oersted to the 1840s

The discovery of the magnetic effects of electric currents, announced by Hans Christian Oersted in July 1820, defined the agenda for the study of electricity and magnetism during the following decades. Viewed in broad terms, Oersted's discovery had implications in two directions: First, Oersted's research had been undertaken in the context of widespread preoccupation – following Alessandro Volta's invention of the voltaic pile in 1800 – with the connections between electric currents and other physical phenomena, including chemical effects, heat, and light.

As he pursued the task of constructing a unified account of electromagnetic phenomena from a field-theoretic point of view – from his initial explorations under Thomson's tutelage in 1854 through his work on a second edition of the Treatise on Electricity and Magnetism in the months before his death in 1879 – Maxwell was unwavering in his basic commitment to a broad mechanical framework, within the confines of which this task was to be carried out. Within this broad mechanical framework, however, there were various methodological options at Maxwell's disposal – traceable to his experiences at both Edinburgh and Cambridge, and also to his interaction with William Thomson – and Maxwell was to make full use of this variety of options, in response to the shifting needs of his evolving research program. In brief, Maxwell started out using an analogical approach to mechanical representation, rooted in Scottish skepticism and reflecting a desire to proceed with minimal physical commitment at the outset; in this context, he presented the mechanical images in his first major paper on electromagnetic theory, “On Faraday's Lines of Force” (1855–6), as purely illustrative, with no claim whatever to realistic status. Subsequently, responding to William Thomson's judgment that the time had come to go beyond mere analogy in electromagnetic theory and to begin the task of constructing a realistic mechanical theory, Maxwell developed his molecular-vortex representation of the electromagnetic field in the paper entitled “On Physical Lines of Force” (1861–2).

The historian of science cannot be unmindful of the fact that, for better or for worse, science, as practiced now and in the past, furnishes one of our central models of rational thought and judgment. Some would use history to demonstrate the worthiness of science as a model for rationality; others would use history to demonstrate the limitations of science in this respect. Above all, however, awareness of the paradigmatic role of science urges the historian of science to seriousness of purpose in trying to delineate and understand the practice of science in the past. In particular, I here endeavor to delineate and understand, in historical context, James Clerk Maxwell's seminal work in electromagnetic theory. Understanding of matters of any significance, however, seems never to come easily: This book has been long in gestation, and it makes some demands of its reader.

I was introduced to the historical study of Maxwell by Martin Klein, whose work and counsel have been seminal for me.

Jed Buchwald, Francis Everitt, Peter Harman, John Heilbron, Ole Knudsen, and David Wilson have furnished ideas, sources, and encouragement beyond what scholarly citations can acknowledge.

My work has been enriched by the conversation as well as the scholarly publications of Joan Bromberg, Geoffrey Cantor, Alan Chalmers, Michael Crowe, Gregory Good, John Hendry, Jonathan Hodge, Robert Kargon, Donald Moyer, Richard Olson, Paul Theerman, and Norton Wise.

James Clerk Maxwell made momentous contributions to the development of electromagnetic theory: In formulating the set of equations that bear his name, he established a systematic and enduring foundation for modern electromagnetic theory; in developing the formalism to embrace optics, he demonstrated the range and power of his mathematized field theory, adumbrating its profound implications for subsequent developments ranging from relativity theory to communications technology. Maxwell's activity in this area spanned a period of twenty-five years – from the mid-1850s until his death in 1879 – and his thinking on the subject was developing and changing throughout that period. It is possible, nevertheless, to identify one crucial period of innovation: a period of about one year, centering on the summer of 1861, during which Maxwell was working on, and publishing in successive installments, a paper entitled “On Physical Lines of Force.” It was during that period that Maxwell modified one of the fundamental electromagnetic equations through the introduction of a new term called the displacement current, thereby rendering the set of foundational equations complete and consistent; and it was also during that period, in conjunction with the introduction of the displacement current, that Maxwell took the crucial first steps toward the unification of electromagnetism and optics.

The molecular-vortex model provided the context for the first appearance of both the displacement current and the electromagnetic theory of light. The first form of the electromagnetic theory of light – which differs from the modern form no less than the first form of the displacement current differs from its modern counterpart – appeared in Part III of “Physical Lines,” published in January 1862. In brief, the newly introduced elastic property of the magnetoelectric medium allowed for the propagation of transverse shear waves in that medium; calculating, from the parameters of the model, the velocity of such waves – and finding close agreement with the measured velocity of light – Maxwell identified these waves in the magnetoelectric medium as light waves, and he concluded that the magnetoelectric and luminiferous media were one and the same. The broad nineteenth-century background bearing on such a connection between electromagnetism and light is taken up in Section 1 of this chapter. Section 2 then broaches the question of the precise role of the molecular-vortex model in the origin of the electromagnetic theory of light: Was the electromagnetic theory of light, as textbook accounts might suggest, from the outset basically a matter of deriving wavelike solutions from the equations of electricity and magnetism – in which case the mechanical model could have played at most an ancillary role in the genesis of the theory – or did the molecular-vortex model in fact play a more essential role in the initial formulation of the electromagnetic theory of light, as suggested by the fact that the theory made its first appearance in Part III of “Physical Lines”?

The immediate context for Maxwell's initial modification of Ampère's law (Ampère's circuital law in differential form), through the introduction of a new term to be known as the “displacement current,” was, as we have seen, his work on the theory of molecular vortices: His proximate aim in modifying Ampère's law was to extend the theory of molecular vortices to electrostatics, and his explicit interpretation at that point of the modified equation was as a mechanical calculation in the theory of molecular vortices, with the new term expressing the flux of the small idle-wheel particles owing to progressive elastic deformation of the vortices. All of the principal symbols and equations in “Physical Lines,” however, had dual significance – mechanical and electromagnetic – and the modified Ampère's law, in its electromagnetic character, had broader connections and significance, transcending its proximate matrix in the theory of molecular vortices. That broader context must be taken into account if we are to achieve a full understanding of the origin of the displacement current and its significance in the history of electromagnetic theory.

The question of the origin of the displacement current has been, and continues to be, the object of much interest and concern: Each year many thousands of students in physics courses throughout the world learn that Maxwell, on the basis of theoretical considerations, modified Ampère's law, through the introduction of a new term called the displacement current, and thereby perfected the enduring foundation for modern electromagnetic theory.

Few working papers survive from the period when Maxwell was working on “Physical Lines.” My own search of relevant archives, as well as the more exhaustive search conducted by Peter Harman in connection with his edition in progress of The Scientific Letters and Papers of James Clerk Maxwell, 3 vols. (Cambridge University Press, 1990–), turned up nothing beyond the material to which attention is directed by A. E. B. Owens's handlist to Add. MSS 7655 at the University Library, Cambridge. Of a set of five folios constituting Add. MSS 7655, V, c/8, two folios clearly correspond to the period when Maxwell was working on “Dynamical Theory” (see Appendix 2), and a third, dealing with “Helmholtz's Wirbelfäden [Vortex Filaments],” appears to date from a later period as well (1864–70 – Peter Harman, private communication; see also Letters and Papers of Maxwell, 2). The two remaining folios (both blank verso) evidently are associated with “Physical Lines.” One of these clearly relates to the treatment of motional electromotive forces that appears in “Physical Lines,” Part II, 476–85; this draft fragment refers explicitly to “equations (55),” which appear on p. 476 of the published version, and arrives at a form of the published equations (77), on p. 482.

The remaining folio is quite informative concerning various aspects of Maxwell's work on the molecular-vortex model; a photograph of it is presented in Fig. A 1.1, and I here transcribe it in full (cf. also the transcription in Harman, ed., Letters and Papers of Maxwell, 1 693).

Manuscript material relating to “Dynamical Theory” and of interest in connection with Chapter 6, Section 1, includes four pages in the Maxwell manuscript materials at the University Library, Cambridge (in Add. MSS 7655, V, c/8; c/11; and V, f/4), to which I shall make reference as follows: Three pages, apparently representing parts of an early draft of “Dynamical Theory,” and corresponding to pp. 559–61, 568, and 569, I shall denote “[DT, A],” “[DT, B],” and “[DT, C]”; a fourth, apparently representing part of a later draft of “Dynamical Theory,” and corresponding to p. 578, I shall denote “[DT, D].” The correspondences to the cited parts of “Dymanical Theory” are not in doubt; that A, B, and C are earlier is suggested by the numbering of the equations, which differs from the published version, whereas D agrees with the published version in numbering of equations. A and B are pages numbered 22 and 23 (evidently in Maxwell's hand), in V, c/8; C is a page numbered 24, in V, f/4, but helpfully identified in the handlist to Add. MSS 7655 as belonging with A and B. D is in V, c/11. Also of interest is the manuscript of “Dynamical Theory” that was submitted to the Royal Society and is preserved there – PT. 72.7 – to which I shall refer as “Dynamical Theory [MS].” (I am informed that much of this material will be published in Harman, ed., Letters and Papers of Maxwell, 2.)

As we have seen, Maxwell took the molecular-vortex model quite seriously – with ontological intent – when he first presented it in 1861–2, and although he later lost confidence in certain aspects of the model and removed it to the periphery of his research program, he continued in his allegiance to the core hypothesis of the model – that is, to the hypothesis of molecular vortices. The centrality of the molecular-vortex model in Maxwell's general thinking about electromagnetic theory and the particular importance of this model in the background of the displacement current and the electromagnetic theory of light together provide motivation for a careful study of this intricate mechanical model of the electromagnetic field.

Maxwell's work on the molecular-vortex model was guided, above all, by his desire – his commitment – to fashion a coherent and comprehensive theory unifying the full range of electromagnetic phenomena from the field-primacy point of view. This was required in order to produce a credible alternative to Wilhelm Weber's unification of electromagnetic theory within the charge-interaction framework; comprehensiveness and coherence were required also in connection with the intended realistic status of the theory – Scottish and Cambridge methodologies converged on this requirement.

Three perennial issues in Maxwell scholarship have woven their way through our study of the origins of the displacement current and the electromagnetic theory of light in the context of the molecular-vortex model: The first concerns the relationship between Maxwell's accomplishments and the mechanical worldview, the second addresses the role of the field-primacy approach in the genesis of Maxwell's innovations, and the third concerns the unity and coherence of Maxwell's mechanical models and mathematical formalisms.

Maxwell and the mechanical worldview

We have seen that Maxwell's stance with respect to mechanical models and his use of them was conditioned by the confluence, in his educational background and scientific training, of Scottish (Edinburgh) and Cambridge traditions, with the former inclining toward an analogical interpretation of mechanical representations, and the latter toward a more ontologically committed approach, in which mechanical hypotheses were viewed as candidates for reality, and evidence of a hypothetico-deductive character was accepted as providing support for their realistic status.

We have examined the movement of Maxwell's own ideas and practices concerning mechanical modeling: from an initial reliance on mechanical models as heuristic physical analogies – in the Scottish, skeptical vein – thence to the installation of the molecular-vortex model as the basis for a realistically intended physical theory – as countenanced in Cambridge methodology – and finally to an attenuated mechanism – representing Maxwell's own, carefully balanced position - in which the physical universe continued to be viewed as ultimately mechanical, but the possibility of coming to know the details of the mechanism receded indefinitely.

The theory of molecular vortices had constituted the focus of Maxwell's research program in electricity and magnetism in the late 1850s and early 1860s, and his two major innovations of that period – the introduction of the displacement current and the treatment of electromagnetism and optics within a single theoretical framework – grew out of the theory of molecular vortices and reflected that context in their initial formulations. In the course of Maxwell's elaboration of the molecular-vortex model, however, problems had accumulated, to the point that he had serious reservations concerning certain parts of the model. In addition, Maxwell's research program in the theory of heat and gases was, in the years around 1860 and thereafter, developing in such a way as to undermine support for the theory of molecular vortices in that area, which had been its original stronghold. Finally, and relatedly, Maxwell's general views on the use of mechanical models in science were developing in a new direction that involved less emphasis on specific and concrete models. All of these factors converged in encouraging Maxwell to begin a measured retreat from the molecular-vortex model.

As part of this general retreat from the model, Maxwell took steps to free his signal innovations in electromagnetic theory from their original matrix in the theory of molecular vortices. The modification of Ampère's law and, more significantly, the incorporation of optics into electromagnetic theory defined new research programs, based on those innovations.

Figures 4.8 and 4.9a depict schematically the vortex rotations and idle-wheel translations associated with a uniform current density inside of a long, straight wire with uniform circular cross section. Inside the wire, the magnetic field grows linearly with distance from the axis; because of this, neighboring vortices rotate with different angular velocities; this engenders motion of the idle-wheel particles interposed between the vortices, constituting a nonzero current density J; and the inhomogeneity of the magnetic field H is associated with a nonzero value for curl H, which is equal to the nonzero current density J.

Outside of the wire, the H field falls off as 1/r, where r is the distance from the axis [E. R. Peck, Electricity and Magnetism (New York: McGraw-Hill, 1953), 214–17]. One might, at first thought, expect that because of this, neighboring vortices would rotate with different angular velocities, and this would engender motion of the idle-wheel particles, constituting a nonzero current density J. Even though the magnetic field H is inhomogeneous, however, curl H and hence curl ω* are zero outside the wire, and Maxwell's calculation leading to equation (3.7a) shows that in this situation there will be no net flux of the idle-wheel particles, and hence no current ι or J.