Classical Electromagnetic Theory: Textbooks, History, Stories and Web 2.0
Fabio Bevilacqua and Lidia Falomo
Dipartimento di Fisica “A.Volta”
Università di Pavia
Classical Electromagnetic Theory (CET) attempts at unifying (static and dynamic) electric and magnetic phenomena interpreted before Einstein’s Special Relativity and Light Quantum theories (1905). Standard textbooks, both at school and at the first years of university, usually present it after Mechanics and Thermodynamics in a “normalized” way. Students present specific learning difficulties in achieving conceptual clarity on CET’s main concepts: e.g. fields, charges, potentials. An effort is made here to contribute with a non-standard approach to CET learning.
A first step of this approach asserts that advanced textbooks for the later University years, written by Nobel prize winners in the 20th century, offer a non-normalized picture. They differ in that underline one or another of the main concepts and one or another of the main principles: e.g. energy conservation or least action. In so doing they relate to 19th century CET foundational debates.
A second step thus asserts that extraordinary pre-paradigmatic science with its competing historical research programmes is unavoidable for an understanding of CET.
As a third step, an effort is made to present a Nature of Science (NoS) image, based on a four-component scheme, which might overcome Kuhn’s separation between normal and extraordinary science.
A fourth step asserts that even if a historical framework is needed, not all the historical intricacies have to be covered in education. Thus a case studies approach is adopted: it underlines a small number of principal conceptual and experimental turning-points. For educational purposes, and in agreement with a growing educational literature, “history” is then transformed into “conceptual stories”. In turn, some of these stories have been transformed into screenplays for short, introductory ten-minute movies than can be downloaded from the web.
But in a fifth and final step, history makes a comeback, through the use of Web 2.0 technologies, specifically a Wiki software which allows users to actively interact with primary and secondary historical sources and with educational materials through tags, threads, and personal contributions (in a Wikipedia style). Movies and 3D animations and simulations now appear only as introductory. In this way a number of web communities can be formed, each at the preferred depth of historic-critical scientific understanding.
What do we mean by “Classical Electromagnetic Theory"?
Electrical and magnetic phenomena have aroused interest for quite a long while in many cultures (Needham, 1962). “Electron” is the ancient Greek word for “amber” and “magnetic” derives from the Aegean region “Magnesia”. In Western culture, the systematic study of electrical and magnetic phenomena started in 1600 with Gilbert’s De Magnete. (Gilbert, 1600). At the beginning of the 20th century, continuous efforts of analysis and synthesis of these phenomena resulted in Lorentz’s Electron Theory (Lorentz, 1909), the archetypical Classical Electromagnetic Theory (CET). The term “classical” indicates that these results conceptually predate Einstein’s Special Relativity (Einstein, 1905a) and Light Quantum (Einstein, 1905b) theories of 1905.
CET is important for various reasons. From a scientific point of view, it is at the root not only of Special Relativity Theory but also of the General one (1916) (Einstein, 1916), which utilised the electromagnetic model of contiguous action to overcome Newtonian action at a distance gravitational theory; after the success of Quantum Mechanics, it has acquired new life as Quantum Electrodynamics (Feynman, 1998; Schwinger 1958); in today’s Standard Model (Cottingham, Greenwood, 2007) the electroweak is, with the strong, one of the basic physical interactions waiting for a further unification with gravitation. From a historical point of view, it is the result of numerous important scientific debates between conflicting research programmes. From an educational point of view, it is supposed to be an example of “normal science” (Kuhn, 1962), still taught today all over the world in schools and in undergraduate courses in physics, mathematics, chemistry, natural sciences, engineering, biology, medicine, etc.
Usually placed after mechanics and thermodynamics, the standard textbooks’ presentation of CET (Giancoli, 2000) starts with electrostatics (Volta’s relation between quantity of charge, tension and capacity, and Coulomb’s Newtonian inverse square action at a distance law). In turn, attention is paid to magnetostatics (stationary currents that according to Oersted’s results produce magnetic effects, described by the laws of Biot-Savart and Ampère), and circuits (Ohm’s, Joule’s and Kirchhoff’s laws). Faraday’s electromagnetic induction follows (electromotive force is produced according to the flux rule). The introduction of the so-called displacement current leads to the four basic equations of Maxwell-Lorentz that establish the relations between sources (charges and currents) and fields, and eventually Lorentz’s force, that establishes the effects of fields on charges.
Basic components of CET are thus corpuscular charges and currents on one side and electromagnetic fields (waves) on the other. The former are the sources of the latter, the fields are autonomous entities that propagate with a finite velocity, the velocity of light, and then interact with other charges and currents. Fields are linked to potentials, usually presented as mathematical quantities, through specific equations.
It is not surprising that teachers and students believe that CET is a difficult subject! (Viard & Khantine-Langlois, 2001; Guisasola et al., 2002; Silva, 2007). Having studied Newtonian mechanics based on action at a distance with infinite speed, students are rapidly introduced to a theoretical framework that in its electrostatics part resembles mechanics (Coulomb’s law is analogous to Newton’s law), while in its magnetostatics part (stationary currents) is based on an interaction between two points that does not act along the line joining them but perpendicularly to it. Moreover, this interaction is interpreted through a contiguous action theory and (in dynamics) is propagated with the velocity of light, a finite one. Thus the concept of “field” is basically different from the one of “force”, but charges and currents still play a relevant role as sources of the fields, despite the fact that classical electromagnetism is usually seen as a wave and not a particle theory. In addition, the principle of energy conservation takes a third form, after the mechanical and thermodynamical ones: in electromagnetism’s dynamical part, the basic mechanical distinction between potential and kinetic energy disappears and the principle assumes the form of a continuity equation (Poynting’s theorem). Last but not least appears Lorentz’s force to express the effects of electrical and magnetic fields on charges, and this force, at variance with Newton’s and Coulomb’s, depends on velocity and not only on distance. Also the role of the potentials is rather disconcerting: are they physical or mathematical quantities?
A vivid and explicit presentation of these conflicting conceptual elements of CET, usually hidden in standard textbooks, can be found in the famous text: “Feynman’s Lectures on Physics” (Feynman, Leighton, & Sands, 1963), usually studied at an advanced university level. A table summarizes a number of results.
The operators divergence, gradient, laplacian, dalambertian utilised in this table are explained by Feynman in the first chapters of the text.
The equations that summarise CET are divided in two columns: in the first the ones that are valid only in the static case, in the second the “true” ones, that is the ones that are valid in general. There is a clear-cut distinction between fields (electric and magnetic) and potentials (scalar and vector). The relations that connect the fields to the potentials are outlined clearly both in the static and dynamic cases.
In the static column we find Coulomb’s law; the laws of the electric static field E both in the elementary form and in the vector one (the rotor of E equals 0), the relation that connects the field E with the gradient of the scalar electric potential; then the laws of the conductors are recalled and with them Volta’s law (Q=CV); in turn there is the law of the production of the magnetic field B through stationary electric currents both in the formulation of Ampère’s elementary law and in the vector form through the relation of the rotor of B and the density of electric current. Next is the expression of Poisson’s equation through the laplacian operator that connect electrostatic potential with charge density, and the similar one that connects vector potential with current density. Finally, the elementary formulations of these two potentials are recalled. In the last line we find the expressions of electrostatic and magnetostatic energy as the products of the density of charge by the scalar potential and of the density of current by the vector potential. In the static case these expressions are equivalent to those of the scalar product of the electric field by itself and of the magnetic field by itself. The first one is identified as potential energy and the second as kinetic.
In the second column of the table, the one dedicated to the “true” laws, we find Lorentz’s law (force depends on the velocity of charges) and Gauss’s law (the divergence of the electric field equals the density of charge). The last one is identified as the first of the four “Maxwell’s equations”. Faraday’s induction law, in the form: the rotor of the electric field equals the derivative of the magnetic field, follows as the second of Maxwell’s equations; and then the link between the electric field and the scalar and vector potentials is outlined. The third of Maxwell’s equations follows: there are no isolated magnetic poles, in the form: the divergence of the magnetic field equals zero; and the fourth, that sees the addition of the “displacement” current to Ampère’s law. Then we have the relation between the magnetic field and the rotor of the vector potential.
Feynman underlines that the equations of the electrical and magnetic fields, E and B, are linked to the sources, density of charge and density of current, but not to the potentials. The fields are here connected to the potentials through other separate equations. This is a main departure from Maxwell’s original equations (Maxwell, 1873; Everitt 1974).
Analysing the potentials, Feynman shows that in the dynamic case both electric and vector potential are now retarded potentials. Thus the potentials too are propagated with a finite velocity, the velocity of light, like the fields. Feynman writes the equations defining the potentials both in the dalambertian and in the elementary form. In a way at this stage both fields and retarded potentials can be considered primary physical quantities: the secondary quantities of choice can be derived through the connecting equations. Coming now to the expression of electromagnetic energy in the general, dynamic case, the distinction between potential and kinetic energy does not hold any more, nor does the equivalence between the expressions with the potentials and those with the fields. The energy is now given by Poynting’s theorem and a global principle of conservation (static case) is transformed into a local one.
CET is not a simple synthesis, it is in fact based on both discontinuous (charges) and continuous (fields) concepts, on action at a distance (statics) and contiguous action (dynamics), on instantaneous interactions (statics) and finite speed (dynamics), on forces depending only on distance (statics) but also on forces depending on velocity (dynamics), on global conservation of energy with a sharp distinction between potential and kinetic energy (statics) and on local conservation which, through the so-called Poynting vector, blurs this distinction (dynamics), and on the role of potentials as mediators between forces and fields.
Feynman’s wonderful synthesis tells us that science cannot really be understood through standard textbooks which offer only a “normalized” view of a scientific theory and avoid the most significant part: research on foundational aspects. “Conservative” teaching, based on the technical solution of pre-defined problems ought to be complemented by “innovative” teaching, which shows how to define problems (Botkin, Elmandjra & Malitza, 1979). In our case: how and why important physicists have conceptualised such antagonistic concepts in electromagnetism and how it happened that these antagonistic concepts have been joined together.
Is the path from history to textbooks univocal?
By CET we usually mean the body of scientific knowledge first synthesized by Lorentz and subsequently less and less well reproduced in standard textbooks. But we will see shortly that this body of knowledge has not been interpreted and presented by the principal 20th century physicists in a standard way in their advanced textbooks, written well after Lorentz’s synthesis. These physicists were well aware of the importance of their own alternative conceptualization and definition of CET: an indication of the fact that “normal” (Kuhn, 1962) science is not that normal after all, and that the (static) presentation of scientific theories is strongly connected to their (dynamic) evolution.