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Maxwells Motor Equation and the Mechanical Power

S. L. Vesely1, A. A. Vesely2, and C. A. Dolci3

1I.T.B.-C.N.R., Italy2Via L. Anelli 13, Milano, Italy3

Liceo Einstein, Milano, Italy

Abstract Maxwells illustration of the Ampere-Maxwell equation as well as his geometricalinterpretation of it by means of the field-of-force concept are analyzed. Then a difficulty associatedwith transferring the same representation to modern electrical machines is clarified, and a possiblereinterpretation of the electromagnetic equations alternative to Einsteins is briefly sketched.

1. INTRODUCTION

The quantities E and B in the electromagnetic equations are assumed to be generated by electriccharges and by currents, perhaps reliant on them, as their sources. The so called direct problem,which consists in reckoning vectors E and B when the sources are known, is taught to be directlyconnected with the ancient way of recording the electromagnetic forces acting on a test body. In fact,

in the late 1700s, Coulomb conducted the experiments about electric and magnetic actions by meansof a torsion balance1 [1], and interpreted them in accordance with the graphical representationsof the static fields of force on a unit mass. That field-of-force notion was not only taken up byelectrostatics and magnetostatics, by defining an electric and a magnetic unit mass, but it wasalso substantially accepted by the pioneers investigating the mutual actions between magnets andconductors. When, at the beginning of 1900, a completely new understanding of the propagatingnature of the fields and of radiation-matter interactions broke through, motivating a profoundrevision of the conceptions of E and B (or H), those new instances led to the modern physicaltheories of relativity and quantum electrodynamics. Today, classical electromagnetism subsists forits usefulness in technical applications. On the one hand, the wish to improve on the classicallydefined fields is considered rather conservative and unaware of later developments. On the otherhand, in order to participate in the new comprehension trends, an electrical engineer is compelledto adopt the new theoretical frameworks together with the physically and mathematically unsolved

problems therein. In our opinion, particularly the need to introduce higher dimensional spacesand the probabilistic interpretation of measurements could hamper practical applications. As afurther option we will try and update the interpretation of paragraphs I and III of the IV partof Maxwells Treatise2 [2]. In fact the experiments described there are reminding of the inverseproblem approach, i.e., of the problem of tracing the received signals back to their sources, whichis more commonly encountered today.

In the 19th century thermodynamics seemed general enough to embrace the whole of the elec-tromagnetic interaction mechanisms. The reason is that the conservation principle allows to takeall contributions into due account by just adding suitably defined energies to the balance3. Greatadvances had been made also in analytical mechanics, where Lagrange, and all the more Hamil-ton, developed the relevant mathematical tools required to generalize the force concept to movingbodies. In spite of those accomplishments, to exhibit more clearly the traits of electromagneticphenomena, Maxwell choose to stick to Newtons geometric style. Hence, besides appreciating thatthe experiments dont indeed fit in with Newtons mechanics, one is struck by the use of a differentgeometry. Those new trends in geometry had come to the fore along with the development of com-plex analysis. Nowadays they have evolved far beyond the geometric description adopted by Maxell,and there are many elegant alternative ways to handle his equations. By contrast, modern electricpower applications continue to be put on a level with those basic experiments. We will try andshow that the experimental setup coming down to the Ampere-Maxwell equation is the one linkingthe electromagnetic fields to the moving electrified bodies in its common meaning. That link failsas soon as power is transmitted, due to the onset of special conversion mechanisms which cannotbe easily represented geometrically. Alternatively, acknowledging that the geometry adopted by

1Since the apparatus is very light, the twisting of the stretched wire in response to an impulse is swift. If the elastic restoringtorque produces a free torsional oscillation with a very low frequency, that twisting action can be compared with a static couple.

2Art. 486491, 530532.3Thermodynamics was supplemented by statistical methods to handle irradiation, conduction and convection phenomena.

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Maxwell is different from Newtons, and thus that the usual link to the mechanical representationsis without foundation, it becomes possible to scrutinize the connection between the electromagneticinduction signal and the degree of electro-mechanical coupling during power conversion.

2. MAXWELLS EQUATIONS SYSTEM

Maxwells field equations summarize Faradays experiments and apparently provide an intuitiveinterpretation for the time-varying electromagnetic quantities involved therein. Yet, electromag-

netism had introduced motion into the static pictures, troubling their geometric representation.Neither the mathematical description was univocally settled any more [3], nor there was agreementabout what physical properties a fields mapping embodies4. Nowadays, if the system of Maxwellsequations is taken to consist of four coupled vector partial differential equations in the four un-known quantities E, B, D and H, the geometric representation of the electromagnetic fields is local,unless the equations are integrable5. With integration in view, the number of variables is usuallyreduced, by expressing the flux densities D and B by means of the field densities E and H, throughthe linear constitutive relations D = E and B = H6, where and are real constants.

As the interpretation of the relevant quantities was debated already at Maxwells times, hehimself illustrated each of the fundamental equations by means of typical experiments. Accordingto the examples provided, two of the equations are transferred literally from electrostatics andmagnetostatics7, implying that even the phenomenology associated with motion shall become staticwhenever the movement ceases. Regardless of the differential equations D = ( being definedas the actual free electric charge density) and B = 0 satisfying Faradays request that there bea close by action, it can be shown that finite equations of Coulombs kind can be recovered, owingto the fact that Gauss theorem doesnt depend on true boundary conditions. Although the staticphenomenology is decidedly not electromagnetic, the corresponding equations are convenient forthe simplification of the coupled ones, which are the Ampere-Maxwell law H = J + D/t,and Faradays electromagnetic induction law E = B/t. Therefore we first give a lookat mathematics. In the static limit the electromagnetic equations involve irrotational solutions,which can be expressed as gradients of some scalar functions and . Maxwell accepts that thepotential function, say, of H can become multivalued in the time-dependent cases, and definesit on a Riemann surface. For those cases, he adds a term D/t to Amperes formula H = J,bringing the stationary solutions to his equations back to those of potential flow analysis and toconformal mappings. Of course if, on account of physical applications, all of the four variables

were needed, the solutions could no longer be restricted to surfaces. Anyway, in the complexplane it is easy to relate E + iB to equipotential lines and streamlines respectively, and thatcan be upheld in geometric transformations, according to the right hand sides of the equations:[2E 00

2E/t2] = 10 [ + 00J/t]8 and [2B 00

2B/t2] = 0[ J]. In fact,since the solutions of the homogeneous equations are harmonic functions, the elements E andB can be made harmonic conjugates in every simply connected domain. Lets now come to thephysical interpretation. To give a geometric representation, the time dependent equations oughtto be exemplified in a strictly linear range, i.e., when the experimental behavior is independent offrequency and intensity9. The electromagnetic induction law is taken to describe the small currenttransient elicited in an electric circuit (receiver) by the movement of a magnet (source)10, leaving

4The meaning apparently attributed by Maxwell to the electromagnetic variables is taken from statics representations ofthe fields of force. His dynamical picture requires the static frame to be displaced in time according to the convective derivative: /t / (v ) + v( ). As opposed to Maxwells dynamics, an electromagnetism in the Newtonianstyle represents the solutions of the equations of motion in the potential fields. The electromagnetic potentials are written asA(t r/c) and (t r/c), and the electric field becomes Eem = Es A/t.

5As an alternative, the Erlanger Program, still unknown in Maxwells times, permits a global imaging interpretation ofthe electromagnetic field coordinates, which is obtained by transforming the space coordinates into the (finite) field variables.It is different from transformation optics in that the geometry models the imaging properties, i.e., the transforming functions,without entailing (mechanical) space properties.

6In the SI units of measure the relations D = 0E+P andB = 0(H+M) hold, where P andM are the electric and magneticpolarizations respectively. Physically the introduction ofD and B besides E and H complies with Faradays observations aboutthe non-inertness of the interposed media.

7They are Faradays cage and the broken magnet experiments, respectively.8This equation is obtained from B = 0J + 00E/t, making use of one static solution, by the steps: B/t =

(E) = 2E 10(D) = 2E 1

0 = 0J/t + c22E/t2. The second one is obtained in a similar way.

9In practice the induction phenomena appear to be much stronger in the magnets then in the conductors. Hence magnets alsoshow more impressive deviations from the linear behavior. Nevertheless, in our opinion, neither of both displays the propertiesof the demodulated electromagnetic signals. Rather the magnetic and conduction properties could be understood as resonancebehaviors of the metals near = 0.

10Or by the movement of the circuit with respect to the magnet. Besides, a mechanical movement is not truly necessary to

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any feeding in aside. Choosing decibel units, this law gets an independent geometric meaning,whatever the meaning of the induction signal. The Ampere-Maxwell law is taken to describe anapplication of rsteds effect carried out by Faraday. rsted had noticed that when a conductorthread (source) is connected to a battery, a magnetized needle (receiver) gets deflected, apartfrom the way it is moved toward the thread. Faraday exploited that jerky deflection to obtaincomplete turning cycles of a part of a fed circuit around a magnetic cylinder, or reciprocally. Ifthe distinction between electromagnetic induction and rsteds effect is considered a matter of

interchanging the term receiver with source, then the second equation means that a needlesensor is being used for detection. Although the detection method is very different, indeed, theequation H = D/t gets the same interpretation as the induction equation, and the constant Jresults from circle integration. If, as currently assumed, H = J is taken to be a magnetostaticsequation, then closed loops carrying a direct electric current are taken as the outer/inner boundingedges of magnetic shells, and the current circuits substitute for the magnetic dipoles as field sources.All the same, on account of the circulation the space topology differs from Newtons. For instance,if one current thread pierces the plane, the surface becomes homeomorphous to a torus, and torepresent the electromagnetic variables as well as the motion variables on the same diagram, theinterchangeability of some circular paths and straight lines has to be allowed for. According toMaxwell there are such inertial circular paths11. However Newtons gravitational theory does notadmit them. Finally the geometric picture of the conservation of angular momentum breaks down,when the motive force operating an engine starts changing, i.e., when there is power delivery.

3. MAXWELLS ILLUSTRATION OF THE MOTOR EQUATION

To give a Euclidean illustration of a machine functioning is like telling that a micro-motor is similarto a marine propulsion engine. The similarity pertains to both their size and their kinematics. Infact, there is no way to distinguish geometrically between the circling of a big electric engineand that of an electrostatic whirl, though evidently the magnetic energy component drops out forthe latter. The crucial point is that, on the one hand no linear representation can account for thedifferent behaviors of conductors versus magnetic materials that arise at terrestrial conditions12. Onthe other hand, circuital schemata are useful to analyze the steady electrical behavior of electricalmachines, such as transformers, reversible DC motors, synchronous motors, and alternators 13. Oncethe analysis is led back to linear circuits, the electromagnetic force gets a symmetric role to an emf,as Einstein remarked, whereas Maxwell in the end had tried to tie Amperes law to an unuttered

property of magnets.The device described by Maxwell shows an electromagnetically driven rotation. The machine

is actually reversible: the motor is called unipolar or homopolar [4], while the dynamo is calledFaradays disc, or unipolar inductor, to stress that the current is generated by induction. Currently,neither the motor nor the generator enjoy relevant applications, albeit a quite big generator hasbeen deployed for some electrolytic process [5]. According to Maxwells description, a cylindricalbar magnet is inserted on a shaft so as to be capable of revolving about its vertical axis. Attachedto it there is one end of a horizontal conducting arm, whose other end dips into a mercury-filledannular trough laying around a ring, which is concentric with the magnets section and fastenedto the support at about midway between the poles. When the circuit is connected to a battery,as soon as the magnet starts rotating the current around the ring is split into two branches whosepath lengths vary cyclically. The direction of the current depends on the polarity of the battery,whereas the sense of rotation of the magnet also depends on whether the upper pole is north or

south. Notwithstanding the fact that the split circuit is not exactly like a commutator, once therevolving motion has been started, it is apparently driven by current commutation, as Maxwellnotes14. Since to model the experiment geometrically one has to proceed along a different lineof thought, he represents the upper magnetic pole in the magnetic force field that the currentgenerates, and suggests that the magnet exerts a reaction force on the circuit. In fact, if the circuit

elicit induction.11It is clarified in the Treatise that the battery only feeds the circuit, without the emf being used to maintain the rotation.12Under moistly conditions the electric effects are much smaller than the magnetic ones, but they seem to affect all substances

equally. (Ferro-)magnetic properties, on the other side, belong only to peculiar substances.13We point out that in this case the rotational speed matches the electric frequency, thus granting the use of filter analysis

to determine the maximum mechanical power transmitted to a load. Although then the electric frequency is understood to bea resonance frequency, the machine usually doesnt assume a state of resonance.

14rsteds experiment is delicate. By definition, the battery supplies a direct current. However, to protect it, the needlesdeflection is usually obtained by switching the circuit on and off.

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arm is detached from the body of the magnet, and is provided with a sliding contact, it can be madeto rotate around the bar magnet. By leaving the field-of-force representation for a kinematical one,the reciprocity can be accounted for by the relativity of motion. However, if the field-of-force ismapped on the same space where the movement is represented, the simultaneous involvement oftwo reference frames rotating with respect to one another renders the interpretation ambiguous [6].

Einstein faced the problem of the drag of the lines of force by the frame in 1905. He remarkedthat the magnetostatic force supports an electric current within the circuit of a generator without

doing any work, and that therefore the current should be induced therein due to a reaction force [7].He postulated the invariance of physical laws under coordinate transformations related to his prin-ciple of relativity. Furthermore, he required that electric current be identified with convection ofelectrified particles endowed with a rest mass. Thus the Lorentz force expresses an equivalencebetween the electromagnetic and the mechanical force15, that depends on the reference frame, andreduces to the electrostatic force in a frame instantaneously at rest relative to an electrically chargedparticle. Reciprocally a magnetic field reduces to an electrostatic field under relativity transfor-mations [8]. The last statement is at odds with Maxwells own illustration of the Ampere-Maxwellequation, and with his belief of describing something different from Faradays induction.

4. IMAGING VERSUS POWER TRANSFER DIAGRAMS

Maxwells equations system can be considered mathematically coherent as is. Perhaps, his theory

is called phenomenological because it illustrates the facts, without implying a unique, generalinterpretation for the field variables. A major problem is to clarify what is meant by a geometricrepresentation of E, D, H and B. There are many instances, such as medical or SAR imaging,where the induction phenomenon can be linearized, so as to become altogether similar to TVimaging, with due care. By contrast, the circuit diagrams of electrical machines performances arerather conceived as graphical means to evaluate energy flow. But, while the circuits parametersare linearly scalable, the evaluated power refers to fixed values of the electrical and mechanicalparameters of the machine. In turn, that means that the watts computed using the scalableschematics as a base can by no means represent the absolute values of the converted power, andthe volt-amperes belong to the resistive and reactive parts of the power factor of the associatedelectrical circuits.

One way to escape the interpretive problem of the fields consists in considering the variables todescribe in every case received signals, and interpreting signals in a different fashion, depending

on the process involved in their detection. Actually, while images can often be reconstructed fromthe recorded electromagnetic input16, a geometric representation of power machines can only beendorsed at steady state. To show it, lets examine what kind of picture a circuit analysis conveysfor the energy flow trough a machine. Usually, the transmitted power is not evaluated during therun-up cycle, but rather at full load conditions, when, except for the dynamical frictions, themotors torque is exactly balancing the loads torque. At steady conditions, the relation betweenthe rotation frequency of the moving parts and the sinusoidal electric current has hardly anythingto do with resonance. In fact, the functioning regime is supposed to be stable, i.e., either themachine is working far away from any resonating condition, or any vibrations, which could havearisen because of couplings with the power supply or the load, have been damped. In that prettylinear regime circuit analysis allows to decompose the electrical signal into its Fourier components,and to evaluate either the transferred power or the efficiency. However, the electro-mechanicalcouplings responsible for power transfer can arise at the moment when supply conditions or load

start changing. For instance, that happens during the time required by an induction motors slipto stabilize itself. In brushless DC motors the drive electronics controls the speed by sensing therotors position, instead. Going back to Maxwells analysis, we notice that the illustrations of thetwo electromagnetic equations differ in one respect. Whereas Faradays induction is an electricaltransient, the Faradays disc is the earliest example of a generator. To describe in every case receivedsignals, we substitute rsteds jerky excitation of the magnetic needle for the electromagneticrotation. By doing so, the improvements to the motional conversion brought about by Faradaybecome apparent, too: they testify how strongly the motor design contributes to the parametricmechanical amplification of rsteds effect. Exactly that kind of amplification is never used for

15This is tantamount to a nonradiation condition.16We stress again that, according to the Erlanger Program, the four relations E = f1(x,y,z,t), D = f2(x,y,z,t), H =

f3(x,y,z,t), and B = f4(x,y,z,t) mean a transform (x,y,z,t) (E,D,H,B) of the space into itself (without necessarilydescribing focused images).

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imaging purposes, even if power amplification is sometimes allowed for in electrical schematics.Thus far, we have been telling that the mechanical power amplification of rsteds effect obtained

by means of electrical machines is non-linear, without endeavoring in the reduction of macroscopicmechanisms to microscopic processes. As we ascribe the power conversion capability to a greatdeal to the machine design, we expect the construction of actuators and MEMS to open up newexperimental opportunities, which may also rely on non-linear phenomena. In our opinion theelectromechanical performance cannot be assessed by modeling the structure of matter so as to

reduce the transduction to linear processes. Still, it is hypothesized by quantum mechanics thatthe functioning of simple systems is energetically compatible with very few different mechanicalstructures. It is assumed that functioning is linear at the atomic level, and that coherence, self-organization, or cooperativity takes over at the macroscopic level. That can be considered a valuableenumeration of the possible modelling mechanisms. We owe to quantum mechanics the insight thata peaked signal entrains a transition from a stationary state to another, instead of evidencing arevolution frequency. However, the atomic hypothesis has given rise to an inherently probabilistictheory, which cannot characterize an individual macroscopic motor as such. In addition the aim ofthat theory at verifying the hypothesized simple functioning on small scale engines fails due to itsprobabilistic mathematical basis.

5. CONCLUSIONS

Maxwell explains the Ampere-Maxwell equation by means of Faradays revolving motor, uninten-tionally suggesting two things: that his description concerns a power generator/motor, and thatthe geometric representation used to represent dynamics may display electromagnetism as well. Inretrospective, his view deserves some revising. A first shortcoming arises because he was unawareof the distinction between a signal and a carrier wave. Today the broadcasted signal is usuallyobtained by modulating a carrier. It is known that there is no energy intrinsic in the act of com-municating, and that the fluctuations of stabilized generators can be held very small. A secondshortcoming is due to the general belief that physical quantities require absolute units of measure-ment. When the chosen geometry is thought of as being the physical space rather than just ageometric model alternative to the mechanical one, electromagnetic fields are measured with thesame scale used for describing the mechanical processes. However, such stance implies that theelectro-mechanical conversion factor is fixed by geometrical considerations, and the possibility ofbettering power conversion by monitoring the associated induction and using it as a broadcasted

(radiated) signal is precluded. We have tried and shown that the usual descriptions of electricpower machines apply to steady state functioning only, whereas, in examining power changes, theAmpere-Maxwell equation could be considered by analogy with Faradays law.

REFERENCES

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3. Silva, C. C. and R. de Andrade Martins, Polar and axial vectors versus quaternions, Am. J.Phys., Vol. 70, No. 9, 958963, 2002.

4. Zeleny, J. and L. Page, The problem of the rotating magnet, Phys. Rev., Vol. 40, 299306,1932.

5. Miller, A. I., Unipolar induction: a case study of the interaction between science and tech-nology, Ann. Sci., Vol. 30, No. 2, 155189, 1981.

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