2.1 Proponents of the Poynting Theorem as Keystone of a Bridge between Classical Electromagnetic and Circuit Theories

Richard B. Adler, Lan Jen Chu, and Robert M. Fano, introducing the first chapter of their book Electromagnetic Energy Transmission and Radiation (Adler et al., 1960:1), state: “It is helpful…to be able to express both lumped-circuit and field behavior in a similar language. The concept of energy …[is] important for achieving this goal.” In their book Electromagnetic Fields, Energy, and Forces (Fano et al., 1968), these authors state that the formulae:

$P=\Sigma I_k{V}_k$

and

$P={\displaystyle \oint \left(E\times H\right)\cdot nda}$

are expressions for power that can be reconciled (p. 294). They add, “The representation of field vectors in the sinusoidal steady state is an extension of the representation of voltages and currents used in circuit theory,” i.e., V̇ and İ ⇔ Ė and Ḣ (p. 317), and on p. 324-35, that “…the complex Poynting’s vector can be identified with the complex power output to the system as defined in circuit theory:

${\displaystyle \oint S\cdot nda=P+jQ=\frac{1}{2}VI*}$

J.M. Aller, A. Bueno, and M.E. Jimenez (1999:54-56) state, “In a three-phase transmission line…the instantaneous active power p(t) corresponds to the longitudinal component of the Poynting vector, and the reactive power q(t) is related to the tangential or rotational component.”

M.E. Balci, M.H. Hocaoglu, and S. Aksoy (2006) state that instantaneous power is ‘exactly’ derived from the flux of the Poynting vector.

R. Becker (1944:195) equates the real part of the complex Poynting vector with the Joule losses and states that the imaginary part of the complex Poynting vector is identical with Slepian’s expression, i.e.

$2iw=\left(U_{mg}-U_{el}\right)$

in which U_mg ≡ energy of the magnetic field, and U_el ≡ energy of the electric field.

Z. Cakareski and A.E. Emanuel (1999, 2001) consider that the Poynting vector interprets the physical process of power transmission geometrically.

N. Calamaro, Y. Beck, and D. Shmilovicz (2015) review the Poynting vector theorem and consider it relevant to the concept of power in circuit theory.

I. Campos and J.L. Jimenez (1992) state that the Poynting vector is an expression of energy-momentum conservation; however, they consider that the Poynting vector theory should be re-examined in relation to cases where electrical charges are present. They stress the importance of understanding the relationship between fields and matter, and they show that electromagnetic fields and charges do not constitute a closed system.

L.S. Czarnecki (2003) considers the Poynting vector relevant to power theory for balanced AC circuits, but not to power theory for unbalanced AC circuits. He and A.E. Emmanuel carried on a long scientific debate on the significance of the Poynting theorem for power theory.

F. de Leon and J. Cohen (2008, 2010), as well as A.E. Emmanuel, are some of the strongest supporters of the viewpoint that the Poynting vector is essential for power theory in electrical circuits.

F. Emde (1902, 1923) distinguishes between two hypotheses concerning power transport in electrical circuits: the first, that power is transferred through the conductor, and the second, that power is transferred around the conductor. He links the mathematical expression of the Poynting theorem with the mathematical expression of reactive power.

J.A.B. Faria (2013:367) equates the complex power expression VI* with the complex expression of the Poynting vector, using the common formula E × H*. In his book, Electromagnetic Foundations in Electrical Engineering (2008), he states (p. 259), “…for time-harmonic regimes, the active power is to be physically identified with Joule losses averaged over time.” He considers that “…the complex Poynting theorem… is an absolutely general theorem” (p. 260) as shown in the formula: $p={\displaystyle {\int }_{S_T}S\cdot nds=ui.}$ (p. 315). “The Poynting vector is the carrier of electromagnetic energy. Electromagnetic energy is not carried by wires. Wires are simply used to guide the electromagnetic waves. Apart from wire losses, the energy flow is essentially external to the wires” (p. 312).

J.A. Ferreira (1988) supports the thesis that the Poynting theorem is relevant for power theory. Ferrero et al. (2000, 2001) state that the Poynting theorem represents the bridge between electromagnetism and circuit theories.

A. Föppl (1894) questions the assumption that the energy flows from the external electromagnetic field into the electrical conductors.

W.S. Franklin (1901, 1903, 1912) refers to J.J. Thomson’s critical view on the Poynting vector (i.e., that it is not uniquely defined). Franklin contradicts the ideas that voltage and current are physically equivalent to electric field and magnetic field intensities and that energy flows perpendicular to the surface of the conductor.

I. Galili and D. Kaplan (1996) underline the unity of electric and magnetic fields. They represent two sides of the same entity (electromagnetic phenomenon) that could co-exist in different proportions, depending on the observer’s frame of reference. They are facets of the same object observed from different perspectives.

M. Guarnieri (2011) introduces the mathematical equivalence between voltage, current, and electric and magnetic fields as:

$\begin{array}{l}E=-\nabla V\\ \nabla \times V=J\end{array}$

J.D. Kraus (1991:569) equates voltage and current of the circuit with electric and magnetic fields.

$p=VI=EHA$

These equations are based on the following equalities:

$\begin{array}{l}V={\displaystyle \int E\cdot dL}\\ I={\displaystyle \oint H\cdot dL}\end{array}$

R. Loudon, L. Allen, and D.F. Nelson (1997:1071) state: “For the electromagnetic momentum in material media, it is necessary to take account of contributions from both the electromagnetic field and the dielectric medium.”

C.G. Montgomery, R.H. Dicke, and E.M. Purcell (1948) consider the circuit theory as a part of electromagnetism; the Poynting vector is the connecting link between terminal quantities (voltages, currents) and field quantities (electrical field E, magnetic field H).

D.F. Nelson (1996:4713) concludes, “… E × H cannot be assumed to be the energy propagation vector and, in fact, is not.” He adds, “…the Poynting theorem does not apply to non-linear interactions” and asserts that questions regarding energy propagation in material media “can be definitively answered only when the matter is treated on as fundamental a basis as the electromagnetic fields are…”

S. Ramo, J.R. Whinnery, and T.V. Duzer (2013:139) state that power expression in electrical circuits is given in terms of voltage and current, while the energy expression in electromagnetic fields is given by the Poynting expression in terms of electric and magnetic field intensities.

P. Russer (2003) considers that the imaginary part of the complex Poynting vector (expression) represents the reactive power that is radiated.

S.A. Schelkunoff (1948) stresses the points of contact between circuit theory and electromagnetic theory. He defines field theory as focused on electromagnetic state as a function of space, whereas circuit theory is focused on electromagnetic state as a function of time. Both field theories and circuit theories are mathematical theories, which should not be confused with circuits or fields as physical concepts. Schelkunoff supports the idea of unifying circuit theory and electromagnetic field theory.

K. Simony (1956:31) states:

$P=UI=EHA$

“UI…denselben Wert ergibt als wenn wir die Leistung mit Hilfe des Poyntingschem Vektor ermittelt hätten” [UI gives the same value that we would have obtained by using the Poynting vector – author’s translation].

C.G. Someda (2006:57) states that “…the complex Poynting vector is

$P=\frac{E\times H*}{2}$

not related to the time-domain Poynting vector by the Steinmetz method.”

J.W. Simmons and M.J. Guttman (1970) state that “plane waves cannot carry angular momentum parallel to the direction of propagation” and “perfect plane waves do not exist.”

M. Stone (2000) notes the importance of distinguishing between energy and momentum of electromagnetic fields and pseudo-energy and pseudo-momentum related to waves moving through a medium. He identifies Minkowski’s expression with the pseudo-momentum of waves in matter (e.g., moving fluids).

J.A. Stratton (1941:133) was critical about the Poynting vector in his now classic book, Electromagnetic Theory: “…the validity of Poynting’s theorem is unimpeachable. Its physical interpretation, however, is open to some criticism.”

P.E. Sutherland (2007) states, on the basis of Poynting’s complex theorem, that there are two types of reactive power.

G. Todeschini et al. (2007) and A. Ţugulea (2002) support the thesis that the Poynting vector is relevant to power theory.

The literature investigated above supports the following tenets of the existing paradigm_

1. The Poynting theorem in classical electromagnetic theory is highly relevant for the concept of electrical power in classical circuit theory. The expression E × H signifies that: a) energy is stored in an electromagnetic field, 2) energy and momentum are ‘transported’ by the electromagnetic field, 3) the electric and magnetic fields are conceived as transverse infinite plane waves, 4) the direction of the Poynting vector indicates an outward flow of energy from the electromagnetic field and an inward flow of energy into the conductors of the electrical network. The electrical network is merely a huge antenna that guides the electromagnetic waves carrying the energy flow.
2. The Poynting theorem in classical electromagnetic theory represents the keystone of the conceptual and mathematical bridge between circuit and electromagnetic theories. Circuit theory is merely an approximation of classical electromagnetic theory; it is a sub-theory of electromagnetic theory.
3. The Poynting theorem is a cornerstone of classical electromagnetic theory. The Poynting theorem derives mathematically from Maxwell’s equations. Therefore, as Stratton states, it is unimpeachable and supports the physically ‘crazy’ (Feynman, 1961) idea that energy flows from generators to the loads through space and not along the transmission lines.

The author refutes the existing paradigm and questions the soundness of the Poynting theorem and its relevance to the process of energy transfer in conductors carrying currents. Because the Poynting theorem is based on classical Maxwellian electromagnetic theory, let us ask the question: What is Maxwell’s theory?

Asked the same question, Heinrich Hertz answered, “Die Maxwellsche Theorie ist das System der Maxwellschen Gleichungen” (‘Maxwell’s theory is Maxwell’s system of equations’) (Fölsing, 1997:371). This may be a witty aphorism, but it is also a self-referential tautology or, in plain English, a logical cul-de-sac. Moreover, a scientific theory is more than a mathematical theory; as we know, Maxwell’s equations are an axiomatically defined set of equations. It is true that physics talks in the language of mathematics, but not every mathematical utterance represents a law of physics. Hertz’s witticism is also misleading: in Maxwell’s time, his theory was already represented in more than 20 different mathematical formalisms. J.W. Arthur (2008, 2009, 2011, 2013) gives 24 different versions of Maxwell’s equations! Hertz refers to the four equations known as the Heaviside-Hertz revised version of Maxwell’s equations, written in the mathematical formalism of vector calculus, in which the important physical concept of magnetic vector potential is omitted (‘killed’ by Heaviside).

Nowadays, when we speak about classical electromagnetic theory, we also include Lorentz’s force equation, which introduces the atomistic concept in the purely ‘field’-oriented Faraday-Maxwell electromagnetic theory. More than 150 years after that theory was introduced, we cannot speak anymore about one electromagnetic theory. We have textbooks on quantum electrodynamics, relativistic electrodynamics, and topological electromagnetic theories and perpetual new research and developments in the field of electromagnetism, based on relativity and quantum mechanics, question the validity of the classical Maxwell-Lorentz electromagnetic theory. In a variation of Hertz’s dictum, one could say that nothing has survived from Maxwell’s original theory except four equations.

For this reason, the author felt necessary to investigate the literature that takes a critical approach to Maxwellian electromagnetic theory.

2.2 Opponents’ View: The Poynting Theorem is not the Keystone of a Bridge between Classical Electromagnetic and Circuit Theories

Classical electromagnetic theory considers the conductor as neutral matter. However, A.K.T. Assis and associates (Assis, 1997; Assis and Hernandes, 2007; Assis and La Mesa, 2001; Assis and Torres Silva, 2000; Assis et al., 1997; Bueno & Assis, 2001; Hernandes and Assis, 2003) stated that a conductor carrying a steady current is not neutral because there are electric charges on its surface, which are unequally distributed. The strength of these charges is proportional to the strength of the electric field of the source. The charges create an electric field outside the conductor. A conductor carrying a steady state current possesses both a longitudinal electric field (parallel to the axis of the conductor) and an azimuthal magnetic field (perpendicular to the axis of the conductor). A steady current possesses an electromagnetic field that acts inside and outside the conductor.

One can conclude that the electric charges distributed along the conductor’s surface act as a coating shield that will deflect the Poynting vector. Only in places where the density of the charges is null can the Poynting vector be perpendicular to the conductor’s surface.

Assis and his associates refute the idea of Clausius (also supported by Feynman, Purcell and many other scientists) that a conductor carrying a steady current is electrically neutral. In their view, conductors are not merely passive antennae submitted to electromagnetic force exerted by external media. They agree with Weber (1846, 1872) and Gluckman (1999) that the conductor is also an active element that exerts an electromagnetic force on the external media.

T.W. Barrett (1993, 2000, 2008) points to the existence of electromagnetic phenomena that cannot be explained by the classical Maxwellian electromagnetic theory. These unexplained phenomena are (1) the Aharonov-Bohm (AB) and Altschuler-Aronov-Spivak (AAS) effects, (2) topological phase effects, (3) phenomena related to bulk condensed matter (Ehrenberg’s and Siday’s observations), (4) the Josephson effect, (5) the Hall effect, (6) the de Haas–van Alphen effect, and (7) the Sagnac effect.

Barrett has the following criticisms of classical electromagnetic theory as modified by Heaviside and Hertz:

• Heaviside and Hertz consider scalar and vector potentials as merely mathematical artifacts and therefore, omit them from Maxwell’s equations; in fact, however, electrical and magnetic potentials do possess physical significance. They represent the physical gauge fields and should be expressed mathematically as local-to-global operators.
• As modified by Heaviside-Hertz, Maxwell’s theory became a linear theory characterized by simple Abelian U(1) symmetry. Modern electromagnetic theory extends the classical (Maxwell-Heaviside-Hertz) electromagnetic theory towards a higher, non-Abelian SU(2) symmetry.
• Maxwell’s classical theory is incomplete; it needs to be modified in order to include particles, consider the multiple connectedness of the electromagnetic space, and re-assess the electrotonic state by taking into account the existence of an electric scalar potential (ϕ) and a magnetic vector potential (A).
• The concept of medium is restricted to dielectrics, in which only a displacement current exists. Classical electromagnetic theory ignores the conducting current and the conducting elements.
• Classical electromagnetic theory does not consider electrical or magnetic sources.
• Heaviside and Poynting believed that a wire functions as a sink into which energy passes from the medium (ether) and is converted into heat; for them, wires merely guide energy, with the Poynting vector pointing at right angles to the conducting wire. Barrett contradicts this position.
• When Barrett states that Maxwell’s equations need extension, he is referring to the Heaviside-Hertz interpretation.

A.F. Chalmers (1973 a, b; 1975, 2013) also criticizes Maxwell’s theory for a number of reasons. First, he cites Maxwell’s poor understanding of the conductivity phenomenon. Maxwell perceives conductivity as a discontinuity in the medium (the electrical charge) and interprets electrical current as merely a rapid change of the displacement current. Maxwell assumes incorrectly that all electrical currents correspond to the motion of electrical charges, but this is not the case for a displacement current or an induction current. Only the conduction current is equivalent to a motion of charges. Second, Maxwell did not realize that a varying current radiates. He ignored electromagnetic radiation (later demonstrated experimentally by Hertz). Maxwell restricted his analysis of electromagnetic phenomena to source-free regions of space. Third, Maxwell was a reductionist: he reduced all electromagnetic phenomena to mechanical phenomena. Finally, Maxwell misinterpreted the displacement current as equivalent to a conduction current. However, the displacement current does not involve a motion of electrons or charges.

M. Frisch (2004, 2005, 2008, 2009, 2014) considers Maxwell’s theory to be an inconsistent theory. In addition, Lorentz’s added expression for force does not consider the fact that an accelerating charge experiences an additional force due to the self-field. Classical electrodynamics ignores the interaction of an electrical charge with its own field (self-interaction).

P. Graneau (1984, 1991) analyzes the different expressions for electromagnetic force. He states that the Ampère expression for force between current-carrying elements shows the existence of a longitudinal component, whereas Lorentz’s expression acknowledges only a transversal component. The existence of a longitudinal component explains such phenomena as rail-gun forces. He adds that classical electromagnetic theory neglects the non-local interactions between distant particles (i.e., action at a distance).

Graneau explains the functioning of induction motors as due to non-local electromagnetic effects and to the existence of the electromagnetic vector potential. Induction motor operation cannot be explained by the action of the local field or by the Poynting theorem; the energy needed to run the motor is transported across the air gap by non-local (action-at-a-distance) forces.

The electromagnetic potential provides a connection between distant particles so that the forces experienced by one particle are also felt by the others. In general, quantum mechanics involves non-local actions (e.g., the Aharonov-Bohm phenomenon), which are not explained by the classical electromagnetic theory. Like Assis, Graneau supports the Weberian electromagnetic theory and the action at a distance (non-local interaction) promoted by Ampère-Neumann-Kirchhoff-Weber.

Henning F. Harmuth (l986 a, b, c; 1989, 1991, 2001) criticizes Maxwell’s theory because it does not satisfy the causality principle and consequently cannot be applied to signal theory. Maxwell’s theory assumes that periodic alternating electric and magnetic waves are antecedent-free: they start from zero. However, only in theory it is possible to encounter a fully formed sinusoidal electromagnetic wave that starts from zero (at t = 0) without having any previous values before the starting point (for t < 0).

Harmuth’s critique of electromagnetic theory is also valid for circuit theory. As noted by Carson (1927:1), “…circuit theory explicitly ignores the finite velocity of propagation of electromagnetic disturbances.” Harmuth notes the importance of recognizing that Maxwell’s equations do not yield a wrong solution, but rather a solution that is undefined. Maxwell’s theory fails for signal propagation in a lossy medium (Ivrlač and Nossek, 2010).

The assumption of sinusoidal electromagnetic waves (as in circuit theory) implies a periodic sinusoidal wave within an interval

$-\infty <t<+\infty$

Mathematically, such a wave would have infinite energy, which is physically impossible.

The assumption of planar transverse electromagnetic waves (TEM) means a one-dimensional representation of the waves similar to the voltage-current representation in circuit theory; however, in a real lossy medium, the waves are three-dimensional.

J.A. Heras (1994, 2006, 2007, 2008 a, b, 2009, 2010a, b, 2011, 2016, 2017) gives a new and different explanation of the displacement current and the induction current. They are not determined only by the local and current values of the contiguous electromagnetic field. They also have a component (or a non-local term) determined by the delayed action of the global electromagnetic potential. This position is contrary to the classical Maxwellian interpretation, which states that the displacement current is equivalent to an ordinary conduction current. Heras demonstrates that the displacement current and the induction current are fundamentally different from the conduction current. In mathematical terms, Heras demonstrates that instead of differential equations, we should use integro-differential equations, which will reflect not only the time-dependent phenomena, but also the non-local (global) electromagnetic phenomena acting with time retardation.

S.E. Hill (2010, 2011) addresses a common misunderstanding of both Faraday’s law and the Maxwell-Ampère law. He says that Faraday’s law is often interpreted to mean that a time-varying magnetic field or flux induces a circulating electric field, i.e., that a changing magnetic field somehow causes a change in the electric field. Similarly, the Ampère-Maxwell law is interpreted to mean that a time-varying electric field causes a change in the magnetic field. He says that both laws are misinterpreted as expressing causality, whereas, in fact, there is only correlation. The principle of causality simply asserts that a cause event precedes the effect event, and if the events are separated by space, they must also be separated in time. In fact, we are dealing with a perfect correlation between a time-varying magnetic field and a time-varying electric field; both share a common cause, i.e., a time-varying current density. Since the two variations are simultaneous, it is incorrect to suppose that one could be the cause of the other.

O.D. Jefimenko (1962, 1966 a, b, 2004, 2008) criticizes Maxwell’s electromagnetic theory as being an ‘acausal’ theory. Equations linking E and H or B and D are equations of correlation and not of causation. Jefimenko (2008) supports Heras’ position with regard to reformulating Maxwell’s equations to take into account the magnetic vector potential and the non-Abelian, or SU(2), symmetry.

Jefimenko also stresses the fact that classical electromagnetic theory ignores the element of retardation or time delay. He points out that electromagnetic phenomena are determined not only by synchronous and contiguous conditions, but also by past and remote events.

Jefimenko considers classical electromagnetic theory as an unfinished theory: no physical theory is complete until or unless it provides a clear statement or description of causal links. A causal equation unambiguously relates a quantity representing an effect to one or more quantities representing a cause or causes. The principle of causality states that present phenomena are determined by previous events.

Jefimenko states that none of Maxwell’s four equations defines a causal relationship; each of these equations connects quantities that occur simultaneously, thus contradicting the theory of relativity. He says that Maxwell’s differential equations should be re-written as integro-differential equations. These equations should include quantities as they existed at a time prior to the time for which the quantities representing the effect are calculated.

C. Jeffries (1992, 1994) states that while the classical electromagnetic theory is based only on fields (E and H), modern electromagnetic theory is based on potentials (ϕ and A). The particle-field interaction represents energy exchange between charged particles mediated by gauge fields ϕ and A. The concept of particles (electrons) was foreign to Maxwell’s classical theory, which envisaged the electrical field as a continuum.

Jeffries also notes that steady-state current does not mean that the charges are moving with constant velocity: the charges inside a conductor experience acceleration. Therefore, the existing model for current transmission is inadequate.

Jeffries also considers that although the Poynting theorem is a mathematically correct expression for energy conservation, it cannot be interpreted as a physical law. Poynting’s vector fails to vanish in static electromagnetic fields and therefore cannot be a correct expression for energy flux. The expression

$u_p=\frac{1}{2}{\epsilon }_0\left(E\cdot E+c^{2}B\cdot B\right)$

is not a measure of physical energy, just as S_p = ε₀ E × B is not a measure of energy flux.

G. Kaiser (2004, 2011, 2012, 2015, 2016), R. Karlsson (Kaiser and Karlsson, 2005), and D. Jeltsema (Jeltsema and Kaiser, 2016) demonstrate that the Poynting complex vector (or the Poynting expression in complex vector form) is incomplete. Jeltsema and Kaiser (2016) state that the theory of instantaneous reactive power and energy is incorrect. They note that the expression Im(E x H*), under coordinate transformations, is not an invariant.

E.J. Konopinski (1978) demonstrates the fallacy of the generally held view that the vector potential A has no physical meaning in classical electromagnetism. One of his most important contributions is to show that both modern electromagnetic theory and quantum mechanics should be at the same level regarding the primacy of electric and magnetic potentials.

Konopinski contradicts Heaviside and Hertz, who considered the vector potential A as a mathematical artifact; they ignored Faraday and Maxwell’s position regarding the physical existence of an ‘electrotonic’ or magnetic vector potential.

Konopinski considers that replacing quaternionic algebra with vector algebra in formulating electromagnetic equations was a step backwards because it signified the mathematical transition from a non-Abelian higher symmetry back to an Abelian lower symmetry (from a non-commutative to a commutative algebra). He also stresses the importance of gauge theory for electromagnetism.

Konopinski’s analysis explains the results of the Aharonov-Bohm experiment and illustrates the physical reality of the magnetic vector potential. This position corresponds with the new developments in quantum mechanics.

According to M. Kline (1962), Maxwell assumed that electromagnetic phenomena occur in ideal conditions (a vacuum). His equations do not take into account the initial conditions and the boundary limits. He therefore also assumed that the vibrations of the ether particles are purely transversal, whereas a real elastic medium can have both transverse and longitudinal waves. Therefore, Maxwell’s equations cannot explain the existence of longitudinal waves, which have been proved to exist in an elastic medium.

Maxwell’s theory ignores the interaction of electromagnetic waves with matter. He considered electromagnetic waves as time harmonics and plane waves, but gave no indication of how to solve the propagation of harmonic waves taking into account curved boundary conditions and curved wave fronts. Plane waves possess infinite energy: the plane wave is a highly ideal concept. No real physical source emits or sends out plane waves.

According to D.F. Nelson (1979, 1991, 1995, 1996), charges on the surface of a conductor carrying current could disrupt the tangential H across a surface. The vector E × H cannot be assumed to represent the energy propagation vector (and in fact, it does not). The Poynting theorem is not applicable to non-linear conditions and to non-homogeneous media. Therefore, the Poynting vector is mathematically inadequate as a representation of the energy propagation phenomena. Questions about energy propagation in material media can be definitively answered only when the matter is treated as thoroughly as the electromagnetic fields are treated. The Poynting theorem does not describe the nature of the interaction between fields and matter.

P.T. Pappas (1983) presents a simple experiment that favours Ampère’s original expression for force and contradicts Lorentz’s expression for force. Ampere’s expression for force puts in evidence the existence of longitudinal force; there is mounting evidence (e.g. rail-gun, arc discharges) for such a force. Lorentz’s expression for force and Maxwell’s theory ignore the longitudinal forces that exist together with the transverse force.

W. Pietsch (2012) sees the electrodynamics of the 19th century as a case of under-determination (or double ontology) between a pure field theory and the action-at-a-distance theory. After Hertz discovered electromagnetic waves, it appeared that the field theory was the victor; however, after the discovery of electrons by Joseph John Thomson in 1897, the action-at-a-distance theory experienced a revival.

Pietsch’s thesis, that there are two competing electrodynamics, means that there is a scientific underdetermination perceived as particle-field double ontology. Field theory requires the existence of a continuous medium that allows for the strictly local transfer of physical actions. Action-at-a-distance theory assumes the existence of discrete or even point-like pieces of matter. The two ontologies correspond to different mathematical frameworks; field theory uses partial differential equations, whereas action-at-a-distance theory uses algebraic equations (proportions). A good example is Coulomb’s law describing the force between two particles: the force is proportional to the amount of charges and inversely proportional to the square of the distance between them.

Even Maxwell admitted that the difference between action at a distance and the field view did not arise from either party being wrong. Maxwell-Lorentz electrodynamics is a field-particle theory that accepts both fields and particles as fundamental entities. In summary, following a situation of scientific under-determination, none of the theory had been abandoned. The particle-field ontology has the advantage that it permits working with both particles and fields, depending on the context.

However, the double ontology creates conceptual problems: 1) there is no agreement on the exact expression for the force of a field acting on particles, 2) there is no explanation for the recoil force that charged particles experience when they are accelerated, 3) there are open questions related to the distribution of energy and momentum between fields and particles, 4) one is confronted with two distinct laws, one for action of a particle on the field and the other for the action of the field on the particle, and 5) the Lorentz force must be supplemented by an additional force term that accounts for radiation reaction. Pietsch (2012:144) agrees with Griffiths et al. (2010:391), who regarded such problems as ‘the skeleton in the closet of classical electrodynamics’.

Y. Pierseaux and G. Rousseaux (2006) state that longitudinal electromagnetic waves do, in fact, exist. These authors refer to the Riemann-Lorenz theory, which is based on scalar and vector potentials, and to Poincaré’s theory. Both theories support the existence of longitudinal electromagnetic waves (contradicting the classical Heaviside-Hertz electromagnetism). Their contribution is also important because it contradicts the Poynting theorem and the concept of pure transverse electromagnetic plane waves. On the basis of experiments, both these authors and Rousseau et al. (2008) consider it necessary to reformulate the classical electromagnetic theory with regard to the primary and fundamental magnitudes E and H, which should instead be φ and A, the scalar and the vector electromagnetic potentials.

W.G.V. Rosser (1963, 1968, 1970, 1976) criticizes Maxwell’s theory for considering that electromagnetic processes (actions) take place according to Newton’s law, i.e. instantaneously. The theory of relativity limits the maximum velocity of any process to the velocity of light.

The Poynting theorem appears to rest to a considerable degree on questionable physical hypotheses. However, Stratton (1941:10) proposes to retain the Poynting-Heaviside viewpoint “until a clash with new experimental evidence shall call for its revision.”