The Genesis of Maxwell's Equations

distinguished scientists, drawn from all areas of Cavendish, seventh Duke of Devonshire, who science, engineering, and medicine. The Society served as Chancellor of the University and donated From Electromagnetism to the was founded in 1660 to recognize, promote, and money for the construction of the laboratory. Prof. support excellence in science, and to encourage the James Clerk Maxwell was a founder of the lab, and development and use of science for the benefi t of became the fi rst Cavendish Professor of Physics Electromagnetic Field: humanity. The Society has played a part in some of in 1871. The Duke of Devonshire had given to the most fundamental, signifi cant, and life-chang- Maxwell, as Head of the Laboratory, the manu- The Genesis of Maxwell’s Equations ing discoveries in scientifi c history. Royal Society scripts of Henry Cavendish’s unpublished “Electri- scientists continue to make outstanding contribu- cal Works.” The editing and publishing of these was tions to science in many research areas. The Royal Maxwell’s main scientifi c work while he was at the Ovidio Mario Bucci Society is the national academy of science in the laboratory. Cavendish’s work aroused Maxwell’s UK, and its core are its Fellowship and Foreign intense admiration, and he decided to call the University of Naples Federico II, Naples, Italy Membership, supported by a dedicated staff in laboratory (formerly known as the Devon shire E-mail: [email protected] London and elsewhere. The Fellowship comprises Laboratory) the Cavendish Laboratory, and to thus the most eminent scientists of the UK, Ireland, and commemorate both the Duke and Henry Cavendish. the Commonwealth. 7. James Clerk Maxwell is buried, with his parents and 6. A photo of the Old Cavendish Laboratory, the his wife, within the ruins of the Old Kirk (1592), Department of Physics at the University of Cam- which lies in the graveyard of Parton Parish Church Abstract bridge, part of the University’s School of Physical (1834). The Old Kirk, roofl ess, with just front and Sciences. It was opened in 1874 as a teaching side walls, is shown in the photo. Parton is about labo ratory. It was named to commemorate British seven miles by road from Maxwell House at This contribution outlines the main stages of the path that in ten years led James Clerk Maxwell to the introduction of the chemist and physicist Henry Cavendish, for Glenlair. concept of the electromagnetic ﬁ eld, to the formulation of the electromagnetic theory of light, and to the development of contributions to science, and his relative, William the equations we still adopt for the description of electromagnetic phenomena. 1. Introduction xactly 150 years ago, on December 8, 1864, James Clerk As for Newton before him and Einstein after, the main EMaxwell (1831-1879) read to the Royal Society of Lon- motivation behind Maxwell’s effort was metaphysical, i.e., his don his third and last fundamental memoir on electromagnet- adherence to a “world view” alternative to the dominant one, a ism, entitled “A Dynamical Theory of the Electromagnetic vision introduced by Faraday in connection with his studies on Field,” the abstract of which had been submitted on October electromagnetic induction and polarization. The construction 27. In this memoir, published in the Transactions of the Soci ety of a coherent and satisfactory theory based on this alternative the following year [1], the equations bearing his name appeared conception required ten years, and an exceptional intellectual for the fi rst time. effort. Following [3, 4], this paper outlines the main stages of this enterprise, in order to follow and attempt to clarify the This introduced a way of looking at electromagnetic phe- evolution of Maxwell’s thoughts, which led to the introduction nomena that opened completely new conceptual (and practi- of the concept of the electromagnetic fi eld, to the electromag- cal) horizons. Together with his other seminal contribution, the netic theory of light, and to the formulation of those equations kinetic theory of gases, Maxwell modifi ed the body of accepted we still adopt for the description of electromagnetic phenom- theories and physical conceptions, namely what Kuhn [2] ena. called the scientifi c “paradigms” of “normal” sci ence, laying the foundations of today’s vision of a physical world made of particles interacting through fi elds. 2. “Mathematizing” Faraday It must be stressed that the development of both theories Maxwell was introduced to the study of magnetism by was not motivated by new experimental fi ndings that did not William Thomson (1824-1907, Figure 1), the future Lord fi t in the existing scientifi c body of knowledge. In particular, in Kelvin, when he was still an undergraduate student at the Trinity the case of electromagnetism, all the facts known in Maxwell’s College in Cambridge. However, Maxwell’s explicit interest in time had been satisfactorily interpreted within the Newtonian electromagnetism started just after his successful graduation at paradigm, and incorporated in a theoretical frame work of the fi nal honors examination for the BA in Mathematics (the so excellent predictive power. This was indeed exploited and called “Tripos”) in January 1854, when he was nearly 23 years further developed until Hertz’s experimental verifi cation of the old (Figure 2). As a matter of fact, in a letter to Thomson, dated most striking physical consequence of Maxwell’s the ory, i.e., February 20, 1854, we read [5]: the existence of electromagnetic waves. IEEE Antennas and Propagation Magazine, Vol. 56, No. 6, December 2014 299 AP_Mag_Dec_2014_Final.indd 299 12/27/2014 2:04:56 PM 2 Fr=q q4πε r3 1 −+ 12 c 2() dr dt1 c222 r d r dt 12 0 () (1) with ε0 being the vacuum permittivity, r being the vector pointing from 1 to 2, r being its length, and c being the ratio between the units of charge in the electromagnetic and electro- static systems. The fi rst term on the right-hand side of Equa- tion (1) clearly corresponds to the Coulomb interaction, the second is equivalent to Ampère’s law for the force between current elements, while the third accounts for the electromagnetic induction. Maxwell made rapid progress. In a letter to Thomson, dated November 13, 1854 [7], he wrote, ....Then I tried to make out the theory of attractions of currents but tho’ I could see how the effects could be determined I was not satisfi ed with the form of the theory which treats of elementary currents & their reciprocal action....Now I have heard you speak of “magnetic lines of force” & Faradays seems Figure 1. William Thomson in 1852. Figure 2. James C. Maxwell, about 1854. Figure 4. Wilhelm Eduard Weber. to make great use of them....Now I thought that as every current generated magnetic lines & was acted on in a manner determined by the lines thro(ough) Suppose a man to have a popular knowledge of wh:(ich) it passed that something might be done by electrical show experiments and a little antipathy to considering “magnetic polarization” as a property Murphys Electricity1, how ought he to proceed in of a “magnetic fi eld” or space and developing the reading & working so to get a little insight into the geometric ideas according to this view. subject wh(ich) may be of use in further reading? If he wished to read Ampère Faraday &c how should This clearly shows that since the beginning, Maxwell they be arranged, and in what order might he read was strongly infl uenced by Faraday’s conception that the your articles in the Cambridge Journal? transmission of forces is mediated by the action of contiguous particles of matter in the space between charged or magnetized In choosing as a fi eld of enquiry electromagnetism, an bodies, i.e., through the action of “lines of force” in space. area at the forefront of current research, Maxwell was natu rally This commitment to Faraday’s ideas underlies all Maxwell’s drawn to the work of Michael Faraday (1791-1867, Fig ure 3), subsequent work, and was central to the development of his whose extraordinary series of experimental discoveries formed fi eld theory of electromagnetic phenomena. (in Maxwell’s words) “the nucleus of everything electric since 1830.” Maxwell was also drawn to the work of the German Maxwell immediately realized that the fi rst necessary scientists Franz Newmann, Gustav Kirchhoff, and, above all, step to make Faraday’s conceptions acceptable was to show Wilhelm Eduard Weber (1804-1891, Figure 4). Weber had that contrary to what had been generally thought, they were developed a comprehensive explanation of all electromagnetic compatible with a mathematical theory of electromagnetic phenomena in the classic Newtonian framework of a direct phenomena. In fact, in a letter to Thomson, dated May 15, 1855 action at distance between charges and cur rents. These last [8], we read, were conceived according to a hypothesis fi rst formulated in 1845 by the Leipzig professor Gustav Theodor Fechner, as a I am trying to construct two theories, mathematically streaming of opposite charges, traveling with equal velocities identical, in one of which the elementary conceptions in opposite directions. shall be about fl uid particles attracting at a distance while in the other nothing (mathematical) is In modern vector notation and units, which will be adopted considered but various states of polarization, henceforth, Weber’s expression for the force exerted in vacuo tension, etc., existing at various part of space. by a point charge, q1 , on another point charge, q2 , reads [6] The construction of the envisaged theory was carried out dur- ing the summer and autumn of the same year, and led to the Figure 3.

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