Electromagnetic induction: physics, historical breakthroughs, epistemological issues and textbooks
aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b Electromagnetic induction: physics, historicalbreakthroughs, epistemological issues and textbooks
Giuseppe Giuliani
Formerly, Dipartimento di Fisica, Universit`a degli Studi, via Bassi 6, 27100 Pavia, [email protected]
Abstract.
The discovery of Electromagnetism by Ørsted (1820) initiated an“extraordinary decennium” ended by the discovery of electromagnetic induction byFaraday (1831). During this decennium, in several experiments, the electromagneticinduction was there, but it was not seen or recognized.In 1873, James Clerk Maxwell, within a Lagrangian description of electric currents,wrote down a ‘general law of electromagnetic induction’ given by, in modern form andwith standard symbols: E = I l [ ~E + ( ~v × ~B )] · ~dl ; ~E = −∇ ϕ − ∂ ~A∂t In Maxwell’s derivation, the velocity appearing in this equation, is the velocity of theline element ~dl . A modern reformulation of Maxwell’s general law, starts with thedefinition of the induced emf as the E = H l [ ~E + ( ~v c × ~B )] · ~dl , where ~v c is the velocityof the charge. It is shown that, this apparently minor difference, is fundamental. Thegeneral law is a local law: it correlates what happens in the line at the instant t to thevalues of quantities at the points of the line at the same instant t . For rigid circuits,it is Lorentz invariant. If expressed in terms of the magnetic field, it allows – in theapproximation of low velocities – the derivation of the “flux rule”, for filiform circuits.The “flux rule” is a calculation tool and not a physical law because it not always yieldsthe correct prediction, it does not say where the induced emf is localized, it requires ad hoc choices of the integration paths and – last but not least – because, if physicallyinterpreted, it implies physical interactions with speeds higher than that of light.Maxwell’s general law has been rapidly forgotten; instead, the “flux rule” has deeplytaken root. The reasons appear to be various. One is plausibly related to the ideathat the vector potential does not have physical meaning, a stand clearly assumed byHertz and Heaviside at the end of the Nineteenth century. An analysis of universitytextbooks, spanned over one century, assumed to be representative on the basis of theauthors and/or on their popularity or diffusion, suggests that other reasons are related tothe habit of presenting electromagnetic phenomena following the historical developmentand to the epistemological stand according to which physical laws must be derived fromexperiment without the need of recurring, soon or later, to an axiomatic presentationof the matter. On the other hand, the rooting of the “flux rule” has been certainlyfavored by its calculation utility: this practical feature has overshadowed its predictiveand epistemological weakness.In the first decades of the Twentieth century, it was common the idea that someelectromagnetic induction experiments with rotating cylindrical magnets could beexplained also by assuming that the “lines of magnetic force” introduced by Faradayrotate with the magnet. This is a surprising hypothesis, if one takes into accountthe fact that Faraday’s experiments, as repeatedly stressed by him, prove that the“lines od magnetic force” do not rotate. More surprisingly, this hypothesis has beenresumed recently. It is shown that the hypothesis of rotating “lines of magnetic force”is incompatible with Maxwell - Lorentz - Einstein electromagnetism and is falsified byexperiment. Finally, the electromagnetic induction in some recent research papers isbriefly discussed.
1. Introduction
From the point of view of electromagnetic phenomena, the Nineteenth century can becharacterized by two events separated by almost one hundred years: the invention ofthe pile by Alessandro Volta (1800) and the discovery of the electron by Joseph JohnThomson (1897). Volta’s invention has been a cornerstone in the technological evolutionand furnished scientists an unthought tool for generating (nearly) continuous electricalcurrents, thus allowing the investigation of their properties. Thomson’s discovery showedthe discreteness and material (endowed with mass) nature of the electric charge. Inbetween, there has been an intricate maze of experiments, models, theories, philosophicaland epistemological positions. But, at the end, the physicists found a way out. Theyunified the electric and magnetic phenomena, passed from an action at a distance viewto a field centered one and progressively acquired a discrete description of matter andelectricity, so preparing the abandon of the ether.Electromagnetic induction constitutes a fundamental chapter of this history. It wasdiscovered in 1831 by Michael Faraday (1791 - 1867), at the end of an “extraordinarydecennium” begun with Ørsted’s discovery of the effect of an electric current on amagnetic needle (1820) (section 4). During this decennium, in several experiments, theelectromagnetic induction was there, but it was not seen or recognized (section 5). Overmore than twenty five years, Faraday tried to establish a coherent description of theinduction phenomena, based on the idea that there is an induced current when there isan intersection between a conductor and the “lines of magnetic force” in relative motion.Though not developed in mathematical form, this description can be considered as atheory, capable also of quantitative accounts (section 6). In the following years, manyphysicists tackled the problem both on the experimental and the theoretical side. Onthe experimental side, it was not so easy to add novel knowledge to what Faraday hadalready discovered. An exception was the rule discovered by Heinrich Friedrich EmilLenz (1804 - 1865) that the induced current is such as to oppose the phenomenon thatgenerated it [1]. Though qualitative, this rule was conceptually relevant and guided thefollowing theoretical attempts to build up a theory of electromagnetic induction startingfrom experimental data. Among these attempts, those by Franz Neumann (1798 - 1895)and Wilhelm Weber (1804 - 1891) [2, pp. 222 - 232], [3, chapter 2]. However, we mustwait for Maxwell’s
Treatise for finding a general law of electromagnetic induction, derived within a Lagrangian treatment of the currents (section 7). The astonishing feature ofthis law is that it was obtained without knowing what an electrical current is, apartfrom the recognition that it is a “kinetic process”. This limited knowledge is reflectedby the fact that Maxwell’s derivation considers the velocity of an elementary circuitelement instead of the velocity of the charges contained in it. However, the definitionof the induced emf as: E = I l ( ~E + ~v c × ~B ) · ~dl ; ~E = −∇ ϕ − ∂ ~A∂t (1)where ~v c is the velocity of the charge, leads straightforwardly to Maxwell’s law withthe correct velocity in it (section 8). Maxwell’s general law has been rapidly forgotten;meanwhile, the “flux rule” took root, in spite of the fact that it is just a calculation tooland not a physical law (section 8.1). In section 9, we try to understand the reasons forsuch an oblivion, by taking into account also the role played by textbooks.In the first decades of the Twentieth century, it was common the idea thatsome electromagnetic induction experiments with rotating cylindrical magnets couldbe explained also by assuming that the “lines of magnetic force” introduced by Faradayrotate with the magnet. This is a surprising hypothesis, if one takes into accountthe fact that Faraday’s experiments, as repeatedly stressed by him, prove that the“lines od magnetic force” do not rotate. More surprisingly, this hypothesis has beenresumed recently. It is shown that the hypothesis of rotating “lines of magnetic force”is incompatible with standard electromagnetism and that it is falsified also by recentexperiments. Finally, the electromagnetic induction in some recent papers is brieflydiscussed (section 10).The treatment of this matter is preceded by two other sections. The first one isdedicated to the names used in Electromagnetism and to the shifts of their meanings(section 2). The other, to the meaning of the locution ‘physical meaning’, frequentlyused in physical literature (section 3).The discussion on the explanation of electromagnetic induction phenomena hasbeen periodically reopened in spite of the fact that the experimental foundations arewell established. This constitutes a rare case in the history of Physics. This paper triesto understand why.
2. The names and the things
Along the Nineteenth century, new names have been invented to denote new phenomenaor new theoretical entities or to newly label already known phenomena. Theknowledge inherited from the past centuries about electricity and magnetism, has beenrevolutionized by Ørsted’s discovery of the action of a current carrying wire on amagnetic needle. Electricity and magnetism manifested themselves as intertwined; hencethe new term
Electromagnetism , which shined in the title of an Ørsted’s paper [4]. Soonafter Ørsted’s discovery, Amp`ere coined the term
Electrodynamics for describing theinteractions between currents, between currents and magnets and between magnets.Consequently, the interaction between electrically charged bodies or between magneticpoles at rest, described by Coulomb, was denoted as
Electrostatics and
Magnetostatics .The terms
Electromagnetism and
Electrodynamics appear usually in the titles of ourtextbooks with, maybe, the specification of ‘modern’.Faraday, after the discovery of electromagnetic induction, speaks of “volta - electric”induction and of “magnetic - electric” induction depending on whether the inducingagent is a current carrying circuit or a magnet. The term “electromagnetic induction”comes in later. Faraday’s theory of electromagnetic induction is based on the conceptof “lines of magnetic force”, a concept still in use today for describing graphically theproperties of the magnetic field. The lexical renewal is a slow process, often accompaniedby shifts in meaning. Thus Faraday’s
Experimental Researches are in Electricity andMaxwell’s Treatise is titled
A Treatise on Electricity and Magnetism . Maxwell denotedby the term “electric field” “the portion of space near electrified bodies, considered withreference to electrical phenomena” [5, p. 45]; the term “magnetic field” had also a similarmeaning. Our electric field ~E is denoted by Maxwell as “electromotive intensity”, andthe force exerted by ~E on a charged body is called “electromotive force” [5, p. 46]. Theline integral R l ~E · ~dl is called “total electromotive force”. There is some ambiguity here,because the adjective ‘total’ is often omitted and, throughout the Treatise , the term“electromotive force” is used to denote both the “electric field ” and its line integral.In the manuals of the Twentieth century and in the contemporaries ones, it isfound that the magnetic field ~B , recovering a Nineteenth century’s denotation, is called“magnetic induction vector” and the field ~H whose sources are the current densities ~J is called “magnetic field”. To then have to stress that what appears in the expressionof the Lorentz force is the magnetic induction vector and not the magnetic field. Notto mention the conceptual confusion created by the attempt to recall contributions bydifferent researchers in the denotation of a formula, often distorting the history. Thenwe read of the “Faraday - Neumann - Lenz law”, or variants at will. The presence ofLenz is justified by the sign (-) that appears in the formula; but Faraday, with the “fluxrule” has nothing to do. In fact, as we shall see (section 6.1), Faraday developed a local theory of electromagnetic induction, based on the idea that there is an inducedcurrent if there is an intersection between the ‘lines of magnetic force’ and a conductor inrelative motion. The same is true for Franz Neumann (1798 - 1895) whose contributionto the understanding of electromagnetic induction consisted of a formula obtained inthe framework of Amp`ere Electrodynamics, within an action at a distance point ofview. It is true that, with hindsight, one can pick up the vector potential in Neumann’sformula; however, Neumann did not identify it as such. This reminds us that thenames of the things take root in spite of the shift of their meanings. This phenomenonpresents two risks. The conservation of old names in a changed conceptual frameworkmay induce confusion; conversely, to neglect the shift in meaning of a name, mightlead, retrospectively, to wrongly attribute a law or the first use of a concept. Notto neglect the Lorentz force. Often, only the component due to the magnetic field isdenoted by this name; or, sometimes, we speak of the “Coulomb - Lorentz law”. And,considering the subject of this paper, we cannot fail to conclude with Electromagnetism.If it is true that Maxwell’s equations we use today are the same as Maxwell’s (thoughwritten in another form), we can not forget that the Nineteenth century interpretationsof Maxwell’s equations are no longer our’s. In an axiomatic presentation of Maxwell’sequations, the physical dimensions of the fields is given by the Lorentz force, while theinherently relativistic nature of the theory has been highlighted by Einstein. Therefore,not to recall various contributions in one name, but to underline our interpretation of thetheory, we should use the term of Maxwell - Lorentz - Einstein (MLE) Electromagnetism.This term will be used from now on.
3. The meaning of ‘physical meaning’
In physical literature, we often encounter the expression ‘physical meaning’, generallyused without specifying its meaning. In these cases, the meaning of ‘physical meaning’is suggested by the context. A ‘physical meaning’ can be attributed to different things:theoretical entities, physical quantities, concepts, single statements, sets of statements.Here, we propose to use the expression in a technical way, based on clear definitions.Let us begin by briefly recalling that examples of theoretical entities are: materialpoint, rigid body, gas, electron. Instead, physical quantities describe properties oftheoretical entities, or are attributed to theoretical entities, or describe interactionsbetween theoretical entities. For example: the mass describes a property of a materialpoint, a rigid body, a gas or an electron; the velocity, with respect to a reference system,is attributed to a material point, a rigid body or an electron; the force describes theinteraction between two rigid bodies or between two electrons.Some theoretical entities used in the past are no more in use; for example: caloric,ether. It is reasonable to think that, in those times, these theoretical entities hadphysical meaning. Therefore, we should find out criteria that are, at the same time,prescriptive but historically flexible. The following seem to satisfy these conditions: • A theoretical entity has physical meaning if its elimination reduces the predictivepower of a theory (strong condition). • And/Or: a theoretical entity has physical meaning if its elimination reduces thedescriptive or the heuristic capacity of a theory (weak condition).This criterion can be considered as a variant of the one enunciated by Hertz:
I have further endeavoured in the exposition to limit as far as possible the numberof those conceptions which are arbitrarily introduced by us, and only to admit suchelements as cannot be removed or altered without at the same time altering possibleexperimental results [6, p. 28].
A significant example for the application of these conditions is provided by the conceptof ether. In Maxwell’s times, the electromagnetic field was conceived as a mechanicaldeformation of the ether. Maxwell’s theory apart, light was conceived as a wavephenomenon that develops in an ether. So, the ether had physical meaning becauseits elimination would have reduced the predictive power of those theories. For the weakcondition, we can consider the concept of space - time. Special relativity was bornand can still be presented axiomatically without the use of the space - time formalism.However, it is evident that the use of this formalism increases the descriptive capacityof the theory. On the other hand, it can not be denied the heuristic role played by thespace - time concept in the birth of general relativity.For physical quantities, a similar criterion is the following. A physical quantity hasphysical meaning if: • Its elimination reduces the predictive power of a theory (strong condition). • And/or if its elimination reduces the descriptive capacity of a theory (weakcondition).In general, a physical quantity can be measured, at least in principle. An interpretedtheory indicates or suggests which physical quantities are to be measured. Usually,this feature is considered as a sufficient condition for attributing physical meaning to aphysical quantity. However, it is easy to see that this is not the case. For example, inthe item
Ether written for the ninth edition of the
Enciclopædia Britannica , Maxwellshowed how to measure the rigidity of the ether by measuring the intensity of the Sun’slight impinging on the Earth [7]. However, the physical meaning of the ether’s rigidityis not assured by the fact that it can be measured (within Maxwell’s interpretation ofhis theory), but by the first condition above. The above criterion connects intimatelythe physical meaning of a physical quantity to the theories that use it. So, the etherand its rigidity had physical meaning within Maxwell’s theory; but they have lost anymeaning in MLE electromagnetism.In the light of the above criteria, it is worth considering the case of the wave -function. It is a physical quantity that can describe – in principle – the properties ofevery physical system. If one finds the expression of the wave - function of, say, theelectron of the Hydrogen atom, then the wave - function allows to predict the expectationvalues of some specific physical quantities of the electron. However, the wave - functioncan not be measured as such. For this reason, the possibility of being measured can notbe considered as a necessary condition for attributing physical meaning to a physicalquantity.We shall use the above criteria in due time.
4. The discovery of Electromagnetism
At the onset of the Nineteenth century, the idea that electricity and magnetism couldbe in some way connected was in the air. In 1805, Jean Nicolas Pierre Hachette (1769 -1834) and Charles Bernard Desormes (1777 - 1862) after having seen a magnet bar placedon a wooden boat floating on quiet water to align along the magnetic meridian, lookedfor a similar behavior of a large voltaic pile [8]. But they did not observed any rotation.Hachette recalled this experiment also in his (September, 1820) report on the discoveriesby Ørsted and Amp`ere [9]. This experiment appears to us very naive; however, itmust be taken into account that the concept of electrical circuit and that of electricalcurrent as electricity in motion began to emerge only after Amp`ere’s interpretation ofØrsted’s experiment [10]. This situation is unambiguously reflected by the fact that our conducting wire connecting the two poles of a battery, was systematically denoted asthe connecting wire. The static conception of electricity was still undiscussed.The turning point was due to Hans Christian Ørsted (1777 - 1851). His discovery(July, 1820) of the deflection of a magnetic needle by a current carrying wire openedthe way towards the unification of electric and magnetic phenomena [11]. Ørsted’sdiscovery has been by no means accidental. He was convinced of a connection betweenelectricity and magnetism, within an unitary view of the chemical, thermal, electricaland magnetic forces [12, 13]. The idea that Ørsted’s discovery was accidental has beensuggested by Ludwig Wilhelm Gilbert (1769 - 1824), then editor of the
Annalen derPhysik , in the presentation of the German translation of Ørsted’s Latin text. Withthese words: “What all research and effort had not produced, happened by chanceto Professor Ørsted in Copenhagen, during his lectures on electricity and magnetismlast winter” [14, p. 292]. This claim was groundless: in fact, Ørsted had providedno indication as to how the initial observation had been made. This drastic positionhas been traced back to Gilbert’s long cultural battle against a speculative approachto science, typical of the supporters of
Naturphilosophie , among whom, evidently, wasconsidered also Ørsted [12, 13, 15].An year after his discovery, Ørsted, in a work entitled
Considerations onElectromagnetism , tries to disprove Gilbert’s version and dedicates, to this aim, the firstsection entitled
To serve the history of my previous works on this topic [4]. In particular,Ørsted recalls that, in 1812, he had written: “One should try to see if electricity in itsmost latent state can produce some effect on a magnet”. However, between this verygeneric indication and the conception and realization of the experiment, about eightyears had passed by. Indeed, Ørsted admits that: “I wrote this, during a journey;therefore I could not easily start the experiments; on the other hand, the way of doingthem was not at all clear to me at that moment , all my attention being focused onthe development of a chemistry system [4, p. 162, french transl.; italics mine].” And,recalling how the first experiment took place during the lesson, Ørsted writes:
My old belief about the unity of electric and magnetic forces had developed with newclarity and I decided to submit my opinion to experiment. Preparations were madeone day when I was supposed to give a lesson that same evening. Then, I showedCanton’s experiment on the influence of chemical effects on the magnetic state ofiron; I called attention to the changes in the magnetized needle during a storm and conjectured that an electric discharge might have some effect on a magnetic needleplaced outside the galvanic circuit . I decided to do the experiment right away. SinceI was expecting the maximum [effect] from a discharge that produces incandescence,I inserted a very thin platinum wire at the position of the needle, placed under [the wire]. Although the effect has been indisputable, it nevertheless seemed so confusingto me that I decided to investigate the matter further when I had more time available[4, p. 163, french transl.; italics mine].
Although Ørsted gives some details about the apparatus used - the glowing platinumwire - he gives no indication as to what he actually observed. The last sentence adds, ifpossible, more ambiguity: the effect was “indisputable” but “confusing”.A recollection by Christopher Hansteen (1784 - 1873) containing rather interestingdetails, absent from Ørsted’s account, appeared in 1870. It brought water to themill of an accidental discovery. In 1857, that is almost forty years after the eventsnarrated, Hansteen wrote a letter to Faraday. This was published in the volume
Lifeand Letters of Faraday by Henry Bence Jones (1813 - 1873) [16, p. 390]. In theletter, Hansteen describes Ørsted as an “unhappy experimenter”, always in need ofhelp; he then describes a lesson during which Ørsted would have performed the famousexperiment without success. The whole passage is worth reading:
Professor Ørsted was a man of genius, but he was a very unhappy experimenter;he could not manipulate instruments. He must always have an assistant, or oneof his auditors who had easy hands, to arrange the experiment; I have often inthis way assisted him as his auditor. Already in the former century there was ageneral thought that there was a great conformity, and perhaps identity, betweenthe electrical and magnetical force; it was only the question how to demonstrate itby experiments. Ørsted tried to place the wire of his galvanic battery perpendicular(at right angles) over the magnetic needle, but remarked no sensible motion. Once,after the end of his lecture, as as he had used a strong galvanic battery to otherexperiments, he said, “Let us now once, as the battery is in activity, try to placethe wire parallel with the needle;” as this was made, he was quite struck withperplexity by seeing the needle making a great oscillation (almost at right angleswith the magnetic meridian). Then he said, “Let us now invert the direction of thecurrent,” and the needle deviated in the contrary direction. Thus the great detectionwas made; and it has been said, not without reason, that “he tumbled over it byaccident.” He had not before any more idea than any other person that the forceshould be transversal . But as Lagrange had said of Newton in a similar occasion,“such accidents only meet persons who deserve them [16, pp. 390 - 391].”
For a historical reconstruction of this testimony see, for example, [12, 17]. In particular,Stauffer argues that Hansteen’s “testimony” is not direct because, at the time of theevents narrated, he was not in Copenhagen [12, p. 309]; while Kipnis describes ingreater detail the changes, over time, in the appreciation of Ørsted’s work [17, pp. 3- 4]. The motivation that led Hansteen to include this recollection in a rather genericletter addressed to Faraday is not clear. From the text, no motivation emerges for tellingthis episode to Faraday. However, the sentence following the text quoted above reads:
You completed the detection by inverting the experiment by demonstrating thatan electrical current can be excited by a magnet, and this was no accident, but aconsequence of a clear idea. I pretermit your many later important detections, whichwill conserve your name with golden letters in the history of magnetism [16, p. 391].
This sentence, contrasting Ørsted’s “casual” discovery with that rationally sought byFaraday, suggests that Hansteen’s aim was to earn Faraday’s benevolence. Whateverthe case, let us assume that Hansteen’s account is a faithful report of what happened.That Ørsted was a clumsy experimenter is a typical ad hominem argument, irrelevant toany evaluation of what Ørsted discovered. The testimony that, initially, Ørsted wouldhave placed the wire perpendicular to the direction of the magnetic needle appearsplausible for reasons of symmetry. In fact, a correct arrangement - with the wire parallel to the direction of the needle (oriented along the magnetic meridian) - wouldhave implied, if we are allowed to use an anachronistic term, a symmetry breaking: whythe needle should choose between East and West, a priori equivalent, in the absence ofa hardly conceivable hypothesis of an asymmetrical action on the needle by the wire?In the light of these reflections, Ørsted’s reaction of astonishment at the discoverymade with the wire parallel to the needle becomes plausible. If things have unfoldedas Hansteen recounts, Ørsted’s attempt, at the end of the lesson, to try with wire andneedle parallel only demonstrates the experimenter’s tenacity to prove the existence of along - hypothesized phenomenon. If rationally interpreted, Hansteen’s recollection turnsinto a “testimony ” in favor of the thesis that it was not an accidental discovery. It istherefore not clear why Hansteen, at the end inserts the sentence “Thus was made thegreat discovery; and it has been said, not without reason, that “he tumbled over it byaccident”. Anyway, the discrepancy between Hansteen’s detailed account and Ørsted’somissive one - “effect so confused” - of the famous lesson is irreconcilable. Kipnis evensuggests that Hansteen saw nothing and that he simply imagined how a laboratorylesson on the discovery of Ørsted could be carried out [17, p. 9].It seems quite clear that, in these circumstances, it is appropriate to resort toepistemological criteria to find out what Ørsted has discovered. If we agree that the discovery of a phenomenon presupposes:(i) the conscious search for the phenomenon; or its recognition if unexpected(ii) the actual observation of the phenomenon(iii) the subsequent confirmation of the existence of the phenomenonthere can be no doubt that – at the end – Ørsted actually looked for and observed thedeviation of a magnetic needle by a wire connected to a voltaic cell . That the path thatled Ørsted to the discovery was long, uncertain and rhapsodic, is reasonably established;that Ørsted set up an experiment by placing a magnetic needle near a conductor wireconnected to a voltaic pile is beyond any reasonable doubt; that this experiment gavean uncertain outcome during the famous lecture is admitted by Ørsted himself. WhenØrsted repeated and extended his lesson’s experiment, a set of favorable factors notunder the complete control of the experimenter played a fundamental role: sufficientcurrent intensity due in turn to an appropriate combination of the electromotive forceof the battery and the resistance of the wire, associated to an appropriate sensibilityof the magnetic needle. In this sense, Kipnis speaks of the role of casuality, meanwhiledistinguishing this role from the concept of accidental discovery. However, this type of0casuality occurs in many experimental situations.
Ørsted writes:
The first experiments respecting the subject which I mean at present to explain, weremade by me last winter, while lecturing on electricity, galvanism, and magnetism,in the University. It seemed demonstrated by these experiments that the magneticneedle was moved from its position by the galvanic apparatus, but that the galvaniccircle must be complete, and not open, which last method was tried in vain someyears ago by very celebrated philosophers. But as these experiments were made witha feeble apparatus, and were not, therefore, sufficiently conclusive, considering theimportance of the subject, I associated myself with my friend Esmarck to repeatand extend them by means of a very powerful galvanic battery, provided by us incommon [11, p. 273, engl. transl.].
According to Ørsted, these experiments have been carried out in the following way. First,the magnetic needle was left to orient in the Earth’s magnetic field. Then a straightmetal wire connected to a voltaic pile was brought near the needle and parallel to it:the needle’s deviation towards the West or the East depended on the position of thewire, above or under the needle and on the direction of the current flow. The rotationof the wire’s direction in a plane parallel to the South - North direction influenced theamount of the needle deviation; this depended also on the power of the pile and on thedistance between the wire and the needle. From these observations, Ørsted concludedthat it is no matter of attraction or repulsion between the wire and the needle, butthat there must be some rotational effect involved in the interaction. The action of thewire on the needle was not shielded by a series of interposed materials like glass, metal,wood, water and others. Also, the effect did not depend on the material of the wire.Ørsted, attributed the effect to an ‘electric conflict’ which takes place in the conductorand in the surrounding space. Ørsted conceived what we now call an electric currentas an electric conflict , namely as a propagation of decomposition and recomposition ofthe two electric fluids; this electrical conflict somehow diffuses in the surroundings. Ofcourse, this ‘electrical conflict’ is little more than applying a name to a thing.
Ørsted’s experiment could be easily reproduced: it required only a pile, a metal wireand a compass, at disposal in every laboratory. Particularly reactive were the Frenchscientists: among them, Jean - Baptiste Biot (1774 - 1862), F´elix Savart (1791 - 1841)and Andr´e Marie Amp`ere (1775 - 1836). Biot and Savart (October, 1820) refinedØrsted’s experiment giving it a quantitative expression:
Using these methods, MM. Biot and Savart were led to the following result whichrigorously expresses the action experienced by a molecule of austral or borealmagnetism placed at any distance from a very fine straight cylindrical wire, mademagnetic by the voltaic current. Draw from the pole a straight line perpendicular to the axis of the wire: the force exerted on the molecule is perpendicular to this lineand to the axis of wire. Its intensity is proportional to the reciprocal of the distance[18, p. 223; italics mine]. The expression ‘made magnetic’ suggests the idea that the current in some way endowsthe wire with magnetic properties, responsible for the action on the magnetic needle.Soon after, Amp`ere showed that two parallel current carrying wires attract or repeleach other according to the direction (parallel or antiparallel) of the currents. Thisexperiment could have been conceived only outside the common guess: if a currentacts on a magnet, perhaps a magnet acts on a current. Indeed, Ørsted’s discoverysuggested to Amp`ere that the Earth magnetism – which orients a magnetic needle –could be due to currents flowing under the Earth’s crust along the Earth’s parallels.From here, the step towards the hypothesis that the magnetic properties of magnetsare due to native internal currents (molecular currents); these currents are there alsoin magnetizable material, the magnetization consisting in the ordering of otherwisedisordered configurations of molecular currents. This bold, creative hypothesis allowedto unify under the same principle apparently different phenomena: the interactionsbetween magnets and currents reduce to the interactions between currents. Thishypothesis shaped Amp`ere’s later experiments and their interpretation.All these phenomena were about forces exerted by something on something else. ForAmp`ere, this mechanical phenomenology should be described by the same mathematicaltools used by Newton for the gravitational attraction and by Coulomb for the interactionbetween electrical charges or between the poles of magnets.The title of Amp`ere’s most famous work,
Mathematical theory of electro - dynamicphenomena uniquely deduced from experiment appears as a manifesto of how to dophysics on Newton’s track [19]. However, this title - manifesto is an unreliableepistemological account of Amp`ere’s work. In the incipit, Amp`ere writes:
First observe the facts, vary the circumstances as much as is possible, accompanythis first work with precise measurements to deduce general laws based solely onexperiment, and deduce from these laws, regardless of any assumption on the natureof the forces that produce the phenomena, the mathematical value of these forces,that is to say the formula which represents them, such is the course which Newtonfollowed. This course was, in general, adopted in France by the scientists to whomphysics owes the immense progress made in recent times, and it is the one whoserved as a guide in all my research on electrodynamic phenomena. I only consultedexperience to establish the laws of these phenomena, and I deduced the only formulawhich can represent the forces to which they are due; I did not do a research intothe very cause that can be assigned to these forces, well convinced that any researchof this kind should be preceded by a purely experimental knowledge of laws, and bya determination, deduced only from these laws, of the value of the elementary forceswhose direction is necessarily that of the straight line connecting the material pointsbetween which they are exerted [19, p. 2 ].
It is true that Amp`ere wrote the formula of the force exerted by one circuit onanother on the basis of experimental data taken with precise measurements carried2out by varying the circumstances as much as possible. But he did so, within the basichypothesis that these forces are directed along the straight line joining two elementaryportions of the interacting circuits and that these forces obey Newton’s third principle,namely, that they are equal and opposite. Moreover, Amp`ere thought over what acurrent might be in terms of electric fluids. However, his idea of current could notbe expressed in mathematical form, susceptible of experimental test. On the otherhand, the fundamental role of theoretical arguments in framing the experimental datais acknowledged by Amp`ere when, in dealing with his hypothesis of molecular currents inmagnets, he affirms that its reliability is not due to a specific experimental corroborationbut to its power of unifying different phenomena:
The proofs on which I base it, result primarily from the fact that they reduce toa single principle three sorts of actions which all phenomena prove to depend on acommon cause, and which cannot be reduced [to a common cause] in a different way[19, p. 83 - 84].
Here, the three ‘sorts of actions’ are, of course, that of a magnet on another, that of acurrent on a magnet (and viceversa) and that of a current on another one.The fundamental formula written by Amp`ere consists in the expression of the forcedue to the interaction between two infinitesimal elements of current carrying circuits. Itslaborious formulation by Amp`ere has been reconstructed, among others, by Whittaker[2, pp. 87 - 92], Darrigol [3, pp. 6 - 30], Tricker [20, pp. 46 - 55], Graneau [21, pp. 459- 465] and, extensively by Assis and Chaib [22, pp. 27 - 140]. In modern notation, itcan be written as [22, pp. 27 - 29], [23, p. 357]: d ~F = − µ π I I ~r r " ~dl · ~dl ) − r ( ~dl · ~r )( ~dl · ~r ) (2)where d ~F is the force exerted on the element ~dl by the element ~dl and ~r is theposition vector of the element dl relative to the element dl . This formula is usuallycompared with the one, usually referred to as the Biot - Savart law: d ~F = µ π I I r ~dl × ( ~dl × ~r ) (3)These two formulas are different in two basic aspects: in Amp`ere’s formula, the forceis directed along the straight line connecting the two current elements, while in Biot- Savart’s the force is perpendicular to the current element; Amp`ere’s formula issymmetrical with respect the two current elements, Biot - Savart’s is not. However,when the integral forms of the two formulae are considered, i.e. when the force exertedby one circuit on the other (and viceversa) is considered, the two formulae yield thesame result. Both formulae yield the same result also for the force exerted by a completecircuit on an element of another circuit [20, pp. 55 - 61], [23, pp. 361 - 362], [24]. Ofcourse, in this last case, Amp`ere’s formula yields a force that is perpendicular to thecurrent element, as Biot - Savart’s does.This is one of the cases in which two different theories yield the same predictionstestable by experiment. In such cases, the choice between the two theories is made on3the basis of epistemological criteria. Amp`ere’s formula has been constructed on thebasis of experiments within a Newtonian theoretical framework and does not belongto a general theory of electromagnetic phenomena. Instead, Biot - Savart’s formulacan be deduced within an axiomatic development based on Maxwell equations and theexpression of the Lorentz force (Appendix B). Hence, nowadays, Amp`ere’s formula hasonly a historical interest unless one is willing to look for a theory alternative to MLEelectromagnetism [25].A very different fate has been that of Amp`ere’s hypothesis of molecular currents. Inspite of its qualitative feature, Amp`ere’s hypothesis has been quantitatively embeddedin standard Electromagnetism, by translating the magnetic properties of magneticmaterials into volume and superficial current densities, through the equations: ~J m = ∇ × ~M (4) ~J s = ~M × ˆ n (5)where ~M is the magnetic moment per unit volume. It is an ironic twist of fate that – inspite of Amp`ere’s Newtonian epistemology – the part of his theoretical work which havesurvived is the one based on his bold, creative hypothesis and not the other, masterfullycast in mathematical form.
5. Electromagnetic induction: an unseen guest in the laboratories (1820 -1830)
Ørsted’s discovery triggered a series of experiments around Europe. In some of them,electromagnetic induction was there, but it was not noticed or was misinterpreted.Amp`ere was on the forefront. In July 1821 he carried out an experiment to test whether acurrent carrying circuit can “produce by influence” a current in another circuit. Amp`erenever used, in those years, the verb “to induce” or the term “induction”. However, forsake of brevity, we will use them as synonyms of those used by Amp`ere. If the problemis posed in these terms, its formulation is ambiguous; and the answer of the experimentcan only be equally ambiguous. In fact, for Amp`ere, there are two types of current whichwe can denote as voltaic current , i.e. produced by a voltaic pile and as molecular current ,responsible for the behavior of magnets and magnetizable materials. Amp`ere suspended,with a “very fine metal” wire, a copper ring inside a fixed coil connected to a voltaic pileand placed a magnet (of what type?) near the ring (how?). If molecular currents wereinduced in the copper ring, this would have acquired magnetic properties, detectableby a magnet. However, the magnet would have detected also the induction of a voltaiccurrent . The experiment was intrinsically ambiguous, because to the experimenter’squestion ‘there is something induced’ the answer could be ‘yes, molecular currents ’ or‘yes, voltaic current ’. This ambiguity was never explicitly dissolved by Amp`ere beforethe discovery by Faraday of electromagnetic induction. Furthermore, we must take intoaccount that Amp`ere’s hypothesis of molecular currents did not imply the possibilityof creating them in non magnetizable materials like copper. In other words, Amp`ere’s4expectation was for a negative result of the experiment.In a letter sent to Albert van Beck in 1822, Amp`ere, referring to this experiment,writes:
Anyway, I have carried out in July 1821 an experiment which proves only indirectlythat electric currents [molecular] in the magnet take place around each molecule.Instead, this experiment proves directly that the proximity of an electric current [voltaic] does not excite at all any [molecular or voltaic] by influence, in a coppermetallic circuit, even under the best conditions for such an influence [26, p. 447 -448, fonts and labels in square bracktes mine].
This statement is ambiguous, because of the use of the same term ‘electric current’ fordesignating both molecular currents and voltaic currents . In the passage quoted above,the word ‘any’ can be interpreted as referring to molecular currents . This interpretationoffers a consistent reconstruction of Amp`ere’s interpretation of the experiment and ofhis reaction to the positive result of a second experiment performed one year later inGeneva, where a stronger horse magnet was at disposal. Of course, also the other choiceis legitimate; however, it leads to the conclusion that Amp`ere completely overlooked theimportance of the positive result of the second experiment.The only published report of this second experiment was that of Auguste de la Rive(1801 - 1873), Amp`ere’s young collaborator in Geneva:
By presenting a very strong horseshoe magnet to one side of this ring, we sawit sometimes advance between the two branches of the magnet, sometimes to berepelled, according to the direction of the current in the surrounding conductors.This important experiment therefore shows that bodies which are not susceptible,through the influence of electric currents, to acquire permanent magnetization, suchas iron and steel, can at least acquire a sort of temporary magnetization while theyare under this influence [27, p. 47 - 48]. de la Rive’s report, speaking of ‘temporary magnetization’ of the ring, supports theinterpretation according to which the currents induced in the ring are molecular currents.In other words, the copper ring acquires transient properties typical of a magnet, owningto the induction of molecular currents . In a report’s draft, written just after his returnto Paris but never published during his life, Amp`ere writes:
The closed circuit placed under the influence of the redoubled electric current, butwithout any communication with it, was attracted and repelled alternately by themagnet, and this experience would therefore leave no doubt about the productionof electric currents by influence, if we could not suspect the presence of a little ironin the copper from which the circuit was formed. However, there was no actionbetween this circuit and the magnet before the electric current passed through thespiral which surrounded it; this is why I regard this experiment as sufficient toprove this production. Nevertheless, to prevent any objection, I plan to repeat itimmediately, with a circuit made of highly purified non - magnetic metal. The factthat electric currents can be produced by influence is very interesting in itself, andis besides independent of the general theory of electrodynamic action (Quoted, inoriginal French language by [28, p. 65]). molecular currents , thus confirming de la Rive’s interpretation interms of magnetization of the ring. The last comment on the independence of theexperimental result from his theory suggests that Amp`ere’s main concern was to preservehis electrodynamic theory, molecular currents included. If Amp`ere would have found aninduced voltaic current instead of induced molecular currents , he very likely would haverealized the absolute novelty of the discovery and he would have thoroughly investigatedthe new phenomenon. He did not, thus leaving the state of the experimental knowledgelimited to the superficial observations reported by de la Rive and himself. In particular,both reports do not give any indication of a transient effect due to the switching on andoff of the current in the inducing circuit. Since this effect was the only one present, it is inprinciple possible that the copper ring behaved as an overdamped ballistic galvanometer,keeping its deviation for a time interval much greater than that necessary for establishingthe stationary current in the inducing circuit. This is the conclusion reached by a recentreproduction of the experiment [29]. Though reproductions of historical experiments canillustrate the difficulties encountered by the original authors, it is questionable wetherthey can significantly contribute to the historical reconstructions of past events, whenthe experimenters’ reports are inaccurate as in this case. In fact, de la Rive and Amp`eredo not speak of an almost stationary deviation of the galvanometer, nor of a transientone. They only reported of a deviation. This remind us of the epistemological criteriaused above for Ørsted’s discovery (page 9). Amper`e and de la Rive observed a deviationof the ring. But the deviation should have been transient. Since this transient effecthas not been reported, it is clear that the experimenters did not “actually observedthe phenomenon”, thus leaving no doubt that they completely missed the discovery ofelectromagnetic induction. For detailed discussions of Amper`e’s ‘induction’ experimentsee, for instance, [28, 30, 31, 32]. Therefore, it is not surprising that, after the discoveryof electromagnetic induction by Faraday in 1831, Amper`e’s attempt to vindicate thediscovery failed. In a paper published in 1831, immediately after learning of Faraday’sdiscovery, Amper`e specified that, in the Geneva’s experiment:
We presented to this ring a strong horseshoe magnet, so that one of the poles wasinside and the other outside the ring. As soon as we connected the two ends of theconducting wire to the poles of the battery, the ring was attracted or repelled bythe magnet, depending on the pole that was inside the ring; which demonstratedthe existence of the electric current produced by the influence of the current of theconducting wire [33, p. 405].
This important detail was lacking in de la Rive’s and Amper`e’s reports. Furthermore,a statement about what happened to the ring after the establishment of the inducedcurrent in it, was still missing. It is possible that – in the Geneva’s experiment –they did see a transient deviation of the ring; but to have dismissed this observationas insignificant was a gross mistake. If we take Amper`e’s belated reconstruction ofGeneva’s experiment as trustworthy, we are again led to the conclusion that, in the first6Twenties, Amper`e’s main concern was that of preserving his theory of electric currents,molecular currents included. For an in - depth analysis of Amper`e’s 1831 paper and theensuing correspondence with de la Rive and Faraday, see [30, pp. 209 - 213].In September, 1821, Michael Faraday showed that the action of a magnet on acurrent carrying wire can produce a continuous rotatory motion of the wire around themagnet’s axis [34]. Here, the electromagnetic induction is a secondary effect, masked bythe primary one. We now know that when the wire is set in motion, an induced currentflows in the wire in the opposite direction of the primary current: the induced currenttends to slow down the motion of the wire. In a stationary condition, the electric energyfurnished by the pile is distributed between the wire’s kinetic energy and the losses dueto the Joule effect.On November 22, 1824 Fran¸cois Arago reported to the French Academy that arotating copper disc produces a deflection of a magnetic needle suspended above it inthe same direction of disc’s rotation. If the rotation speed is high enough, the needlefollows the disc in its rotation. A detailed account of this experiment, appeared onlytwo years later, when Arago entered a controversy with Liberato Baccelli (1772 - 1835)and Leopoldo Nobili (1784 - 1835) [35]. At that time, Arago’s experiment had alreadybeen carefully repeated and extended by Charles Babbage (1791 - 1871) and FrederickWilliam Herschel (1738 - 1822) [36]: they demonstrated also the reverse phenomenonby rotating a horseshoe magnet around its axis below a suspended copper disc. Thisastonishing phenomenon, which we now know as due to the eddy currents induced inthe disc, could not find an explanation in those times.
6. Electromagnetic induction: the discovery
On November 24 of the year 1831, Michael Faraday presented to the Royal Society ofLondon a research entitled ‘Induction of electric currents’. In May, 1832 Faraday’s paperon electromagnetic induction was published in the
Philosophical Transactions and wasto open the first volume of his monumental
Experimental Researches in Electricity [37] ‡ . The experiments carried out by Faraday constitute the basic experimental evidenceof currents induced in a closed conducting loop by switching on and off the currentin another nearby circuit (volta - electric - induction) [37, pp. 1 - 7], or by moving aloop towards or away from a magnet (magneto - electric - induction) [37, pp. 7 - 16].These experiments are usually presented in textbook or classrooms as the starting pointfor talking about electromagnetic induction. Joseph Henry had obtained some of theseresults one year before, but published them one year later [39].Faraday’s researches can be considered as an archetype of experimental enquire.A significant experiment, namely an experiment capable of producing new knowledge,stems from the acquired knowledge and is designed to ‘interrogate the Nature’ aboutan unresolved issue. The idea of the experiment can not be logically deduced from the ‡ Faraday’s paper was among the firsts to be submitted to referees before publication in the
Transactions [38].
Experimental Researches . With thewords of Maxwell: “The method which Faraday employed in his researches consistedin a constant appeal to experiment as a mean of testing the truth of his ideas, and aconstant cultivation of ideas under the direct influence of experiment” [40, p. 162, § As I proceeded with the study of Faraday, I perceived that his method ofconceiving the phenomena was also a mathematical one, though not exhibited in theconventional form of mathematical symbols. I also found that these methods werecapable of being expressed in the ordinary mathematical forms, and thus comparedwith those of the professed mathematicians. [5, p. X].
Faraday sought to develop a theory of electromagnetic induction over more than twentyyears. While his basic experiments date back to the first Thirties, his mature reflectionshave been developed in the Fifties. We shall try to present Faraday’s theoreticaleffort by taking into account comprehensively his reflections, independently from theirchronological order. Furthermore, on the wake of Maxwell, we shall try to find out ifand how Faraday’s ideas can be cast in mathematical form.Faraday initially thought that the electromagnetic induction was due to a particularstate of matter named ‘electrotonic state’ [37, §
60, p. 16], but he soon realized thatthis hypothesis was too generic for yielding any experimental prediction § . Therefore,he formulated the idea of magnetic curves or lines of magnetic force and found thatthis concept, conveniently used, could describe the observed phenomena. The lines ofmagnetic force are [June, 1852]: . . . those lines which are indicated in a general manner by the disposition of ironfilings or small magnetic needles, around or between magnets; and I have shown,I hope satisfactorily, how these lines may be taken as exact representants of themagnetic power, both as to disposition and amount; also how they may be recognizedby a moving wire in a manner altogether different in principle from the indicationsgiven by a magnetic needle, and in numerous cases with great and peculiar advantage[42, § § In fact, in the same communication, Faraday writes “Thus the reasons which induced me to supposea particular state in the wire ( §
60) have disappeared. . . [37, § The concept of lines of magnetic force is used for describing ‘the magnetic power’, i.e. – in ourlanguage – the intensity (amount) and direction (disposition) of the magnetic field. Then, itcan be translated into the language of fields by carefully substitute to them our ~B .In considering the motion of a wire in a constant and uniform magnetic field, Faradaywrites [October, 1851]:It is also evident, by the results of the rotation of the wire and magnet ( § § § N intersected. We can translate Faraday’s description in a formula by writing:∆ q ∝ v ⊥ ∆ t ∆ N (6)or in terms of the induced current: i = ∆ q ∆ t ∝ vB sin θ (7)where B is our magnetic field. v f B x yP
Figure 1.
Point P represents a rigid conducting wire placed perpendicularly to thelines of magnetic force which describe a constant and uniform magnetic field. If thevelocity vector is parallel to the lines of magnetic force, the wire intersects no lines;if the velocity vector is perpendicular to the lines, the number of lines intersected ismaximum. In general, the number of lines intersected is ∝ v sin θ . The translation of Faraday’s experimental results into the language of fields contains – withhindsight – some features of the magnetic component of Lorentz force. On the other hand, itstresses the limits of Faraday’s inability to use mathematical tools.The relative motion of conductors and lines of magnetic force has been considered byFaraday also in a different and conceptually intriguing experimental situation [October, 1851]:When lines of force are spoken of as crossing a conducting circuit ( § translation of a magnet. No mere rotation of a bar magnet on its axis, produces any induction effect on circuits exterior to it; for then,the conditions above described ( § § § not being dependent on orassociated with matter ( § § would have their changes transmitted with thevelocity of light, or even with that higher velocity or instantaneity which we supposeto belong to the lines of gravitating force, and if so, then a magnetic disturbanceat one place would be felt instantaneously over the whole globe [42, § independency of the lines ofmagnetic force from matter; and, consequently, that their changes are transmitted with thevelocity of light or instantaneously. This property of the lines of magnetic force allows Faradayto describe, at least qualitatively, all the studied phenomena. In fact [January, 1832]:To prove the point with an ordinary magnet, a copper disc was cemented uponthe end of a cylinder magnet, with paper intervening; the magnet and disc wererotated together, and collectors (attached to the galvanometer) brought in contactwith the circumference and the central part of the copper plate. The galvanometerneedle moved as in former cases, and the direction of motion was the same as thatwhich would have resulted, if the copper only had revolved, and the magnet beenfixed. Neither was there any apparent difference in the quantity of deflection. Hence,rotating the magnet causes no difference in the results; for a rotatory and a stationarymagnet produce the same effect upon the moving copper [37, § therefore a little hole made in the centre of each end to receive a drop of mercury,and was then floated pole upwards in the same metal contained in a narrow jar. Onewire from the galvanometer dipped into the mercury of the jar, and the other intothe drop contained in the hole at the upper extremity of the axis. The magnet wasthen revolved by a piece of string passed round it, and the galvanometer - needleimmediately indicated a powerful current of electricity. On reversing the order ofrotation, the electrical current was reversed. The direction of the electricity was thesame as if the copper cylinder ( § singular independence of the magnetism and the bar in which it resides is renderedevident [37, § § § it moved in theopposite direction across them, or towards the wire carrying the current. Hencethe first current induced in such cases was in the contrary direction to the principalcurrent ( § § § ~E = − ∂ ~A/∂t ,where ~A is calculated from the known values of the current circulating in the inducing circuit.Notice that the use of the vector potential satisfies Faraday’s requirement that the physicalinteraction with every point of the induced circuit must be local: this condition is not satisfiedby the use of the magnetic field through the “flux rule” (section 8.1).Finally, an epistemological note. Speaking of lines of magnetic force that move, Faradaystresses that he is dealing only with a theoretical entity, without any ontological commitmentabout their existence in the world. However, in the Fifties, Faraday returned to this issue,again with a strong methodological warning [March, 1852]: Having applied the term line of magnetic force to an abstract idea, which I believerepresents accurately the nature, condition, direction, and comparative amount ofthe magnetic forces, without reference to any physical condition of the force, I havenow applied the term physical line of force to include the further idea of their physicalnature. The first set of lines I affirm upon the evidence of strict experiment ( § §
7. James Clerk Maxwell
Maxwell’s electromagnetic theory was developed within an image of the world in which theether played a key role. Maxwell himself wrote the entry
Ether for the ninth edition of the
Encyclopædia Britannica in which the electromagnetic field was conceived as a mechanicaldeformation of the ether [7]. Nonetheless, Maxwell’s theory survived the emergence of thediscrete nature of the electric charge, the disappearance of the ether, the birth of specialrelativity and the resurgence of the corpuscular description of light. The intimate reason liesin the fact that, as Hertz put it with a conscious simplification, “Maxwell’s theory is Maxwell’ssystem of equations [6, p. 21]”.The birth of special relativity created no problem for the simple fact that Maxwell’s isa relativistic theory. Indeed, Maxwell’s theory entered implicitly in the axioms of specialrelativity through Einstein’s postulate according to which “light in empty space alwayspropagates with one speed determined c , independent of the state of motion of the bodiesissuers [43, engl. trans. p. 100]”. This postulate can be replaced by: Maxwell’s equationsin vacuum are true. Einstein dedicated to this issue a note in a subsequent paper: “Theprinciple of the constancy of the velocity of light used there [the paper on special relativity]is of course contained in Maxwell’s equations” [44, engl. trans., p. 172]. Indirectly, Maxwell’sElectromagnetism was therefore the cause of the reduction of Newtonian dynamics to anapproximation of relativistic dynamics for low velocities. Furthermore, the predictions ofMaxwell’s Electromagnetism are still valid when the number of photons used is high enough.For instance, this is true for the interference experiments carried out with one photon ata time [45, p. 564]. In particular, the fact that the probability of finding a photon at apoint on the detector is proportional to the electromagnetic intensity at the same point isdue to the same mathematical structure of the electromagnetic and quantum description [46,pp. 12 - 15 ] [47, pp. 166 - 170]. Finally, Maxwell’s equations for vacuum and withoutsources, with the electric and magnetic fields replaced by suitable operators, are at the base ofQuantum Electrodynamics. Perhaps, in the history of physics there is no other theory whichhas remained vital and creative through so many radical transformations. In the introductory and descriptive part of the
Treatise dedicated to electromagnetic induction[40, pp. 163 - 167], Maxwell writes:The whole of these phenomena may be summed up in one law. When the number oflines of magnetic induction which pass through the secondary circuit in the positivedirection is altered, an electromotive force acts round the circuit, which is measuredby the rate of decrease of the magnetic induction through the circuit [40, § E = − d Φ( B ) dt (8)Maxwell does not write this formula, that, instead, can be derived by combining two equationswritten some pages ahead; see below. Equation (8) is the “flux rule”, which is generallyconsidered as the law of electromagnetic induction. However, Maxwell wrote a general law ofelectromagnetic induction . In order to discuss this issue, we need to take a few steps back.As we have seen, physicists learned how to describe mathematically the interactionsbetween current carrying wires without knowing what a current was. This ignorance did notimped Georg Simon Alfred Ohm (1789 - 1854) to establish his laws (in modern notation alongwith Ohm’s): i = E r + R ; X = ab + x (9) i = σ Al ∆ V ; X = k ωl a (10)“where X is the strength of the magnetic action of the conductor whose length is x , and a and b are constants depending on the exciting force and resistance of the other parts ofthe circuit [48, p. 151].” Notice that Ohm speaks of the magnetic action of ‘the conductor’and not of the magnetic action of the ‘current’ and that Ohm measured the current intensitythrough the deviation of a magnetic needle in a torsion balance. Ohm established his laws byusing a Cu - Bi thermocouple as a current source, instead of the voltaic pile used in previousmeasurements. This choice was forced by the instability of the currents produced by a voltaicpile. For discussions of these topics, see, for instance, [49, 50]. These achievements took along time to be recognized, owing to the poor knowledge of the properties of the voltaic pile,in particular of the role played by its internal resistance r ( b in Ohm’s) and owing to thelaborious process of clarification of the meaning of the emf E ( exciting force a in Ohm’s) andcurrent intensity i ( X in Ohm’s). Ohm’s main paper appeared in 1827 [51], but the acceptanceof his laws required many years, as documented also by the delay with which his paper wastranslated in English (1841) [52].Maxwell acknowledges his ignorance about what a current is with these words:The electric current cannot be conceived except as a kinetic phenomenon.[. . . ]The effects of the current, such as electrolysis, and the transfer of electrificationfrom one body to another, are all progressive actions which require time for theiraccomplishment, and are therefore of the nature of motions. As to the velocity of the current, we have shewn that we know nothing about it, itmay be the tenth of an inch in an hour, or a hundred thousand miles in a second. So far are we from knowing its absolute value in any case, that we do not even knowwhether what we call the positive direction is the actual direction of the motion orthe reverse . But all that we assume here is that the electric current involves motion of somekind . That which is the cause of electric currents has been called ElectromotiveForce. This name has long been used with great advantage, and has never led toany inconsistency in the language of science. Electromotive force is always to beunderstood to act on electricity only, not on the bodies in which the electricityresides. It is never to be confounded with ordinary mechanical force, which actson bodies only, not on the electricity in them. If we ever come to know the formalrelation between electricity and ordinary matter, we shall probably also know therelation between electromotive force and ordinary force [40, § F = qE = ma , thus unveiling the ‘the formal relation betweenelectricity and ordinary matter’.The idea that the current is a kinetic phenomenon, allowed Maxwell to treat electricalcircuits with the Lagrangian formalism and to use a mechanical analogy. Considered a systemsof two electrical circuits, he writes that the kinetic energy of the system due to its electricalproperties – i.e its electrokinetic energy – is given by [40, § T = 12 L ˙ y + 12 L ˙ y + M ˙ y ˙ y (11)where ˙ y and ˙ y are the currents in the circuits. In L , L and M we recognize, asMaxwell does, the inductances of the two circuits and their mutual inductance k . Then,the electrokinetic momentum of, for instance, circuit 2 is given by: p = dTd ˙ y = L ˙ y + M ˙ y (12)The physical dimensions of T are those of an energy; the electrokinetic momentum , namely theanalog of mechanical momentum, has instead the dimensions of an electric potential multipliedby time. The role played by T appears clearly when a single, isolated (non interacting withother circuits), circuit is considered. In this case, and with our notations: T = 12 Li ; p = Li (13)Then, T is the magnetic energy of the magnetic field created by the current i and p = Li = Φ,where Φ is the magnetic flux through the circuit. Since Φ = H ~A · ~dl , where ~A is the vectorpotential, we get for the electrokinetic momentum : p = I ~A · ~dl (14) k The analogy between a coil’s self - inductance and the inertial mass of a particle is underlined byFeynman in his
Lectures [53, pp. 17,11 - 17,12]. Feynman considers an ideal inductance L inserted ina circuit. The voltage difference between the two ends of the inductance is given by: V ( t ) = L ( dI/dt ).This equation has the same form of F = m ( dv/dt ). Hence, the analogy between the self - inductanceand the inertial mass of a particle and that between the current and the velocity. Of course, theepistemological stand of the two analogies is very different. For Maxwell, the analogy is a creative one,because it leads to the general law of electromagnetic induction. Instead, for Feynman, the analogyis only an ex post consideration. [Thanks to Biagio Buonaura for having recalled my attention toFeynman’s analogy.] Let us now consider a conducting filiform loop that, at the instant t = 0 is connected to thepoles of a battery. As we now explain to our students, the current which flows in the loop doesnot attain at once its stationary value E /R , owing to the emf self - induced in the loop, emf that contrasts the current increase. The energy supplied by the battery during this process is: E = Z ∞ Ri dt + 12 LI (15)The first term is the energy dissipated in the circuit as heat, the second is the energy T of Maxwell’s equation (13) in steady conditions. Instead of considering the energy balance,Maxwell writes [40, § E = R ˙ y + dpdt ; E = Ri + L didt (16)where we have added our translation of Maxwell’s equation. Maxwell’s comment:The impressed electromotive force E is therefore the sum of two parts. The first, R ˙ y ,is required to maintain the current ˙ y against the resistance R . The second part isrequired to increase the electromagnetic momentum p . This is the electromotive forcewhich must be supplied from sources independent of magneto - electric induction.The electromotive - force arising from magneto - electric induction alone is evidently − dp/dt , or, the rate of decrease of the electrokinetic momentum of the circuit [originalitalics].Therefore, the emf induced in an isolated circuit is given by: E = − dpdt = − ddt I l ~A · ~dl (17)where E has the dimensions of an electric potential, as it should.Naturally, equation (17) is valid also in the case of two or more circuits, as it can be easilyverified. In a section entitled Exploration of the field by means of the secondary circuit [40, p.212], Maxwell treats in detail the case of two circuits in several conditions. Maxwell beginsby recalling that, on the basis of equation (12) the electrokinetic momentum of the secondarycircuit consists of two parts; the interaction part is given by: p = M i (18)Under the assumptions that the primary circuit (circuit 1) is fixed and its current i is constant ,the electrokinetic momentum of the secondary circuit depends only – through the mutualinductance M – on its form and position. Under the above conditions [40, § I l ~A · ~dl = I S ~B · ˆ n dS (19)where ~B is the magnetic field. Now, if the secondary circuit is at rest, by combining equations(17) and (19), we get the “flux rule” (8). The fact that Maxwell does not make this step,confirms that, in his mind, the basic law of electromagnetic induction is given by E = − dp/dt .Indeed, this equation is the starting point for obtaining the General Equations of theElectromotive Force [40, § emf induced in the secondary circuit is given by: E = − dpdt = − ddt I l ~A ( ~r, t ) · ~dl (20) The details of the calculation (omitted by Maxwell) can be found in [54]. It turns out that (inmodern notation): E = − dpdt = I l " ( ~v × ~B ) − ∂ ~A∂t − ∇ ϕ · ~dl = I l ~E v · ~dl (21)where we have added the suffix v to the vector ~E , not present in Maxwell’s text. Maxwell’scomment:The vector ~E v is the electromotive force [electric field] at the moving element dl .[. . . ]The electromotive force [electric field] at a point has already been defined in §
68. It isalso called the resultant electrical force, being the force which would be experiencedby a unit of positive electricity placed at that point. We have now obtained the mostgeneral value of this quantity in the case of a body moving in a magnetic field due toa variable electric system. If the body is a conductor, the electromotive force [electricfield] will produce a current; if it is a dielectric, the electromotive force [electric field]will produce only electric displacement. The electromotive force [electric field] at apoint, or on a particle, must be carefully distinguished from the electromotive forcealong an arc of a curve, the latter quantity being the line - integral of the former.See §
69. [40, § • The first term ~v × ~B is due to “the motion of the particle through the magnetic field [40, § • The second term − ∂ ~A/∂t “depends on the time variation of the magnetic field. This maybe due either to the time - variation of the electric current in the primary circuit, or tomotion of the primary circuit [40, § • The third term ∇ ϕ is introduced “for the sake of giving generality to the expression for ~E v . It disappears from the integral when extended round the closed circuit. The quantity ϕ is therefore indeterminate as far as regards the problem now before us, in which thetotal electromotive force round the circuit is to be determined. We shall find, however,that when we know all the circumstances of the problem, we can assign a definite valueto ϕ , and that it represents, according to a certain definition, the electric potential at thepoint ( x, y, z ) [40, § E = I l [ ~E + ( ~v × ~B )] · ~dl ; ~E = −∇ ϕ − ∂ ~A∂t (22)This means that the induced emf in a filiform circuit is the line integral of the Lorentz forceon the unit positive charge.In commenting his general law (21), Maxwell speaks of the “electromotive force [electricfield] at a point, or on a particle”: we must intend these words as referring to “the force whichwould be experienced by a unit of positive electricity placed at that point”. Therefore, whenhe speaks of the “motion of the particle in the magnetic field”, we are leaned to intend the ‘motion of the unit of positive electricity’ or ‘the motion of the charge’. Anyway, the velocitywhich appears in Maxwell’s deduction of equation (21) is the velocity of the circuit element dl : an interpretation in terms of the velocity of the charge presupposes a model of the electriccurrent which Maxwell did not posses.
8. Electromagnetic induction within Maxwell - Lorentz - EinsteinElectromagnetism
In an axiomatic presentation of MLE electromagnetism, we begin with the set of Maxwell’sequations in vacuum (in the form given to them by Oliver Heaviside): ∇ · ~E = ρε (23) ∇ × ~E = − ∂ ~B∂t (24) ∇ · ~B = 0 (25) ∇ × ~B = µ ~J + ε ∂ ~E∂t ! (26)where: ρ ( x, y, z, t ) is a charge density defined in a finite region of space, ~v is the velocity of ρ , ~J = ρ~v and ε , µ are constants to be specified. This set of equations allows to derivethe values of ~E and ~B , named as the electric and the magnetic field, respectively. In fact, atheorem by Helmoltz assures that a vector field is specified if we know its divergence and itscurl. However, up to now, the electric and the magnetic field are only mathematical symbolswith a name: we must attribute physical dimensions to them. We can do so, by stating that,considered a point charge q , the fields exert a force on the charge given by: ~F = q ( ~E + ~v × ~B ) (27)This assumption establishes the physical dimensions of the two fields, together with those ofthe two constants ε , µ . Equation (27), is the expression of the so - called Lorentz force andleads to new predictions. If the charge is at rest, the Lorentz force reduces to ~F = q ~E . Thislast equation is typical of the approach that starts with the Coulomb interaction between twocharges at rest. Then, by applying the relativistic transformation equations for the fields andthe forces we get the Lorentz force (27).The assumption of the Lorentz force (27) implies that the definition of the inducedelectromotive force in a whatever closed conducting loop l must be [55], [56, pp. 317 - 344,online vers.], [57]: E = I l ( ~E + ~v c × ~B ) · ~dl (28)where ~v c is the velocity of a unit positive point charge and ~E, ~B are solutions of Maxwell’sequations. This integral yields - numerically - the work done by the electromagnetic fieldon a unit positive charge through the entire loop. The emf defined by this equation maybe considered as the translation of Maxwell’s formula (21) into MLE electromagnetism. Thistranslation not only specifies that the velocity appearing in the formula is the velocity of a unitpositive point charge but also grounds the definition of the induced emf on the assumption ofthe Lorentz force. Since: ~E = − grad ϕ − ∂ ~A∂t (29) ( ϕ scalar potential; ~A vector potential), equation (28) assumes the form: E = − I l ∂ ~A∂t · ~dl + I l ( ~v c × ~B ) · ~dl (30)This equation implies that there are two independent contributions to the induced emf : thetime variation of the vector potential and the effect of the magnetic field on moving charges.The terms ∂ ~A/∂t and ~v c × ~B have the dimensions of an electric field. This remark is trivial.However, the two terms imply that an electric field is created at every point of the integrationline and that this electric field is responsible for the electric current that circulates in thecircuit. This physical property makes (30) a local law: it relates the line integral quantity E at the instant t to other physical quantities defined at each point of the integration line at thesame instant t ¶ . If the integration line coincides with a rigid, filiform circuit, equation (28)is Lorentz invariant, as it can be easily seen (Appendix A). By the way, a proper descriptionof the inertial relative motion of a magnet and a rigid conducting loop has been consideredby Einstein as one of the reasons for the foundation of special relativity [41, p. 140, eng.trans.]. Here, we only stress that, while in the reference frame of the magnet is operative thecontribution of the charge moving in the magnetic field, in the reference frame of the loop,the operative term is that due to the time variation of the vector potential. However, bothreference frames apply the same equation: this is a typical relativistic feature.If every element of the circuit is at rest , ~v c = ~v d , where ~v d is the drift velocity of thecharge. (Needless to say, this specification was impossible for Maxwell, because he did nothave a microscopic model for the electrical current). Then, equation (30) assumes the form: E = − I l ∂ ~A∂t · ~dl + I l ( ~v d × ~B ) · ~dl (31)This equation shows that, in general, the drift velocity contributes to the induced emf . Infiliform circuits, the second line integral is null, because, in every line element, ~v d is parallel to ~dl . However, in extended conductors, the drift velocity plays a fundamental role. Particularinteresting is the case of Corbino’s disc, where the application of equation (31) yields themagneto - resistance effect without the need of using microscopic models [57, pp. 3 - 5]. The expressions of the induced emf derived in the previous section contains the vector potential,as in Maxwell’s formula. However, following the tradition, the induced emf can be written alsoin terms of the magnetic field. Starting again from equation (28), we write, in the referenceframe of the laboratory: E = I l ~E · ~dl + I l ( ~v c × ~B ) · ~dl = Z S ∇ × ~E · ˆ n dS + I l ( ~v c × ~B ) · ~dl (32)= − Z S ∂ ~B∂t · ˆ n dS + I l ( ~v c × ~B ) · ~dl where S is any arbitrary surface that has the integration line l as contour. Of course, also thisequation, like equation (30), shows that there are two contributions to the induced emf : the ¶ An equation is local if it relates physical quantities at the same point and at the same instant, or ifit relates physical quantities in two distinct points at two subsequent instants t , t , provided that thedistance d between the two points satisfies the equation: d ≤ c ( t − t ). This locality condition is anecessary, but not sufficient prerequisite for interpreting causally an equation. time variation of the magnetic field and the motion of the charges in the magnetic field. Now,we must use the identity (valid for every vector field with null divergence) [56, pp. 323 - 324,online vers.]: Z S ∂ ~B∂t · ˆ n dS = ddt Z S ~B · ˆ n dS + I l ( ~v l × ~B ) · ~dl (33)where ~v l , the velocity of the line element dl , can be different for each line element. Then,equation (32) becomes: E = − d Φ dt − I l ( ~v l × ~B ) · ~dl + I l ( ~v c × ~B ) · ~dl (34)In the case of a rigid, filiform circuit moving with velocity V along the positive direction ofthe common x ′ ≡ x axis, this equation becomes: E = − d Φ dt − I l ( ~V × ~B ) · ~dl + I l ( ~v c × ~B ) · ~dl (35)Since V ≪ c and v d ≪ c , we put ~v c = ~V + ~v d . This last equation with the equal sign insteadof the ≈ is valid only if c = ∞ . Of course, this position implies the loss of Lorentz invariance.Then, finally: E = − d Φ dt + I l ( ~v d × ~B ) · ~dl = − d Φ dt (36)i.e. the “flux rule”.On the other hand, in the reference frame of the circuit, we have: E ′ = − Z S ′ ∂ ~B ′ ∂t ′ · ˆ n ′ dS ′ + I l ′ ( ~v ′ d × ~B ′ ) · ~dl ′ = − ddt ′ Z S ′ ~B ′ · ˆ n ′ dS ′ = − d Φ ′ dt ′ (37)Hence, the “flux rule” is valid in both reference frames – laboratory’s and circuit’s – only if oneuses the Galilean transformation of velocities ( c = ∞ ). This suggests that the flux rule – forrigid and filiform circuits – may be invariant under a Galilean transformation of coordinates.As a matter of fact, if c = ∞ , the magnetic field has the same value in every inertial frameand we see at a glance that the “flux rule” is invariant under a Galilean transformation ofcoordinates + . It is worth adding that, if the circuit is at rest but not filiform, we get, insteadof equation (36) the equation: E = − d Φ dt + I l ( ~v d × ~B ) · ~dl (38)Up to now, we have just performed mathematical calculations. Their physical interpretationis as follows. The “flux rule” does not satisfy the locality condition. In fact it relates the emf induced in a conducting filiform circuit at the instant t to the time variation – at the sameinstant t – of the flux of the magnetic field through an arbitrary surface that has the circuitas contour. As a consequence, the “flux rule” can not be interpreted causally because whathappens at the surface at the instant t can not influence what happens at the circuit at thesame instant t , unless physical interactions can propagate with infinite speed. Moreover, sincethe integration surface can be arbitrarily chosen among those that have the circuit as contour,we should have infinite causes of the same effect.All the above considerations apart, a comparison between the deduction based on thevector potential and that based on the magnetic field is impressive. The former requires only + As pointed out in [64], there are two Galilean limits of Electromagnetism. For some more details,see on page 43. one passage; the latter is long, cumbersome and full of conceptual pitfalls. Sometimes (often?),historical developments go through bumpy paths.The “flux rule” not always yields the correct prediction. In 1914, Andr´e Blondel showedthat there can be a flux variation without induced emf . Blondel used a solenoid rolled on awooden cylinder placed between the circular ending plates of an electromagnet. An end ofthe coil was connected to another parallel wooden cylinder, placed outside the electromagnet(i.e. in a null magnetic field), in such a way that the coil could be unrolled by maintaining theunrolling wire tangent to the two cylinders [65], [66, p. 873 - 876]. Blondel performed severalexperiments; the one which interests us here is that in which the unrolling of the coil – whileproducing the variation of the flux of the magnetic field trough the solenoid placed betweenthe plates of the electromagnet – does not produce any induced emf . This experiment showsthat the “the flux rule” is only a calculation tool that must be handled with care and not aphysical law. The “flux rule” can not say where the induced emf is localized (section 8.3);in many cases, it yields the correct answer only by choosing ad hoc the integration line [67,pp. 704 - 708], [57, p. 3]; it can not describe phenomena in which the drift velocity plays anessential role.The fact that the “flux rule” is only a calculation tool can be illustrated also by anexperiment in which the induced circuit is placed in a region of space where there is a vectorpotential field but no magnetic field (fig. 2). Though this experiment does not yield newsupport to the epistemological stand attributed to the “flux rule” in the present paper, it isinteresting because it can be considered the classical analog of the Aharonov - Bohm effectand it is denoted as the Maxwell - Lodge effect [68]. ~ S D Figure 2. S is a long solenoid of radius a in which flows a low frequency alternatecurrent I . In the ring D of radius R , placed symmetrically in a plane perpendicularto the solenoid axis, an emf is induced. In the ideal case of an infinitely long solenoid, the magnetic field outside the solenoid is zero.According to the general law of induction (30), the emf induced in the filiform ring is givenby: E = − I ring ∂ ~A∂t · ~dl = − ddt I ring ~A · ~dl = − πR dAdt (39)where R is the radius of the ring. If dA/dt > E <
0: this means that the current in the ringcirculates in the opposite direction of the current in the solenoid. The values of the vectorpotential can be calculated directly from the known value of the current in the solenoid [68].Alternatively, the induced emf (39) can be calculated more easily by applying the “flux rule”: E = − ddt Z S ~B · ˆ n dS = − µ nπa dIdt (40)where S is an arbitrary surface that has the ring as a contour and n is the number of turns perunit length of the solenoid. In the ideal case, the induced emf does not depend on the radius R of the ring. The vector potential outside the solenoid is a vector tangent to a circumference centered on the solenoid axis and directed as the current in the solenoid. Its value can becalculated by equating the last members of the two equations above: A = µ na R I (41)where the constant of integration has been put equal to zero, in order to obtain a vector fieldthat vanishes far away from the source. Also in this case, the “flux rule” relates what happensin the ring at an instant to what happens at the same instant in the portion of the integrationsurface S which intersects the solenoid. Also in this case, this relation can not be interpretedas a causal one because no physical interaction can propagate with infinite speed. If one takesinto account the fact that the length of the solenoid is finite, the magnetic field outside thesolenoid is not null. This implies that the value of the vector potential is smaller than thatcalculated by equation (41) and, consequently, the measured emf is smaller than that givenby equation (39) [68, pp. 253 - 254]. emf Often, instead of (28), the induced emf is defined as: E C = I l ~E · ~dl (42)This definition (‘C’ stays for Coulomb) follows from the definition of the force on a charge atrest: ~F = q ~E (43)While equation (43) is correct, the definition (42), if improperly used, leads to wrongpredictions. In fact, the definition (42) can be used only in the reference frame of a rigidand filiform circuit. In particular, it can not be applied when the circuit or part of it is inmotion.Since: ∇ × ~E = − ∂ ~B∂t (44)it follows that: E C = I l ~E · ~dl = I s ∇ × ~E · ˆ n dS = − I S ∂ ~B∂t · ˆ n dS (45)where S is any (arbitrary) surface that has the integration line l as contour. If the integrationline does not change with time (circuit at rest), we get: E C = − ddt Z S ~B · ˆ n dS = − d Φ dt (46)i.e. the “flux rule”. At this point, one might be tempted to ask what happens if we let thecircuit (or part of it) move. Then, by re - starting from equation (45) and by using equation(33), one gets immediately: E C = − ddt Z S ~B · ˆ n dS − I l ( ~v l × ~B ) · ~dl = − d Φ dt − I l ( ~v l × ~B ) · ~dl (47)This equation is wrong, the correct one being equation (38). The reason is that the definitionof the induced emf (42) lacks the term I l ( ~v c × ~B ) · ~dl (48) coming from the second term of the correct definition of induced emf (28). If we add this term,equation (47) transforms into the correct one (38) or in its approximation for low velocities(36). The above considerations will be useful in the discussion of how electromagnetic inductionis treated in textbooks (section 9). emf ? Einstein, among others, held that this question is meaningless: “Questions as to the ‘seat’ ofelectrodynamic electromotive forces (unipolar machines) also becomes pointless [41, p. 159,eng. transl.].” Instead, we shall show that this question has physical meaning, in the sensediscussed in section 3, by considering the illustrative example of a rigid wire moving along anU shaped conducting frame (fig. 3). T v I B xz ABCD + - Figure 3.
The filiform, rigid wire AB of length a moves with constant velocity ~v alongthe U - shaped frame T in a uniform and constant magnetic field ~B – produced by asource at rest in the reference frame of T – and directed along the positive directionof the y axis. This thought experiment goes back to Maxwell [40, pp. 218 - 219, § emf is due to the magnetic componentof the Lorentz force and is given by E = B y va : the emf is localized in the bar, as it can beshown [56, pp. 325 - 330, online vers.]. Here, we recall only the results. The bar acts as abattery, with the positive pole at the point B . Experimentally, this implies that, in the wireAB, the current circulates from A to B , i.e. from the point at lower potential to that at higherpotential, like in a battery. Instead, taken a pair of points on the frame, the current circulatesfrom the point at higher potential to that at lower potential. The concept of localization of theinduced emf has physical meaning because it allows the prediction – testable by experiment –of how the potential difference between two points is related to the direction of circulation ofthe current. If we drop this concept, we decrease the predictive power of the theory.As it can be easily seen, the flux rule predicts the correct induced emf . The flux of themagnetic field through the area ABCD is given by Φ = B y avt , if at the instant t = 0 ABcoincides with DC and the circulation direction is clockwise. Then E = − B y av , where theminus sign indicates that the induced current circulates counterclockwise, as indicated in thefigure. The “flux rule” can not say where the emf is localized; it can only guess that it mightbe localized in the wire AB because it is moving (on the ground that it is the wire’s movementthat produces the variation of the magnetic field flux and, hence, the induced emf ). However,in the reference frame of AB, the “flux rule” will say that the emf is localized in the oppositevertical arm CD of the frame, because it is moving: this statement is false, because, also in thereference frame of AB, the emf is localized in AB. In fact, in every point of the AB’s frame, there is an electric field – due to the fields’ transformation equations – given by: E ′ z = Γ vB y (49)Therefore, in AB, an emf E ′ = Γ vB y a = Γ E is induced. On the other hand, in CD, theelectric field E ′ z is exactly balanced by the magnetic component of the Lorentz force. Therelation E ′ = Γ E is a particular case of the more general one treated in Appendix A. Ofcourse, for maintaining the wire AB in motion with constant velocity, a mechanical powermust be supplied. It is immediately found that this power is equal to the electrical powerdissipated in the circuit. In spite of Faraday’s statement that the so called unipolar induction is only a particular caseof Faraday’s disc, there is a rather ample literature which holds that the unipolar induction isstill a problematic case (see sections 8.5, 9). For this reason, we shall treat here in some detailthe case of Faraday’s disc (fig 4) and its unipolar version. The Faraday’s disc is easily treatedby using the general law of electromagnetic induction (30). A C’ ABCD N E S I+ - Figure 4.
Faraday’s disc. A conducting disc rotates about its axis with constantangular velocity Ω in the clockwise direction. The disc is immersed in a uniformmagnetic field generated by the cylindrical magnet, electrically isolated from the disc.The magnetic field at the disc is pointing upward. C and D are two sliding contacts.A current I flows in the circuit DEABCD. The radius DC acts as a battery. The sameresults are obtained if the magnet rotates with the disc. If we neglect the drift velocity of the charges in the disc, the general law (30) yields for theinduced emf (the vector potential is constant): E = Z a Ω rBdr = 12 Ω Ba (50)where a is the radius of the disc. The current I is then given by I = E /R where R is theelectrical resistance of the entire circuit. The emf is localized in the radius CD of the disc,with V D > V C . Then, the radius CD acts as a battery. The current in it circulates from thepoint at lower potential C to that at higher potential D. Instead, taken a pair of points inthe external circuit, the current circulates from the point at higher potential to that at lowerpotential, as it should be in an Ohmic element. This treatment is a quantitative one andallows the experimental verification of all the physical quantities involved. Exactly the sameresult is obtained if the magnet rotates with the disc. On the other hand, if only the magnetrotates, the induced emf is null, since is null each of the two line integrals of the general law.The fact that the unipolar induction is only and simply a particular case of the Faraday’sdisc, can be seen by considering figure 5. AB v NS + - Figure 5.
Unipolar induction. The cylindrical, conducting magnet NS rotates aboutits axis with angular velocity Ω. An emf is induced in the radius AB of length a , andits value is given by E = (1 / Ba . In this arrangement, the upper surface of the conducting magnet plays the role of the copperdisc in the Faraday’s disc arrangement. If we move the sliding contact B to a point on thelateral surface of the magnet, nothing changes from the conceptual point of view. If we takeinto account the fact that the magnet is an extended conductor, and – consequently – we takeinto account the drift velocity of the charges, the calculations become much more complex, asshown in [57, p. 4] for the simple case of Faraday’s disc with circular symmetry.Recently, some basic Faraday’s experiments has been reproduced by M¨uller with anarrangement that avoids the complications due to extended materials [58]. The cylindricalmagnet is made up by a stack of ceramic ring magnets: an equatorial gap allows the use ofa rectangular loop, as shown in figure 6. The setup allows to swing linearly or rotationally,singularly or together, several parts of the apparatus. In this way, many Faraday’s experimentscan be reproduced with only one apparatus. to voltmeter E R C mercury cup
I D
Figure 6.
A cylindrical magnet is made up by a stack of ceramic ring magnets:an equatorial gap allows the use of the loop EI + IR + ECD . IR and ECD areconnected through a mercury cup. The cylindrical magnet and the circuits IR and ECD can perform, independently or together, small rotational oscillations. The systemis enclosed in a ferromagnetic frame (not shown in the figure) that essentially shieldsthe circuit
ECD from the magnetic field of the cylindrical magnet. Therefore, nocontribution to the measured emf comes from the circuit
ECD [58].
Let us see how. With small rotational oscillations of IR or IR + magnet or of the magnet,all experiments shown in table 1 are carried out.In particular: • rotational oscillations of IR correspond to experiment I of table 1 • rotational oscillations of the magnet correspond to experiment II What Relative motion Inducedis rotating? wire IR - magnet current
I. Wire IR Yes YesII. Magnet Yes NoIII. Wire IR and magnet No Yes
Table 1.
Phenomena that can observed with M¨uller’s apparatus (fig. 6). The referenceframe is that of the laboratory. • rotational oscillations of IR and of the magnet correspond to experiment III and, atthe same time, to the unipolar induction experiment (since IR is “embedded” into themagnet)The experimental results confirm all Faraday’s ones carried out with the disc or the conductingmagnet (unipolar induction). In particular, the experiment with only the magnet in oscillation(experiment II of table 1) shows that the lines of magnetic force do not rotate , as Faradayshowed about two centuries before. Quantitatively, the emf induced in the radius IR is givenby: E = 12 B Ω( R − R ) (51)where B is some average value of the magnetic field in the gap containing IR . The measuredand the theoretical values are in reasonable agreement. It must be recalled that also Blondelperformed an experiment conceptually equivalent to the Faraday’s disc and verified the validityof equation (51). Precisely, it was experiment IV discussed in [65, pp. 89 - 90] and in [66, p.874]. Finally, also the experiments with the various components oscillating linearly, confirmFaraday’s fundamental distinction between linear and rotational relative motion of magnetand conductor. (do not) move and (do not) rotate Within MLE electromagnetism, the lines of magnetic force are only a way of illustratinggraphically the spatial variations of the magnetic field. Indeed, every prediction of MLEelectromagnetism can be made without using the concept of lines of magnetic force. To putit plainly: Maxwell’s equations are about fields, not about lines of force. This is a case inwhich the criteria enunciated in section 3 can be usefully employed. According to the strongcriterion, the concept of lines of magnetic force had physical meaning in Faraday’s theory ofelectromagnetic induction because its elimination would have reduced the predictive powerof the theory (to zero in this case). Instead, in MLE electromagnetism, the same conceptsatisfies only the weak criterion. In fact, in MLE electromagnetism, while the concept of linesof magnetic force can be dropped without diminishing the predictive power of the theory, itselimination diminishes the descriptive capacity of the theory.Given this, if somewhere in the literature we encounter expressions like ‘rotation of linesof magnetic force’ we should at once recognize an improper use of Faraday’s concept. Improperbecause Faraday repeatedly stressed that – on the basis of his experimental findings – the linesof magnetic force do not rotate.In three popular textbooks of the first decades of Nineteenth century, written by EduardRiecke (1845 - 1915), Henri Bouasse (1866 - 1953) and Orest Danilovich Chwolson (1852 - also by supposing that the linesof magnetic force rotate with the magnet [59, pp. 213 - 214], [61, pp. 374 - 377], [62, pp. 120 -122]. This assertion implies to attribute to the lines of magnetic force a velocity different fromthat of light. This assumption strikingly contrasts with Faraday’s view according to whichthe changes of lines of magnetic force propagate with the velocity of light, or instantaneously.Unexpectedly, the idea of rotating or moving lines of magnetic force has been resumed recently:see, for instance, [63]; or [90, p. 6], where it is considered, in a secondary passage, only as apossibility not excluded in principle. Therefore, it seems worth trying to further clarify thispoint.If a cylindrical magnet rotates in the laboratory frame with angular velocity Ω, we assume– in contrast with Faraday’s view and in contrast with MLE electromagnetism – that at a pointwhose distance from the magnet’s axis is r , a velocity Ω r must be attributed to the field, withthe understanding that the field rotates in the same direction of the magnet: the magnetdrags its magnetic field in its motion. Of course, for appropriate values of Ω and r , Ω r may begreater than c . But we keep going ahead. Next, we must decide which is the effect of the fieldmoving with velocity ~v B on a charge at rest . This means that, instead of the Lorentz force,we could write, for instance: F = q [ ~E + ( ~v c × ~B ) + ( ~B × ~v B )] (52)where ~v c is the velocity of the charge and ~v B that of the magnetic field. For satisfying therelativity principle, this equation should be valid in every inertial frame. Of course, equation(52) is incompatible with MLE electromagnetism. For two reasons: because it modifies thefundamental equation which gives physical meaning to the fields; and because it modifies anequation that has been confirmed by innumerable experiments. Equation (52) has anotherweird property: it contains two terms, those in round brackets, that contradict each other.The former says that the magnetic field exerts no action on a charge at rest; the latter saysthat the magnetic field does. It seems as a sleight of hand. Equation (52) is never written bythe supporters of this heterodox idea, though they use, at will, some pieces of it.We shall see how equation (52) works in two exemplar cases. Firstly, let us consider –in the laboratory reference frame – a positive charge q that is moving with velocity ~v c alongthe positive direction of the x axis, in a uniform and constant magnetic field directed alongthe positive direction of the z axis; the magnetic field is generated by a source at rest inthe laboratory. According to the formula (52), on the charge is exerted a force given by F = q ( ~v c × ~B ) which has only the component F y = − qv c B z different from zero. This force isdue to first term in round brackets; the contribution of the second term is null, since, in thereference frame of the magnet the magnetic field is at rest. In the inertial reference frame co- moving with the charge, the source of the magnetic field and the magnetic field move withvelocity ~v ′ B = − ~v c . Then, on the charge is exerted a force given by: F ′ = q [ ~E ′ + ( ~B ′ × ~v ′ B )] = q [ ~E ′ − ( ~B ′ × ~v c )] = q [ ~E ′ + ( ~v c × ~B ′ )] (53)that – taking into account the fields transformations – has only the y component different fromzero and this is given by F ′ y = − Γ v c B z − Γ v c B z = − v c B z . This force is twice the correctone. Therefore, as expected, the assumption (52) is incompatible with MLE electromagnetism.Of course, if we drop the second term in the round brackets in equation (52), we come back tothe description of MLE electromagnetism. Instead, if we drop the first term in round brackets(i.e. the magnetic component of the Lorentz force) we see that the hypothesis of the rotatingmagnetic field mimics the correct result. However, this is accomplished at the cost of applyingequation (52) only in the reference frame of the moving charge, i.e. at the cost of violating the relativity principle. Therefore, the hypothesis of rotating magnetic field is without anytheoretical foundation.As a second example, let us consider the case of Faraday’s disc in the configuration inwhich the magnet rotates with the disc (fig. 4). As we have seen, the unipolar inductionis only a particular case of Faraday’s disc. So, what is valid for Faraday’s disc is valid alsofor its unipolar variant, considered by Riecke, Bouasse and Chwolson. According to MLEelectromagnetism, there is an induced emf that is localized in the radius CD with V D > V C .Instead, if we consider the alternative description based only on the second term in roundbrackets of equation (52), we can proceed in the following way. We suppose AE to be very closeto BC: then the value of the magnetic field and of its velocity will be approximately the samein every pair of corresponding points of AC ′ and BC. As a consequence, their contributionsto the induced emf will be the same but opposite to each other. The contribution from EDis null, since the magnetic field is at rest in every point and that from AB can be neglectedbecause its length can be reduced at will. Hence, we are left with the contributions comingfrom CD and C ′ E. The contribution coming from CD is null, since the charges in it are not atrest. Therefore, there is only the contribution coming from C ′ E: C ′ E acts as a battery withthe positive pole at C ′ : hence the circulation of the current is clockwise, as it must, and theinduced emf is approximately equal to that calculated by considering the magnetic componentof the Lorentz force operating in the radius CD of the disc. In this way, we see how – inthe limit considered – the idea of rotating magnetic field can predict the correct value of theinduced emf and the correct direction of the induced current. Of course, as we know, a simplemeasurement discriminates between the two descriptions: the emf is localized in the radiusCD of the disc and not in the segment C ′ E. Instead, if we keep both terms in round brackets,the predicted emf will be twice the correct one. Let us now consider the case in which onlythe magnet rotates. In this case, let us assume that AB coincides with CC ′ an C ′ E is faraway from CD. Then, the contributions to the induced emf due to the second term in roundbrackets of equation (52) from the various segments are: zero from DE, because in every point ~v B = 0, zero from CC ′ , because the vector ~B × ~v B is perpendicular to the wire CC ′ ; zerofrom C ′ E because the magnetic field is vanishingly small; equal to (1 / B Ω a from CD. Thisprediction is falsified by the experiment of Faraday and M¨uller: there is no induced emf . Asalready noticed, the null result of this experiment is predicted by the general law (30): eachof its two line integrals is null.If the appearance of the idea of rotating lines of magnetic force in the first decades of thetwentieth century can be figured out in a period in which MLE electromagnetism was still to becompletely assimilated, its nowadays reappearance is hard to comprehend [63]. The aim of thepaper by Leus and Taylor is that of finding an experimental support – support, not proof – ofthe hypothesis of rotating magnetic field. The experimental setup is conceptually identical tothat of Faraday’s disc with the magnet co - rotating with the disc, but with the drawback thatthe values of the magnetic field in a region in which there is a part of the rotating circuit arenot known. These missing data impede a reliable quantitative prediction of the experimentalresults, unless one is intended to ignore the magnetic component of the Lorentz force, as theauthors do. In the introduction of the paper, we find a standard formula according to whichthere are two contributions to the induced emf : the time variation of the magnetic field andthe movement of the charges in the magnetic field. This last term contains, obviously, themagnetic component of the Lorentz force. However, from this point on, the second term inround brackets of equation (52) is introduced as an independent postulate, without noticingits incompatibility with MLE electromagnetism. The paper does not – and could have not –contain new experimental results, with respect to those obtained with the Faraday’s disc co - rotating with the magnet. The description of the induced emf is given by using only the secondterm in round brackets of equation (52), as it has been done above for the Faraday’s disc co- rotating with the magnet, but without considering the alternative description based on themagnetic component of the Lorentz force. Therefore, the authors missed the only possibleexperimental test capable of discriminating between the two description: to measure wherethe induced emf is localized.It may be of some utility to summarize the content of this section in few, basic points: • The concept of lines of magnetic force, fundamental for Faraday’s theory ofelectromagnetic induction, is not a basic one within MLE electromagnetism. It servesonly as a graphical illustration tool. • Faraday conceived the lines of magnetic force as independent from matter. Therefore, thelines of magnetic force do not participate in the motion of matter. Instead, their changespropagate with the velocity of light or instantaneously. • The hypothesis of attributing to the magnetic field a velocity typical of a materialentity is completely extraneous to Faraday’s theory of electromagnetic induction andit is incompatible with MLE electromagnetism. • This hypothesis is falsified by experiment.
9. Why Maxwell’s general law has been forgotten?
We shall try to understand why Maxwell’s general law of electromagnetic induction has beenforgotten and why, instead, the “flux rule” has taken root. Before all, it must be stressedthat Maxwell, after having deduced his general law, did not made any comment about therelation between the general law and the “flux rule”. This lacking comment, has probably notfavored the acceptance of the general law, because the “flux rule” and the general law couldhave been considered by an inattentive reader as somehow equivalent. It has been suggestedthat the elimination of the vector potential by Hertz and Heaviside from the presentation ofMaxwell’s Electromagnetism could have contributed to the oblivion of Maxwell’s general law[66, pp. 871 - 872], [47, p. 91]. This is a plausible hypothesis. However, other factors mayhave contributed. To find out these other factors, we shall look at some university textbooks,considered as significant on the basis of their authors and/or of their diffusion or popularity.Let us begin by looking at the first decades of Nineteenth century, namely far enoughfrom Maxwell’s
Treatise , but only few years after Einstein’s reformulation of Maxwell’selectromagnetism. In those years, three textbooks have reached a wide appreciation andpopularity. The authors were Eduard Riecke, Henri Bouasse and Orest Danilovich Chwolson ∗ . Riecke presents the phenomena of electromagnetic induction by discussing a series ofexperiments, most of them taken from Faraday’s Researches . He discusses the phenomena of‘magnetic induction’, due to the relative motion of a magnet and a conducting loop and thoseof ‘electric induction’, due to the time variation of the current in a circuit, in two different, notconsecutive chapters. The names used are the same as Faraday’s. The theoretical treatmentof the ‘magnetic induction’ is based on the idea that there is induced current in a closed loop if ∗ The unpublished working notes of my coworker Paolantonio Marazzini (1940 - 2013) have guided mein the choice of these three texts. Marazzini, in 2009 began a study on the theoretical description ofelectromagnetic induction from Maxwell’s
Treatise to about 1940. As acknowledged in [66], from thisMarazzini’s research, I came to know about Blondel’s work. the number of lines of magnetic force across the loop changes. This passage signs the abandonof Faraday’s local theory towards the “flux rule”. Quantitatively, this statement is specified bysaying that “If a closed loop moves in a magnetic field, the integral value of the electromotiveforce is proportional to the algebraic difference between the number of lines of force across theloop at the beginning and at the end of the motion [59, § Nature [69]. Its closing sentence was:In taking leave of this treatise we wish to say that students owe much to Prof. Rieckefor giving them a readable, not too abstruse, and yet thoroughly sound and fairlyfull discussion of the elements of physics. To many German students who have nottime to struggle through the larger treatises this book must be very welcome [69, p.567].Riecke’s textbook on Electromagnetism has been reprinted in 1928 [60].The book by Henri Bouasse (1866 - 1953) is more sophisticated, owing to the role reservedto the mathematical treatments. It is also peculiar, for the epistemological and didactic choices.The title of the book is: “
Magnetism and Electricity - Part one - Study of the Magnetic Field ”.Nowhere in the book one can find the word “electric field”, because:. . . in no application, you hear well, in no one, the electric field steps in. Why talkabout it first? In order to avoid any dispute on this subject, the second volume ofthis
Course on Magnetism and Electricity is devoted to applications, the third tothe electric field. The reader will verify that the word electric field is not printedonce in the thousand pages contained in the first two volumes [61, p. XIV; originalitalics].It is true. However, the word is simply omitted, because the electric field is there in theequations. Throughout the entire volume, the term “electromotive force” is used with ourmeaning: it has the dimensions of an electric potential. However, as in Maxwell’s Treatise,the same term denotes what we call “electric field”. After having discussed some typicalexperiments of electromagnetic induction, Bouasse writes:A closed circuit (of invariable or variable shape) moves in a [magnetic] field; thevariation of the flux of the [magnetic] induction through an arbitrary surface, withthe circuit as a contour, creates an electromotive force, whose absolute value in givenin volts by the relation: E = 10 − d Φ dt (54) t represents the temp. The direction of this electromotive force is such that thecurrent which it tends to produce will be the cause of the electromagnetic forceswhich oppose the displacement [of the circuit] [61, p. 275].(I have changed the symbols for uniformity with those used in this paper). Some pages ahead,in a paragraph entitled “Full expression of the electromotive force” [electric field], Bouassewrites the expression of the electromotive force [electric field] at a point of a circuit as the sumof three terms (in modern symbols) [61, p. 322 - 323]: • −∇ ϕ : electrostatic electromotive force [electric field] • − ∂ ~A/∂t : induced electromotive force [electric field] • the electromotive force [electric field] of non - homogeneity (as in the batteries)When the circuit is moving, we must add a fourth term ~v × ~B due to the cut flux of the‘magnetic induction’. Of course, the total electromotive force acting in a circuit, is the lineintegral of the sum of the four terms. Differently from Maxwell – who derived the general lawin a deductive way within a Lagrangian treatment of the currents – Bouasse builds up thegeneral law in a constructive way, piece by piece. Bouasse does not cite Maxwell, coherentlywith his choice of avoiding any quotation.Daniel Abramovich Chwolson (1819 - 1911) wrote a physics textbook composed byfive volumes. It has been translated in French, German and Spanish. The treatment ofelectromagnetic induction appears in the second chapter of the fifth volume, written byAlexander Antonovich Dobiash (1875 - 1932) [62]. The treatment begins with this statement:Consider a wire placed in a magnetic field. When, for any reason, the number oflines of magnetic induction which cross the surface limited by this wire comes tochange, a current start flowing in the wire. Such a current is called induced current.It ceases as soon as the cause which produced it disappears, that is to say as soon asthe number of magnetic induction lines, which pass through its contour, no longervaries [62, p. 44 - 45].This is the “flux rule”. The way in which it is presented strongly suggests a causal relationbetween the flux variation and the induced electromotive force. The following pages arecharacterized by a somewhat blurred discussion of some experiments. Mainly, the induced emf is related to the relative motion between the circuit and the lines of magnetic force, as inFaraday’s theory. From this discussion, emerges the formula ~v × ~B , just for finding out thedirection of the induced emf [62, p. 47]. The case of a time varying magnetic field is cited, butnot treated. At last, the “flux rule” is derived – again – by considering the motion of a circuitin a magnetic field [62, p. 49 - 50]. The chapter, after the treatment of some applications likeself - induction, alternate and eddy currents, ends with the discussion of the unipolar induction[62, p. 120 - 122]. As already recalled and discussed in section 8.5, all three textbooks, indiscussing the unipolar induction, hold that it can be explained both by supposing the linesof magnetic force at rest or rotating with the magnet (Riecke, pp. 213 - 214; Bouasse, pp. 374- 377). For a historical reconstruction of unipolar induction’s issue, see, for instance, [70].After the second world war, the number of textbooks on Electromagnetism has increasedin an impressive way. Therefore, the only viable choice is that of picking up a few texts, mainlyon the basis of their authors and/or their diffusion.The book by John Slater (1900 - 1976) and Nathaniel Frank (1904 - 1984), entitled Electromagnetism , was published in 1947. Electromagnetic induction is treated in three pages,half of them of introductory character. At the beginning of the theoretical treatment, we finda general and careful definition of the electromotive force:By definition, the emf around a circuit equals the total work done, both by electricand magnetic forces and by any other sort of forces, such as those concerned inchemical processes, per unit charge, in carrying a charge around the circuit [71, p.79].However, this premise does not lead the authors to define the induced emf as the line integralof the Lorentz force on a unit positive charge as in equation (28). In fact, soon after, it is stated that the phenomena of electromagnetic induction are described by “Faraday’s law inintegral form” (I have changed the symbols for uniformity with those used in this paper): I ~E · ~dl = − ddt Z S ~B · ˆ n dS (55)When the circuit is fixed and the magnetic field depends on time, it is straightforward toderive from (55), by using Stokes’ theorem, the Maxwell’s equation: ∇ × ~E = − ∂ ~B∂t (56)The path from the ‘Faraday’s law’ (55) (considered as an experimental result) to Maxwell’sequation (56) is typical in many textbooks. Also the reverse path is very common: assumedthe validity of Maxwell’s equation (56), the ‘Faraday’s law’ (55) is obtained. Clearly, for Slaterand Frank, the electromagnetic induction is – from a teaching viewpoint – an unproblematicissue, which does not deserve particular attention. This stand seems typical of textbooks thatare considered by their authors as an exemplification of Theoretical Physics.In order to find a critical approach to electromagnetic induction, we must wait for thesecond volume of Feynman’s Lectures on Physics [53]. Chapter XVII is entitled “The lawsof induction” and is dedicated to the theoretical treatment of the experiments and technicaldevices presented in the previous chapter (“Induced currents”). In this chapter, the “flux rule”is enunciated: the inverted commas are Feynman’s. According to Feynman, we are dealingwith a rule, not with a physical law. Though Feynman does not use our term of “calculationtool”, it is the same thing. The reason lies in the fact that there are “exceptions to the “fluxrule””, discussed in a special paragraph (17.2). Feynman considers two cases: a version ofFaraday’s disc and that of the “rocking plates” and conclude that the “flux rule” does notwork in these cases because:
It must be applied to circuits in which the material of the circuit remains the same .When the material of the circuit is changing, we must return to the basic laws. Thecorrect physics is always given by the two basic laws: ~F = q ( ~E + ~v × ~B ) (57) ∇ × ~E = − ∂ ~B∂t (58)[53, p. 17.3; italics mine].Feynman holds that the “flux rule” can be applied only when the material to which it isapplied does not change. This condition encloses – ex post – the case in which the “flux rule”makes wrong predictions (Blondel’s experiment) and those that need an ad hoc choice of theintegration path. However, the basic reason appears to be more profound and due to the factthat, as shown in section 8.1, the “flux rule” – if physically interpreted – implies physicalinteractions with velocity higher than that of light. Therefore, it can not be a physical law.These Feynman’s pages are quoted and discussed in many (all?) of the following papers dealingwith electromagnetic induction: see, just for a superficial check, two papers separated by morethan forty years [67, 72].Clearly, Feynman and co - authors were unaware of Maxwell’s general law nor did theyknew about Blondel’s experiment. Furthermore, they did not fully develop the consequencesof their last sentence by writing down the definition of the induced emf as the line integral of the Lorentz force on a unit positive charge. Nonetheless, in [73, p. 138], this definition isattributed to them.Feynman’s reflections stimulated directly or indirectly a series of papers onelectromagnetic induction, starting with the one by Scanlon et al [67]. This paper isparticularly interesting owing to its central thesis, taken up also by others [74]. The thesis isthat there are no “exceptions to the flux rule” if only one chooses adequately the integrationline. The search of an integration path that saves the “flux rule” is clearly an ad hoc choicethat intrinsically recognizes that we are dealing with a calculation tool and not with a physicallaw. In fact, a physical law must be valid for any integration path, as (28) does. The authorsare aware of the fact that one should take into account also the drift velocity of the charge,but they dismiss it on account of its smallness [67, p. 699]. However, if we disregard the driftvelocity, we can not deduce – starting from the definition of induced emf (28) – the “flux rule”for filiform circuits in the approximation of low velocities (section 8.1). Nor can we deal withelectromagnetic induction in extended materials [57, pp. 3 - 4].Lev Davidoviˇc Landau (1908 - 1968) and Evgeny Mikhailovich Lifshitz (1915 - 1985),in their Theoretical Physics Series , treated electromagnetic induction in the eight volume(
Electrodynamics of Continuous Media ), in a paragraph dedicated to “The motion of aconductor in a magnetic field” [75]. If a conductor is in motion in a magnetic field withvelocity ~v , the authors consider the reference frame “in which the conductor, or some partof it, is at rest at the instant considered”. Then, taking into account the transformationequations of the fields, they write, in the low velocity approximation: E = I l ( ~E + ~v × ~B ) · ~dl (59)where ( ~E + ~v × ~B ) ≈ ~E ′ and, as usual, the superscript ( ′ ) refers to the reference frame ofthe circuit in motion with respect to the source of the magnetic field, which is at rest in thelaboratory reference frame [75, p. 205]. It is worth seeing in detail the various passages,omitted in the text. We start with the statement that the force on a charge at rest is givenby: ~F ′ = q ~E ′ (60)Consequently, the electromotive force in the reference frame of the circuit is given by: E ′ C = I l ′ ~E ′ · ~dl ′ (61)where the subscript C stays for ‘Coulomb’ and reminds the fact that the force exerted ona point charge is the one contemplated in Electrostatics . Then, by using the transformationequations for the fields and the coordinates, we gets: E ′ = Γ I l ( ~E + ~v × ~B ) · ~dl = Γ E ≈ E (62)At the end of the paragraph, the authors discuss the unipolar induction. They consider thereference frame in which the magnet is at rest, namely a rotating non inertial reference frame.In this frame, the external circuit appears as rotating in the opposite direction. The authorswrite that the induced emf is given by: E = I ext ( ~v × ~B ) · ~dl = − I ext [ ~B × ( ~r × ~ Ω)] · ~dl (63)where ~ Ω is the angular velocity of the magnet in the laboratory reference frame and ~r thedistance of a point of the external circuit from magnet axis [75, p. 209]. Then, the induced emf should be located somewhere in the external circuit and not along a radius of the magnet, as predicted by using the inertial reference frame of the laboratory (figure 5). Furthermore,the induced emf should depend on the form of the external circuit. Instead, the experimentscarried out by M¨uller [58] confirm that the induced emf is localized in a radius of the magnetand its value is given by equation (50) or its variation (51).In Classical Electrodynamics by John David Jackson (1925 - 2016) [76], theelectromagnetic induction is dealt with in a section entitled “Faraday’s Law of Induction”and the treatment is the same as that of the book’s first edition [77, pp. 170 -173]. Afterhaving summarized Faraday’s principal experiments, it is stated that “Faraday attributed thetransient current flow to a changing magnetic flux linked by the circuit” [76, p. 208]. Thishistorical falsehood is common to many texts and is accompanied by the improper denotationof the “flux rule” as the “Faraday’s law”. Then, “Faraday’s law” is written as: E = − k d Φ dt (64)where k is a constant depending only on the choice of the units for the electric and magneticquantities ( k = 1 in the SI units System). Jackson adopts the definition of induced emf (42);as shown in section 8.2, this definition, if applied in the laboratory reference frame to movingcircuits, together with equation (33), leads to the wrong prediction of equation (47). In fact,by applying equation (33), one gets: E = − k d Φ dt = − k Z S ∂ ~B∂t · ˆ n dS + k I l ( ~v × ~B ) · ~dl (65)Since it is assumed that: E = I l ~E · ~dl (66)we have: I l [ ~E − k ( ~v × ~B )] · ~dl = − k Z S ∂ ~B∂t · ˆ n dS (67)Let us further assume that “Faraday’s law” is valid in any inertial reference frame, that theGalilean coordinates transformations are valid and that ~B ′ = ~B [76, p. 209]. Then, in thereference frame co - moving with the circuit, we can write: I l ′ ~E ′ · ~dl ′ = − k Z S ′ ∂ ~B ′ ∂t ′ · ˆ n dS ′ = − k Z S ∂ ~B∂t · ˆ n dS (68)Therefore, by equating the first members of equations (67) and (68): ~E ′ = ~E − k ( ~v × ~B ) (69)This approximated relation between the values of the electric field in the two referenceframes is wrong. However, instead of (69), Jackson writes the correct approximated equation ~E ′ = ~E + k ( ~v × ~B ), owing to an odd interchange of the vectors ~E and ~E ′ . This interchange isjustified by saying that: a) in equation (67) the field is ~E ′ and not ~E since “. . . it is that fieldthat causes current to flow if a circuit is actually present”; and b) in equation (68) the fieldis ~E and not ~E ′ since “Since we can think of the circuit C and surface S as instantaneouslyat a certain position in space in the laboratory” [76, p. 210]. Hence, it is worth developingan alternative derivation based on the definition of the induced emf as the line integral of theLorentz force on a unit positive charge. As shown in section 8, the “flux rule” can be derivedin the approximation of low velocities for filiform circuits (in which the drift velocity is parallelto any elementary part ~dl of the circuit): E = I l [ ~E + ( ~v × ~B )] · ~dl = − d Φ dt (70) Under Jackson’s Galilean approximation we can write, in the reference frame co - moving withthe circuit: E ′ = I l ′ ~E ′ · ~dl ′ = − d Φ ′ dt ′ = − d Φ dt (71)Hence, since ~dl ′ = ~dl : ~E ′ = ~E + ~v × ~B (72)Jackson’s Galilean approximation made at the beginning of the calculation reminds us thatLe Bellac and L´evy - Leblond have shown that, indeed, there are two Galilean limits ofElectromagnetism [64]. These limits come out by letting c = 1( √ ε µ ) → ∞ . This can bedone by keeping ε and eliminating µ = ε /c by letting c → ∞ (electric limit); or, by keeping µ and eliminating ε = µ /c by letting c → ∞ (magnetic limit). More physically, in theelectric limit | ρ | c ≫ | j | , while, in the magnetic limit | ρ | c ≪ | j | . Jackson’s approximation, ifdeveloped by starting with the definition of the induced emf as the line integral of the Lorentzforce on a unit positive charge, falls in the magnetic limit. Recently, this topic has beenresumed and expanded by several authors [78, 79, 80, 81].In Wolfgang Panofsky (1919 - 2007) and Melba Phillips (1907 - 2004) Classical Electricityand Magnetism , “Faraday’s law” of induction is treated in chapter IX, entitled “Maxwell’sequations” [82, p. 158]. This treatment is the same as that contained in the first edition ofthe book [83, 142 - 146]. Considered a circuit of resistance R with an emf E , “Faraday’s law”is stated by the equation: IR − E = − d Φ dt (73)“This means that the current in the circuit differs from that predicted by Ohm’s law by anamount which can be attributed to an additional electromotive force equal to the negativetime rate of change of flux through the circuit. Note that equation (73) is an independentexperimental law and is in no way derivable from any of the relations that have been previouslyused [82, p. 158; italics mine].” From this experimental law, putting − d Φ /dt = H l ~E · ~dl andarguing that this relation holds also in vacuum, Maxwell’s equation for the curl of the electricfield is derived.The discussion of the case of a moving circuit is treated as in Jackson’s textbook, namelyby oddly interchanging the values of the electric fields in the two reference frames [82, pp. 160- 163].The book Electricity and Magnetism by Edward Purcell (1912 - 1997) and David Morin[84] is the third edition of the renowned book by Purcell, published after his death. As inthe cases of Jackson’s and of Panowski and Phillips’ texts, the treatment of electromagneticinduction is the same as in the previous editions. Purcell’s treatment is founded on the ideathat physical laws should be derived from experiment. Therefore, the “flux rule” – Faraday’slaw in Purcell’s denotation – is “derived” step by step from three types of thought experiments,in principle reproducible in laboratory. The first is that of a conducting bar moving in anuniform and constant magnetic field. This thought experiment is treated by using the magneticcomponent of the Lorentz force and it is described in both reference frames: the laboratory’sand the frame co - moving with the bar. The fields transformations, previously derived – theytoo, within thought experiments – are used in the low velocities approximation [84, pp. 345 -346]. Then the motion of a rectangular loop in a non - uniform magnetic field is considered. Inthis way, by using again the magnetic component of the Lorentz force, the “flux rule” is derived [84, pp. 346 - 350]. However, since this derivation has been carried out in a particular case, it isnecessary to state and prove the “flux rule” as a theorem, valid for any constant magnetic field.This can be done by deriving our equation (33) with ∂ ~B/∂t = 0 [84, pp. 350 - 352]. Of course,this can be accomplished by implicitly assuming that the charge velocity coincides with thethe velocity of the circuit element that contains the charge. The more general case, i.e. that inwhich the magnetic field is non uniform and changes with time, is treated by considering threethought experiments [84, p. 355 – 358]. Two laboratory’s tables, separated by a curtain, canbe moved and are setup in the following way [84, fig. 7.15, p. 355]. On table 1 we find a coilconnected to a battery through a variable resistance. On table 2, there is only a conductingloop connected to a galvanometer. If table 2 is moved with velocity v away from table 1, thegalvanometer’s needle on table 2 deviates. If, reestablished the initial configuration, table 1is moved away from table 2 at the same speed v , the galvanometer deviates in the same way.Now, after having reestablished the initial configuration, the current in the coil of table 1 isvaried by varying the variable resistance in such a way that the magnetic field at the coil 2varies as in the experiments I and II. This implies that in coil 2 an emf is induced and this emf is identical to that induced during experiments I and II. Since table 2 does not knowwhat happens to table 1, table 2 sees only the same deflection of the galvanometer in allthree experiments. Since table 2 has seen the same variation of the magnetic field in all threeexperiments, he concludes that they all are described by Faraday’s law: E = I l ~E · ~dl = − d Φ dt (74)if the integration line is stationary , as it is in table 2.It is worth noting that what is measured by table 2 is transparently illustrated by thegeneral law (30) written for table 2: E = − I l ∂ ~A∂t · ~dl + I l ( ~v c × ~B ) · ~dl (75)Since the second line integral is null (for filiform circuits), it turns out that, in table 2, what ismeasured is due to the time variation of the vector potential: no matter if this time variationis due to a variation of the current in table 1 or/and to a relative motion between the twotables.As for the other textbooks so far discussed, in the text by Corrado Mencuccini andVittorio Silvestrini the “flux rule” is considered as the “law of electromagnetic induction”,here denoted as “Faraday - Neumann law”. The induced emf is defined as: E = I l ~E i · ~dl (76)where ~E i is the “induced electromotive field” and its value is given by [85, p. 353]: ~E i = ~Fq = ~E + ~v × ~B (77)where ~F is, of course, the Lorentz force. Moreover, it is specified that we should write ~v = ~v l + ~v d ; however, since in filiform circuits ~v d is always parallel to ~dl , we can safelyuse equation (77), where ~v is the velocity of the line element dl . Overall, the treatment ofelectromagnetic induction follows standard patterns, characterized by the assumption of the“flux rule”, followed by a series of illustrations that consider different experimental setups.The conceptual difference between Mencuccini and Silvestrini treatment and those of othertextbooks, is the fundamental specification that the induced electromotive field is given byequation (77). This specification implies that all written equations are correct, though their theoretical hierarchic stand is obscured, owing to the missing straightforward application ofequations (76, 77).The recent Modern Electrodynamics by Andrew Zangwill [86], is characterized by thefact that there are many references, thus recalling that the matters dealt with in textbooksrely an a large bulk of investigations. The Introduction to Chapter II, entitled “The MaxwellEquations” starts with the statement:All physical phenomena in our Universe derive from four fundamental forces. Gravitybinds stars and creates the tides. The strong force binds baryons and mesonsand controls nuclear reactions. The weak force mediates neutrino interactions andchanges the flavor of quarks. The fourth force, the Coulomb - Lorentz force, animatesa particle with charge q and velocity ~v in the presence of an electric field ~E and amagnetic field ~B : ~F = q ( ~E + ~v × B ) (78)The subject we call electromagnetism concerns the origin and behavior of the fields ~E ( r, t ) and ~B ( r, t ) responsible for the force (78) [86, p. 29].Notwithstanding the recognition of the basic role of Lorentz force, the approach toelectromagnetic induction does not start with the definition of the induced emf (28), butwith the integration of Maxwell’s equation (44) [86, p. 462]. This procedure is similar to thatof section 8.2, but with the fundamental difference that the line integral H l ~E · ~dl is not assumedas the definition of the induced emf in a circuit. This entails that the following equations arecorrect. The correct definition of what Zangwill denotes as “Faraday’s electromotive force”is resumed soon after within the approximation of low velocities and the introduction of thedrift velocity of the charges, getting in this way equation (38) for the induced emf [86, p. 463].This treatment privileges the basic Maxwell’s equation (44) as the starting point. In doing so,it gives up the straightforwardness of the path which starts with the definition of the induced emf as the line integral of the Lorentz force on a unit positive charge.The textbooks taken into consideration confirm the longstanding tradition of presentingelectromagnetic phenomena by following their historical development. In teaching practicesat university level, this approach constitutes an exception. Newtonian mechanics andthermodynamics are usually presented starting from a set of postulates. Even so more, theaxiomatic approach is used for special and general relativity and quantum mechanics, perhapsafter some introductory pages recalling the basic historical steps of the matter. Hertz, inreferring to Maxwell’s equations, wrote:These statements form, as far as the ether is concerned, the essential parts ofMaxwell’s theory. Maxwell arrived at them by starting with the idea of action -at - a - distance and attributing to the ether the properties of a highly polarisabledielectric medium. We can also arrive at them in other ways. But in no way can adirect proof of these equations be deduced from experience . It appears most logical,therefore, to regard them independently of the way in which they have been arrivedat, to consider them as hypothetical assumptions, and to let their probability dependupon the very large number of natural laws which they embrace. If we take up thispoint of view we can dispense with a number of auxiliary ideas which render theunderstanding of Maxwell’s theory . more difficult, partly for no other reason thanthat they really possess no meaning if we finally exclude the notion of direct action- at - a - distance [6, p. 138, italics mine]. In contrast with Hertz’s position, the historical approach implies, intentionally or not, the ideathat physical laws must be induced cumulatively from experiment without need of recurring,soon or later, to their derivation within an axiomatic theory. This approach is exposedto the risk, not always avoided in the texts considered, of dragging electrostatic conceptsand definitions into the domain of Electromagnetism. The definition of the induced emf as E = H ~E · ~dl is a crucial example. This definition is legitimate, but with the warning that itcan be used only in the reference frame of the induced, rigid and filiform , circuit. When thecircuit is in motion, the force exerted on a unit positive charge is not ~E but ( ~E + ~v c × ~B ) and,therefore, the induced emf must be written as H [ ~E + ( ~v c × ~B )] · ~dl . Moreover, the definitioninherited from Electrostatics ignores the drift velocities of the charges: coherently, since inElectrostatics all charges are at rest.The introduction of the vector potential is not a formal choice but the only way of writingdown an equation for the induced emf that is local. The violation of this locality conditionappears to be the basic reason for the absence of the vector potential in the treatment ofelectromagnetic induction. The other face of the medal is the attribution of the rank ofphysical law to a calculation tool such as the “flux rule”. The subsidiary role attributed tothe vector potential may have contributed to the oblivion of Maxwell’s general law. Thissubsidiary role might be also an heritage left by Heaviside and Hertz who, at the end of theNineteenth century proposed to exclude the vector potential from the fundamental equationsof Maxwell’s theory. With the words of Hertz:In the construction of the new theory the potential served as a scaffolding; by itsintroduction the distance forces which appeared discontinuously at particular pointwere replaced by magnitudes which at every point in space were determined onlyby the condition at the neighbouring points. But after we have learnt to regard theforces themselves as magnitudes of the latter kind, there is no object in replacingthem by potentials unless a mathematical advantage is thereby gained. And it doesnot appear to me that any such advantage is attained by the introduction of thevector potential in the fundamental equations; furthermore, one would expect tofind in these equations relations between the physical magnitudes which are actuallyobserved, and not between magnitudes which serve for calculations only [6, p. 196].This Hertz’s criterion can be substituted by the less restrictive one – enunciated on page 6and derived by another, most general, Hertz’s criterion – for attributing physical meaning to aphysical quantity. The elimination of the vector potential impedes the derivation of a generallaw of electromagnetic induction that satisfies the locality condition. Therefore, the vectorpotential satisfies the strong condition of the criterion. Not to mention the weak condition: atthe turn between the Nineteenth and the Twentieth centuries, the role of the potentials hasbeen vindicated by Alfred - Marie Li´enard (1869 - 1958) [87] and Emil Wiechert (1861 - 1928)[88] in dealing with the electromagnetic field produced by a point charge in arbitrary motion(Appendix B). Furthermore, after the unveiling of the relativistic nature of Electromagnetismby Einstein, Maxwell’s equations written in terms of the potentials have proved to be the moresuited for dealing with this fundamental property.As for the physical meaning of the vector potential, the only one comment is found –among the texts considered – in Feynman’s Lectures . Feynman speaks of the vector potentialas a ‘real field’, where for ‘real field’ it is intended “a mathematical function we use foravoiding the idea of action at a distance [53, p. 15 - 7].” However, Feynman, while speakingof the vector potential as a ‘real field’ in connection with Quantum Mechanics (and QuantumElectrodynamics), denies it such a feature in classical Electromagnetism. “In any region where ~B = 0 even if ~A is not zero, such as outside a solenoid, there is no discernible effect of ~A .Therefore, for a long time it was believed that ~A was not a ‘real field’. It turns out, however,that there are phenomena involving quantum mechanics which show that the field ~A is in facta “real” field in the sense we have defined it [53, p. 15 - 8]”. Feynman’s concept of ‘real field’corresponds to – but not coincides with – our conditions for attributing a physical meaning toa physical quantity.With the exception of Feynman’s and Zangwill’s texts, all the textbooks hold that the“flux rule” is the law of electromagnetic induction, thus attributing the rank of a physical lawto a calculation tool. The paths followed for holding such a position are various. It can be saidthat the “flux rule” is, explicitly or implicitly, assumed as the law of electromagnetic induction,leaving to the mathematical developments only the task of supporting the starting assumption.No textbook recognizes explicitly that the “flux rule” violates the locality condition. Indirectly,only Jackson recognizes it by noticing that the “flux rule” obeys a kind of Galilean invariance.The analysis of the textbooks reminds us that the precious task attributed to them is thatof contributing to the transmission of the acquired knowledge to the new generations. However,the acquired knowledge is written nowhere and, hopefully, no one will ever be appointed towrite it.
10. Electromagnetic induction in some recent research papers
Recently, Chyba and Hand made a proposal for obtaining electrical energy throughelectromagnetic induction in a circuit at rest on the Earth surface [89]. The circuit, viewed froman inertial frame K centered at the Earth’s center, is rotating in the Earth’s magnetic field.The proposal has been subjected to an experimental verification with a substantially negativeresult [90]. The proposal has also triggered a discussion about its theoretical foundation[90, 91, 92].The basic idea of the proposal consists in the magnetic manipulation of a part of thecircuit in order to unbalance the contribution to the induced emf by the Earth’s magnetic field,supposed uniform at the position of the circuit. Chyba and Hand start with the definition ofinduced emf as: E = I l ( ~E + ~v × ~B ) · ~dl (79)where it is implicitly assumed that the charge velocity ~v coincides with that of the circuitelement that contains the charge [89, p. 2]. In Appendix A it is shown that this definitionleads to a law of electromagnetic induction Lorentz invariant for rigid circuits. In particular,considered the relative inertial motion of a magnet and a circuit, it is shown that E circuit =Γ E magnet , where Γ is the time dilation factor and E magnet and E circuit is the induced emf in thereference frame of the magnet and circuit, respectively.This relation allows to prove in a simple way that the proposed setup can not yield anyelectrical power if the Earth’s magnetic field is uniform at the position of the circuit.Let us consider the case in which the positive direction of the x axis of the Earth centeredinertial frame K coincides – instantaneously – with the direction of the velocity vector ~V ofthe circuit. Using the Lorentz invariance of the law of electromagnetic induction for rigidcircuits, it is convenient and sufficient to use the inertial reference frame K ′ instantaneouslyco - moving with the circuit. In this frame, we have: E ′ = I BA [ ~E ′ + ~v ′ × ( ~B ′ Earth + ~B ′ Shield )] · ~dl ′ + I AB [ ~E ′ + ~v ′ × ~B ′ Earth ] · ~dl ′ (80) where AB is the magnetically manipulated part of the circuit, ~v ′ is the velocity of the circuitelement dl ′ and ~E ′ is coming from the relativistic transformations of the fields between thetwo reference frames K and K ′ . Since ~v ′ ≡
0, equation (80) reduces to: E ′ = I l ′ ~E ′ · ~dl ′ (81)Since, ~E ′ has the same value at every point of the circuit (because the Earth’s magnetic fieldis uniform at the position of the circuit), the integral is null and we have E ′ = 0 and, hence,also E = E ′ / Γ = 0. Equation (80) shows at glance that the magnetic manipulation of part ofthe circuit is useless, while equation (81) shows that the emf induced in the circuit is null.The proof presented here is conceptually equivalent to that of Veltkamp and Wijngaarden[90], with the difference of relying uniquely on the general relation between the induced emf inthe reference frames of a magnet and a circuit in relative inertial motion, without the necessityof developing detailed calculations in both reference frames.Veltkamp and Wijgaarden stress that their proof does not exclude a possible effectdue to an Earth’s magnetic field co - rotating with the Earth [90, p. 6]. However, insection 8.5, we have shown that the hypothesis of a magnetic field co - rotating with a solidmagnet is incompatible with MLE electromagnetism and is falsified by Faraday’s and M¨uller’sexperiments (section 8.4).Finally it is worth stressing that in all the papers cited in this section, the correct definitionof induced emf (28) is used.
11. Conclusions
The discussion about the theoretical description of electromagnetic induction has periodicallyrekindled through about two centuries, in spite of the fact that the basic experiments go back toFaraday’s
Experimental Researches and notwithstanding that a ‘general law’ has been derivedby Maxwell within a Lagrangian description of electrical currents (section 7.1). This situationis very peculiar and the present paper tries to understand why. A modern reformulation ofMaxwell’s law is based on the definition of the induced electromotive force as the line integralof the Lorentz’s force on a unit positive charge: E = I l [ ~E + ( ~v c × ~B )] · ~dl ; ~E = −∇ ϕ − ∂ ~A∂t (82)This equation is, formally, the same as Maxwell’s, but with the fundamental difference thatthe velocity appearing in it is the velocity of the charge and not the velocity of the line elementcontaining it. The general law is a local law: it correlates what happens in the integrationline at the instant t to the values of quantities at the points of the line at the same instant t .For rigid circuits, it is Lorentz invariant (Appendix A). If expressed in terms of the magneticfield, it allows – in the approximation of low velocities – the derivation of the “flux rule”,for filiform circuits. The “flux rule” is a calculation tool and not a physical law, because, ifphysically interpreted, it implies physical interactions with velocities higher than that of light.Not always it predicts the correct result; it does not say where the induced emf is localized;it requires ad hoc choices of the integration paths, thus revealing its being a calculation toolthat must be handled with care (section 8.1).Maxwell’s general law has been rapidly forgotten; instead, the “flux rule” has deeplytaken root. An analysis of university textbooks, spanned over one century, assumed to berepresentative on the basis of the authors and/or on their popularity or diffusion, suggeststhat one reason for this oblivion may be related to the longstanding tradition of presenting electromagnetic phenomena following their historical development and to the connected,implicit or not, epistemological position according to which physical laws must be cumulativelyderived from experiment without need of recurring, soon or later, to an axiomatic formulationof the matter. But the main reason appears to be the lacking recognition that a physical mustobey the locality condition. The violation of this condition appears to be the basic reason forthe absence of the vector potential in the treatment of electromagnetic induction and for theattribution of the rank of a physical law to a calculation tool such as the “flux rule”. On theother hand, the rooting of the “flux rule” has been certainly favored by its calculation utility:this practical feature has largely overshadowed its predictive and epistemological weakness.In the first decades of the Twentieth century, it was common the idea that someelectromagnetic induction experiments with rotating cylindrical magnets could be explainedalso by assuming that the “lines of magnetic force” introduced by Faraday rotate with themagnet (section 9). This is a surprising hypothesis, if one takes into account the fact thatFaraday’s experiments, as repeatedly stressed by him, prove that the “lines od magnetic force”do not rotate. More surprisingly, this hypothesis has been resumed recently. It is shown thatthe hypothesis of rotating “lines of magnetic force” is incompatible with Maxwell - Lorentz -Einstein electromagnetism and is falsified by experiment (section 8.5). Acknowledgements.
I would like to thank Biagio Buonaura for a critical reading of themanuscript and for his valuable suggestions.
Appendix A. Lorentz invariance of the general law of electromagneticinduction
Let us consider a source of magnetic field at rest in the laboratory reference frame. A filiform,rigid circuit moves with velocity V along the positive direction of the common x ≡ x ′ axis. Inthe circuit’s reference frame, the general law (28) assumes the form: E ′ = I ~E ′ · ~dl ′ + I ( ~v ′ d × ~B ′ ) · ~dl ′ (A.1)where ~v ′ d is the drift velocity of the charges. Since the second integral is null, because in everycircuit element the drift velocity is parallel to ~dl ′ , we are left with the first line integral. Takinginto account the fields transformation: E ′ x = E x = 0 E ′ y = Γ[ E y + ( ~V × ~B ) y ] E ′ z = Γ[ E z + ( ~V × ~B ) z ]and the coordinates transformations: dx ′ = Γ dxdy ′ = dydz ′ = dz we get: E ′ = Γ I E x dx + Γ I [ E y + ( ~V × ~B ) y ] dy + Γ I [ E z + ( ~V × ~B ) z ] dz (A.2)= Γ I [ ~E + ( ~V × ~B ) · ~dl ] In the laboratory reference frame, we have: E = I ( ~E + ~V × ~B ) · ~dl (A.3)Hence: E ′ = Γ E (A.4)We have thus shown that the phenomenon of electromagnetic induction, involving electric andmagnetic fields, must be treated relativistically, as claimed by Einstein. Since the relativevelocity V ≪ c , we can put Γ = 1 because the predicted value of Γ differs from one by anamount experimentally not detectable. Appendix B. Sources and fields Q* Q P r* r O r r* r charge trajectory Figure B1.
Point charge q in arbitrary motion. P is the point in which we want tocalculate the values of the fields at the instant t . Q is the position of the charge at thesame instant t . Q ∗ is the position of the charge at the retarded instant t ∗ = t − r ∗ /c . The electromagnetic potentials produced by a point charge in arbitrary motion, originally andindependently studied by Li´enard [87] and Wiechert [88], can be written as (fig. B1) [53, p.21 - 11]: ϕ ( x , y , z , t ) = 14 πε qr ∗ − ~v ∗ · ~r ∗ /c (B.1) ~A ( x , y , z , t ) = µ π q~v ∗ r ∗ − ~v ∗ · ~r ∗ /c (B.2)where the asterisk indicates the retarded values of the quantities, i.e. the values at the retardedinstant t ∗ = t − r ∗ /c . By using the basic relations: ~E = −∇ ϕ − ∂ ~A∂t ; ~B = ∇ × ~A (B.3)one calculates the fields. The rather laborious calculation yields, for the electric field [56, pp.101 - 103, online vers.]: ~E = q πε d ∗ ( − v ∗ c ! ~r ∗ − r ∗ ~v ∗ c ! ++ 1 c ~r ∗ × " ~r ∗ − r ∗ ~v ∗ c ! × ~a ∗ (B.4) where: d ∗ = r ∗ − ~v ∗ c · ~r ∗ (B.5) d ∗ can be denoted as the reduced retarded distance between the field point and the retardedposition of the charge (of course, depending on the sign of the scalar product, this reduceddistance can be greater than r ∗ ). A similar equation is obtained for the magnetic field: ~B = q µ π ( d ∗ − v ∗ c ! ( ~v ∗ × ~r ∗ )++ (cid:20) cd ∗ ( ~a ∗ × ~r ∗ ) + 1 c d ∗ ( ~a ∗ · ~r ∗ )( ~v ∗ × ~r ∗ ) (cid:21)(cid:27) (B.6)The electric and the magnetic field depend on the retarded values of the velocity and of theacceleration of the point charge. They are independently produced by the charge. These fieldspropagate with the velocity of light: their values at the point ~r at the instant t depend onthe values of the velocity and of the acceleration of the charge at the retarded position ~r ∗ andat the retarded instant t ∗ = t − r ∗ /c . From these two equations, it can be proved that thetwo fields are related by the equation: ~B = 1 c ˆ r ∗ × ~E (B.7)This equation can not be interpreted in any way as a causal relation. It only reflects a propertyof the two fields as they are independently produced by the charge.This reminds us of an another subtle question. Let us consider Maxwell’s equation: ∇ × ~E = − ∂ ~B∂t (B.8)It is easy to find statements according to which this equation shows that a time varyingmagnetic field causes an electric field (and symmetrical statements for the equation of the curl of the magnetic field). The causes are not intrinsic properties of mathematical equations,but are superimposed to them by us [47, pp. 11 - 15] in agreement with the entireacquired knowledge ♯ . Furthermore, the causal chain attributed to an equation must havea correspondence in the experiment. This means that, in the experiment, a causal chaincorresponding to that contemplated in the theory must operate. The nowadays interpretationof Maxwell’s equations is that the charges produce (cause) the fields †† ; from this point on, noequation can be interpreted by saying that a field (or one its variation) produces another field. ♯ For instance, let us consider the law ~F = d~p/dt . As the momentum varies over time, we are inclined tointerpret this equation by saying that the force ~F ‘causes’ the variation of the momentum ~p . However,there are situations in which the change in momentum ‘causes’ a force. Consider, for example, acompletely absorbing surface S , hit by a monochromatic beam of light perpendicular to the surfaceand directed along the negative direction of the x axis. In this case, we write for the momentum P x ofthe surface, if N is the number of photons absorbed in a unit time: dP x /dt = − N hν/c = F x and wecan say that the variation of the momentum of photons has produced a force F x on the surface S . Asimilar situation is found in the kinetic theory of gases: the pressure (force per unit area) on the wallsis due the exchange of momentum with the particles. †† It was not so in Nineteenth century. For Hertz, the fields were the primary entities (cause) and thecharges the secondary ones (effect). This position is clearly outlined in his discussion of the physics ofthe capacitor [6, pp. 20 - 28]. Hertz’s position was not isolated. For instance, in Italy, Galileo Ferraris,shared Hertz’s position [93, pp. 42 - 48, online vers.]. 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