1 Introduction

Classical electrodynamics is traditionally regarded as a well-established and complete area of physics, yet there are some difficult questions that have persisted unanswered for more than \(\sim\) 120 years and have thus far been the subject of much controversy and discussion. One such class of problems involves the transfer of energy and momentum by electromagnetic field. The formal description of these processes was essentially given by Poynting [1], while the importance of electromagnetic momentum for the comprehension of various complex effects was emphasized by Thomson [2].

Among such effects, the mechanical forces arising from the interaction of light with opaque surfaces play an important role. They are now viewed as evidence that quickly oscillating electromagnetic waves carry momentum. The existence of such forces was previously predicted within the framework of electrodynamics by Maxwell ([3], sec. 792) and based on thermodynamic arguments by Bartoli [4]; nonetheless, obtaining experimental proof of their existence was no easy task. The first serious attempts to experimentally prove the existence of the pressure of light, treated as an electromagnetic wave, were made at the beginning of the twentieth century by Lebedew [5] and, somewhat later, by Nichols and Hull [6]. However, the first credible evidence would not appear until 1924 and is attributed to Golsen [7]. Today, after numerous experiments [8] (and the invention of optical tweezers [9] in the 1980s, for example), there is no question about the existence of the ponderomotive effect of electromagnetic waves.

An essentially different and much more difficult situation, however, occurs in cases where the electromagnetic fields are static. The behavior of the Poynting vector and related electromagnetic momentum is quite counterintuitive and reveals almost mysterious features. For example, a battery and a piece of permanent magnet placed together in a plastic jar [10] should generate a permanent energy flow in the surrounding space with uncompensated momentum. In another example, we can estimate the flow of electromagnetic energy arising from static electric and magnetic fields existing near the surface of the earth [11]. The mean value of the vertical fair-weather electric field is approximately E =  − 120 V/m, and the mean value of the horizontal component of the magnetic field is B = 40 μT. Using formula (1) below, we determine that near the surface of the earth there must be a massive permanent eastward energy flow with a density of S = 3.8 kW/m2 and having no apparent effect. In theoretically addressing the similar problem of a charged magnetized sphere (see Fig. 1), Heaviside [12] commented on the existence of such a flow with the words: “What is the use of it? On the other hand, what harm does it do?” What is the main reason that such a seemingly significant phenomenon is not observed experimentally? Among other things it is due to an enormously small factor \(\sim\) 1/c2 (≈ 1.1 × 10−17 s2/m2, c is the speed of light, see Eq. (5) below) controlling the relation between the electromagnetic energy flow and the corresponding electromagnetic momentum representing possible observable forces. Thus, we are dealing with very tiny effects or, in the words of some authors, relativistic effects that are extremely difficult to demonstrate experimentally. As far as we know, the only credible observation of an exchange between angular electromagnetic momentum generated by independent quasi-static electric and magnetic fields and a ponderable system is attributed to Graham and Lahoz of the University of Toronto (see [13] and references therein). There is, however, a complete lack of experimental evidence for a similar interaction between linear electromagnetic momentum and ponderable matter. That is why in this area of electrodynamics we must turn to thought experiments and theoretical speculations with varying degrees of reliability. Of course, such a state of the art provides a fertile substrate for many controversies, arbitrary constructs, and the introduction of rather peculiar concepts as well.

Fig. 1
figure 1

Illustration of the problem of a charged magnetized sphere—a model of the earth’s globe [12]. B is the horizontal component of the terrestrial magnetic field and E the vector of the fair-weather electric field

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To eliminate this serious flaw, which is evidently due to the absence of relevant experimenta crucis in this field, we have been looking for effects operating in quasi-static EM fields that are sensitive enough to reveal essential features of momentum exchanges.

Therefore, in this paper we reanalyze the ordinary Hall effect, a well-known and robust phenomenon operating nominally in static electromagnetic fields, where the conversion of electromagnetic energy and momentum can be quantitatively evaluated and compared with experimental data. We are convinced that a detailed study of this phenomenon can be very helpful in the resolution of some ambiguities in contemporary electrodynamics and may also contribute to the elucidation of the experimentally poorly manageable problem of so-called hidden momentum.

The present paper uses SI units throughout and is organized as follows: we begin by providing a brief formal description of flows of electromagnetic power and momentum as well as a concise overview of the problem of hidden momentum in electromechanical systems within a static electromagnetic field; next, we review the basics of the phenomenology of the Hall effect applied to the simplest possible metallic system with electrons as exclusive charge carriers, followed by a short critical discussion on the traditional interpretation of the ordinary Hall effect in terms of the Lorentz force; we then move on to a rather uncommon analysis of the Hall effect using concepts of the Poynting vector and linear electromagnetic momentum of static electromagnetic fields. In this section, which represents the core of the paper, we use quantitative estimates to show that the conversion of electromagnetic momentum into mechanical momentum and vice versa can be dependably traced. In Sect. 7, the general concepts of thought experiment and isolated system are discussed in connection with real Hall effect measurements. In addition, the significance of the time factor represented by the period of system assembly is also addressed. Our conclusions are then briefly summarized in the last section. For the sake of convenience and the flow of the text, some data and quantitative estimates appear in the Appendix.

2 Description of static electromagnetic fields

The conventional physical meaning of the Poynting vector, which in media with no intrinsic magnetization is defined by the formula [1, 14, 15]

$$\mathbf{S} = \frac{1}{{\mu_{0} }}(\mathbf{E} \times \mathbf{B}),$$
(1)

where E and B are vectors of an electric and a magnetic field, respectively, in a given point and μ0 is the permeability of vacuum (μ0 = 4π × 10−7 H/m), is the density of electromagnetic power flow measured in W/m2. As mentioned above, the Poynting vector is only seldomly applied to static electromagnetic fields, but it is very useful for describing quickly oscillating electromagnetic waves. Application of formula (1) to static fields is sometimes considered to be even unphysical [16]. Namely it may happen that S is generally non-zero, while its divergence identically vanishes (S ≠ 0, div S = 0). In such a case we can thus imagine that the electromagnetic energy circulates persistently in a closed path without any apparent effect. According to Jeans, [16] “it is difficult to believe that this predicted circulation of energy can have any physical reality”. Of a somewhat different opinion, however, was Feynman [17], who claimed that such a “mystic circulating flow of energy, which is seemed ridiculous, is absolutely necessary. There is really momentum flow needed to maintain the (angular) momentum conservation of the whole world”.

To resolve problems of this type satisfactorily, one has to be more specific about relations between electromagnetic forces and electromagnetic energy fluxes existing under various conditions. Surely there is an intimate connection between the volume force F exerted by an electromagnetic field on a system of charged particles, e.g. electrons (also, of course, on ponderable matter containing these electrons) and the flow of electromagnetic energy penetrating this system. According to H. A. Lorentz [18, 19], putting aside for a while the contact stresses mediated by the surface of the system, force F is given by the following integral taken over the volume V of the whole system:

$$\mathbf{F} = - \frac{1}{{c^{2} }}\int_{V} {\dot{\mathbf{S}}} {\text{ d}}V.$$
(2)

Here, Ṡ is the time derivative of the Poynting vector incident to the volume element dV, (cf. also [20]). This fundamental equation may be rewritten using a concept of quantity introduced by Abraham [21] under the name of electromagnetic momentum G:

$$\mathbf{G} = - \frac{1}{{c^{2} }}\int_{V} \mathbf{S} {\text{ d}}V.$$
(3)

The law of a purely electromagnetic force exerted on a ponderable electronic system then reads.

$$\mathbf{F} = - \dot{\mathbf{G}},$$
(4)

where Ġ again means the time derivative of electromagnetic momentum. This equation is, however, in full accord with the law of force known from classical mechanics, thus it can be inferred that the electromagnetic momentum is of the same nature as the mechanical one. Equation (3) can also be reformulated locally. For the volume density of electromagnetic momentum g we obtain the following simple relation:

$$\mathbf{g} = \frac{\mathbf{S}}{{c^{2} }}.$$
(5)

Considering formula (1), this quantity may be alternatively expressed as.

$$\mathbf{g} = \varepsilon_{0} \left( {\mathbf{E} \times \mathbf{B}} \right),$$
(6)

where ε0 is the permittivity of vacuum (ε0 = 8.85 × 10−12 F/m).

Applying Eq. (2) to the case of purely static electromagnetic fields, e.g. to a crossed electric and magnetic field, the time derivative of the Poynting vector Ṡ will be identically zero, meaning that there will be no force exerted by these fields on, and no momentum imparted to, the electronic subsystem. However trivial this conclusion may sound, it is crucial for our further considerations. For example, the macroscopic systems at rest submitted to the static electromagnetic fields, where Ṡ = 0 identically, does not, according to Eq. (2), interact with these fields, although its parts, i.e. electrons, may “feel” these fields. For the same reason, there is also no integral electromagnetic momentum imparted to the system or stored within it in some form. Incidentally, this fact fully accounts for the absence of mechanical effects induced by the electromagnetic energy flow in Heaviside’s problem mentioned above. Since the interactions may take place only if the Poynting vector entering the system is changing in time (Ṡ ≠ 0), it is only the period during which the external fields are set into their final static state that is decisive for correct analysis of systems in static electromagnetic fields. This period (or process) will be tentatively called the period of system assembly (cf. [22]). The key significance of this time factor for correct interpretation of systems submitted to static electromagnetic fields is discussed in detail in Sect. 7.

As will be further shown, the simple fact that an electromagnetic field may carry momentum, inevitably entering into the momentum balance of electromechanical systems, frequently generates problems and paradoxes that have yet to be satisfactorily resolved, including the problem of hidden momentum; therefore, the following section looks at some leading opinions concerning this puzzling entity and related concepts.

3 Hidden momentum

In his estimable papers [2, 23], Thomson investigated the behavior of the momentum of an electric (i.e. electromagnetic) field corresponding to various configurations of electric charges. Studying the momentum balance of these systems, he produced a remarkable paradoxical result, namely that an electromagnetic system at rest may contain non-zero electromagnetic momentum, and in doing so launched a lively debate that continues in the present day. His further computations give rise to the possibility of constructing an electromagnetic (bootstrap) spaceship, a contrivance that would propel itself in some direction without transferring momentum to its surroundings [24]. He also discovered that the angular electromagnetic momentum of a magnetic monopole plus an electric point-charge is independent of the separation between the two, anticipating in this way Dirac’s model of electron spin [25]. Essentially the same conclusion was later arrived at by Page [26], who in connection with Gunn’s theory of double star formation, which was popular at the time, investigated the behavior of ionic atmospheres in static electromagnetic fields.

The subject experienced a boom in the 1960s when Shockley and James [27] interpreted their highly idealized conceptual thought experiment using a special model device involving a combination of electrostatic forces and momenta induced by magnetic dipoles of either a Gilbertian or Amperian nature. Among other things, they showed that an electrically neutral current loop in an electric field experiences a special force, and that a body whose center is at rest can carry a non-zero mechanical momentum. These surprising discoveries compelled them to coin the term hidden momentum, which they associated with the lacking mechanical momentum balancing electromagnetic momentum in systems at rest. They conjectured that the effect was due to the presence of reactive forces, likely of a relativistic origin, acting on the sources of the external fields.

Their pioneering paper was followed by one from Coleman and Van Vleck [28], in which the authors investigated a similar hypothetic system consisting of an electric point charge and a magnet. Their computations, which were model independent, showed that if the magnetic moment is changed, the magnet exerts a force on the distant electric charge without apparent back action. Since evidence for this was prevailingly given in terms of the relativistic center-of-mass/energy theorem, they considered such a discrepancy in the action-reaction balance, which was in full agreement with Shockley and James, a relativistic correction. It should be stressed that the relativistic center-of-mass/energy theorem traditionally provides a basis for many argumentations concerning the problem of hidden momentum. The theorem as proposed by Taylor [29] reads:

Center-of-mass/energy theorem: The total momentum of an isolated system is zero in that inertial frame in which the center of mass/energy is at rest.

Unfortunately, the straightforward application of this theorem to various hypothetical electromechanical models is not always correct (cf. [22]). Namely in cases where a part of the electromagnetic momentum imparted to the system, which was originally at rest, is transformed into some hidden form and not an overt form, the momentum balance is uncertain, and the center of mass/energy of the system will no longer be at rest. Thus, the conditions of the theorem are violated, making its usage questionable. This very fact is often ignored in the literature or left without an adequate explanation.

General considerations concerning the linear electromagnetic momentum of stationary distributions of charges and currents and corresponding static electromagnetic fields are credited to Calkin [30, 31]. Consequently, separating field net momentum and source net momentum requires a description of electromagnetic momenta that is slightly different than the one used in Sect. 2. While electrostatic fields are as usually described by the Coulomb law in terms of charge density ρ, the static magnetic field generated by the stationary distribution of electric currents is defined by vector potential Λ:

$${{\varvec{\Lambda}}} = \frac{{\mu_{0} }}{4\pi }\int_{V} {\frac{\mathbf{i}}{r}} {\text{ d}}V.$$
(7)

Since qΛ(x) physically represents the linear momentum acquired by the electromagnetic field when charge q is brought from infinity to point x, we can write the linear electromagnetic momentum within a certain volume ν as.

$$\mathbf{G} = \int_{V} \rho {{\varvec{\Lambda}}}{\text{d}}V.$$
(8)

Nonetheless, this formula, which provides an alternative to (3), directly connects the electromagnetic momentum with characteristics of the sources. If we now apply the center-of-mass/energy theorem, we find that the absolute magnitude of linear electromagnetic momentum (8) and that of the mechanical momentum of the sources must be equal and opposite to one another; however, when the source current realized by a circulating charge (i.e. an Amperian loop) is treated not classically but in the framework of special relativity, this argument is shown to be not exactly valid. This is because in relativistic mechanics, the mass of a charge carrier varies with velocity, so the momentum is no longer directly proportional to it.

A good illustration of this is provided by the following instructive but somewhat non-realistic model (ignoring the unavoidable existence of surface charges on the conductor [32, 33]) suggested by Babson et al. [34]: a rectangular current loop made of a hypothetical conductor having no screening ability and no internal scattering, and which is subjected to an external electric field, was treated either classically or relativistically. It was shown that in the latter case, due to the difference of charge carrier speeds in different parts of the loop, an extra relativistic correction to the momentum occurs that is not associated with the macroscopic movement of the loop as a whole. Evidently, then, the momentum has the character not of overt momentum but of hidden momentum. Moreover, the value of this relativistic correction makes it possible to distinguish whether the involved charged particles have a dynamical character as Amperian current loops or are pairs of static Gilbertian magnetic charges. In this way, for example, hyperfine splitting of the hydrogen ground state confirms that the magnetic momentum of the proton is due to an Amperian current loop [35].

A rather discordant contribution to the problem of electromagnetic forces and hidden momentum was published by Mansuripur [36, 37]. Analyzing anew the behavior of electromagnetic forces exerted by a magnetic dipole on an electric charge in a magnetically polarizable medium, he inferred that the Lorentz force law is incompatible with special relativity and conservation laws. Therefore, he formulated the radical claim that the Lorentz force law should be abandoned in favor of the one proposed by Einstein and Laub [20, 38], which is in conformity with these conditions. What is the essence of this alternative law completing the system of Maxwell’s equations? The main difference of this approach, with respect to the Lorentzian one, lies in the introduction of electric and magnetic polarization vectors as independent variables in addition to usual bound electric charges and currents. Consequently, in formulae for electromagnetic energy density and the Poynting vector (see formula (1)), quantity H (magnetic field strength) must appear in place of magnetic vector B. In polarizable materials, there are also distributions of local electromagnetic forces, and momentum densities change correspondingly. Nevertheless, the integral end formulae for finite systems are identical for both approaches. For example, the last equation for the electromagnetic volume force in the Einstein–Laub paper [20] coincides with Lorentz’s relation (2). Quite naturally, both approaches must remain equivalent for materials without intrinsic polarizations, e.g. metallic systems. Considering then the aforesaid differences in the distribution of momentum densities, Mansuripur claimed that if one uses the Einstein–Laub formalism, the introduction of the concept of hidden momentum into problems of the Shockley–James type is redundant. This claim, however, was rebutted by Griffiths and Hnizdo [39, 40], who, referring to an older paper [41], have shown that the solution to the problem is dependent on which magnetic dipole model is used (i.e. the Gilbertian or the Amperian). Moreover, they showed that the Lorentz force law is consistent with special relativity.

In addition to analysis of various idealized thought models, there have been efforts to quantify hidden momentum by using a single compact formula to simply identify hidden momentum in any arbitrary system. Of interest, then, is a definition of hidden momentum formulated by Vanzella [42], cf. [43]: if we consider the electromagnetic force of density fV, which exerts upon the volume element dV of a certain specified subsystem having the relative velocity v, the hidden momentum Gh will be defined by the volume integral over this subsystem, namely,

$$\mathbf{G}_{h} = - \frac{1}{{c^{2} }}\int_{V} {\mathbf{r }(\mathbf{v f}_{V} )} {\text{d}}V,$$
(9)

where r is the position vector of a given volume body element with respect to the center of mass/energy of the entire system. Where the movements of body elements are stationary, the product r (v fV) corresponds to the flow of electromagnetic power mediating their mechanical interactions, and Eq. (9) then represents the hidden mechanical momentum stored in the stationary structural patterns within the system. The main persisting flaw of this definition, however, is the uncertainty as to which subsystem Eq. (9) actually applies.

Torque and forces affecting stationary magnetic dipoles of a various physical constitution were investigated by Vaidman in [33]. A key point of his argumentation was a special lemma derived from the center-of-mass/energy theorem given above, namely:

Any finite stationary distribution of matter has zero total momentum.

In stationary situations, where the Amperian loop carrying constant current is placed in a static external electric field E, the momentum of the resulting electromagnetic field of the whole system can be computed by formula

$$\mathbf{G}_{{{\text{em}}}} = - \varepsilon_{0} \mu_{0} \left( {\mathbf{m} \times \mathbf{E}} \right),$$
(10)

where m is the magnetic dipole moment of the loop. Since the system is finite and stationary with respect to the laboratory frame, the above lemma can be used, and consequently the total momentum of the system must be zero, i.e.

$$\mathbf{G}_{h} + \mathbf{G}_{{{\text{em}}}} = 0.$$
(11)

Therefore, the current loop necessarily contains internal hidden mechanical momentum Gh, which precisely compensates the electromagnetic momentum of the surrounding field. Yet because this argumentation implicitly depends on the application of the center-of-mass/energy theorem, caution towards such a derivation is in order.

A valuable contribution to the problem of hidden momentum was made by Hnizdo [44], who carefully analyzed stationary systems carrying both charges and currents. In his paper, he demonstrates that the existence of hidden momentum in such systems is sine qua non for maintaining correct relativistic transformation properties of total energy and momentum. Consequently, hidden momentum is not an omittable constituent in relativistic momentum analysis of stationary systems; moreover, for ponderable bodies, the constituents of which move stationarily, formula (9) was derived independently. Essential points of this approach are the assumption of validity of Vaidman’s lemma, i.e. particularly the validity of formula (11) and also the natural assumption that the magnetic fields involved are generated dynamically by stationary movements of charged elements of the system (Amperian loops) and not by an arrangement of relativistic invariant magnetic charges (Gilbert’s dipoles). Interestingly enough [43], the force of density fV does not necessarily have to be of an electromagnetic origin—the formula remains valid for gravitational forces as well. This fact sheds new light on the concept of hidden momentum, which is obviously more general than is usually thought.

Previous opinions in the literature concerning the existence of hidden momentum were reexamined by Redfern [45], who turned his attention to a fact of primary importance, namely, that one often tacitly assumes the possibility of making electromagnetic experiments in absolute isolation from the surroundings. Of course, an ad hoc system consisting of, say, a current loop and a point charge can be considered (without concern for how the system came to be) at rest in an arbitrary frame of reference. This approach makes it possible to apply the relativistic center-of-mass/energy theorem without problems; it does, however, ignore the Lorentz forces inevitably involved when the system is built up [22]. Computed examples have shown that when the momentum imparted to the system during its assembly is taken into account, the total momentum is conserved without the need to postulate a hidden form of momentum at all.

Although no more than a concise digest of elementary facts related to hidden momentum, this brief review clearly demonstrates that in spite of the enormous attention devoted to the subject, it remains a vague and controversial entity, something of a façon de parler about perfectly genuine non-zero momentum (largely of a non-electromagnetic nature) that occurs or can be stored in a system whose center-of-mass remains at rest. On the other hand, it convincingly shows that this seemingly marginal problem reaches to the more distant parts of classical electrodynamics and physics. As such, it deserves a satisfactory theoretical resolution supported (in the best-case scenario) by experimental evidence.

4 Phenomenology of the ordinary Hall effect

Let us now recall and summarize the basic experimental facts concerning the ordinary Hall effect, a promising candidate for a future case study on momentum balance in electromechanical systems, which also has very interesting historical and epistemological connotations.

The effect is generally known as a useful practical tool for the study and characterization of electronic conduction processes in metals and semiconductors [46,47,48,49,50]. Its discovery in 1879 [51, 52], however, was closely related to somewhat general questions concerning the physical nature of electric currents [53]. After experimental confirmation of the equivalence between an electric current and a moving charged body (an idea originally proposed by Faraday) by means of Rowland’s famous experiment [54], pressing new questions arose. For Hall, the following passage in Maxwell’s Treatise [3] (section 501) was the most challenging: “It must be carefully remembered, that the mechanical force which urges a conductor carrying a current across the lines of magnetic force, acts, not on the electric current, but on the conductor, which carries it.” Hall disagreed. He regarded the statement to be in conflict with the most natural supposition and believed that a magnet acts upon the current and not primarily on the conductor. Since his lecturer, Rowland, had a similar view, Hall undertook a series of experiments as part of his Ph.D. dissertation intended to confirm an opinion that was precisely opposite to that of Maxwell. Under the supervision of Rowland, who provided him with unusually accurate instrumentation and support, Hall did a series of interesting experiments, though the results were, unfortunately, only negative. In the end, Hall carefully repeated an experiment that had previously been carried out by Rowland, and although he had no foreknowledge of Rowland’s result, he proved the direct influence of an external magnetic field on a current inside a conductor [51].

The original layout of Hall’s successful 1879 experiment was as follows (cf. [51, 52, 55], see Fig. 2): the ends of a strip of thin gold leaf mounted on a glass plate were fitted with massive contacts as wide as the strip itself, forming part of an electric circuit through which a current was passing. The sample was placed between the poles of an electromagnet, with the plane of the strip perpendicular to the lines of magnetic force. A sensitive galvanometer was connected at different points on opposite edges of the strip, and the positions of these points were then changed until they were found to be at the same potential. When the magnetic field was created or destroyed, a permanent deflection of the galvanometer needle was observed, indicating a change in the relative potential of the two points at opposite edges. It was thus shown that the magnetic field produced a new electromotive force in the strip of gold leaf at right angles to the primary electromotive force as well as to the magnetic force and also proportional to the product of these forces. In contemporary notation, the corresponding empirical relation for the observed Hall voltage VH based on these measurements reads

$$V_{H} = R_{H} \frac{IB}{d},$$
(12)

where I is the primary current, B the applied magnetic field, d the thickness of the gold strip, and RH the constant factor (with a dimension of m3/C), now called the Hall coefficient. Interestingly enough, Hall believed that RH should be a universal constant independent of the size and material of the sample, thus in some way anticipating the quantum Hall effect [56].

Fig. 2
figure 2

A sketch of Hall’s original experimental arrangement (according to [52], g—glass substrate, m—metal (gold leaf), b—copper blocks)

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In Hall’s experiment, the length of strip L was much greater than its width w (w ≪ L), which corresponds to the arrangement now known as the long-sample configuration. Since the influence of the current-supplying contacts can be neglected in this configuration [48], which significantly simplifies theoretical considerations and analysis of the experiments, we will use it in this paper exclusively. Moreover, for the sake of definiteness, a gold sheet (Au) with thickness d, length L, and width w is used as a model device. For interpretation and quantitative estimates, we can thus use the simplest version of transport theory, the free electron gas model [57] (see also Appendix) with appreciable credibility.

The phenomenology of the ordinary Hall effect can be summarized as follows: in the experimental arrangement described above, the turning-on of the magnetic field B gives rise to an extra electromotive force, which can be represented by a certain vector EH (Hall field) perpendicular to both—the ohmic current and magnetic field. We can thus express this effect as a small correction (E ≫ EH) to Ohm’s law [55, 58], namely,

$$\mathbf{i} = \gamma (\mathbf{E} + \mathbf{E}_{H} ),$$
(13)

where, as usual, i is current density, E the external field, and γ conductivity. The correction means, however, that the current and the conjugate equipotential lines can no longer be rectangular, and the equipotential lines must be slightly skewed. In cases where the voltage drop VH is measured by means of a high-impedance device, it can be assumed that the normal current component along the sample edges is identically zero. Then the currents driven by EH will charge the strip edges until it is completely canceled by the induced electrostatic field. The result of this process, which is quite analogous to the formation of the flow tube (sphondyloid) structure in conductors [32, 59], will be the reestablishment of a system of current lines in bulk parallel to the sample edges. Absolute values of the vector quantities involved can then be expressed using (12) and other obvious relations as

$$E_{H} \approx \frac{{V_{H} }}{w} = R_{H} \frac{IB}{{dw}} = R_{H} iB = R_{H} \gamma EB.$$
(14)

Considering the relative positions of the corresponding vectors, we obtain the 3-vector formula

$$\mathbf{E}_{H} = R_{H} \gamma \,(\mathbf{E} \times \mathbf{B}),$$
(15)

which explicitly gives the correction term in (13), thus making (15) the most important empirical relationship describing the ordinary Hall effect.

5 The Hall effect as a manifestation of the Lorentz force

This somewhat specific form of Eq. (15) describing the Hall effect immediately gave rise to fundamental questions, such as whether the magnetism is of a rotary or linear nature [60] and how the system of Maxwell’s equations should be completed to involve the forces exerted by a magnetic field on electric charges and currents [18]. One of the results of this effort was the most common explanation of the Hall effect based on the Lorentz electromagnetic force [61], which affects the moving charges and electric current inside the sample.

The position of the Lorentz force within the system of classical electrodynamics is somewhat solitary [14] and reveals ambiguous aspects [36]. While Maxwell’s equations describe how moving charged particles, or equivalently macroscopic currents, give rise to surrounding electromagnetic fields, the Lorentz force law describes the influence of an electromagnetic field on moving charges. Since Maxwell’s equations do not contain any direct information about the effect of fields on charges, it is impossible (without strong additional assumptions) [62] to derive the Lorentz force law from Maxwell’s equations alone. Rather it is a separate physical law that was independently established by generalizing certain experimental results [18]. The corresponding formula can be written as

$$q\,\mathbf{f} = q\{ \mathbf{E} + (\mathbf{v} \times \mathbf{B})\} ,$$
(16)

where f is the Lorentz force per unit charge, E and B are local electric and magnetic fields, respectively, and v is the velocity of exposed charge element q. All quantities are related to the same inertial frame, and the charge element is assumed to be infinitesimally small, q → 0. Formula (16) also enables one to unambiguously decompose the observed force density f into an electric and a magnetic field, provided that the kinematic parameters of the test charge path are known and the Bremsstrahlung for this path is negligible.

The traditional textbook Lorentz force-based derivation of the formula describing the Hall effect is then astonishingly concise. (Throughout this paper the following convention is used: charge of electron q =  − e, where elementary charge 0 < e = 1.602 × 10−19 C). Charge carriers—electrons traveling through a conductor with drift velocity vD—are deflected by the magnetic component of the Lorentz force (16) toward one edge of the sample, which is enriched by charge carriers while the opposite one is depleted. Then, across the sample, this charge redistribution generates the Hall electric field, which in turn counterbalances the magnetic component of the Lorentz force so that the charge carriers in the bulk continue to flow as before. This idea of a dynamical equilibrium between the Hall field EH and magnetic part of the Lorentz force affecting an electron with drift velocity vD (velocity of electricity [51]), can be expressed as

$$e\mathbf{E}_{H} + e\,(\mathbf{v}_{D} \times {\mathbf{B}}) = 0.$$
(17)

Taking into account conventional forms of Ohm’s law and Drude’s relations [63], between current density i and drift velocity vD of electrons, i.e.

$$\mathbf{i} = - ne\,\mathbf{v}_{D} ,$$
(18)

we immediately obtain a relationship that is identical to formula (15) above:

$$\mathbf{E}_{H} = \frac{1}{ne}({\mathbf{i}} \times \mathbf{B}) = R_{H} \gamma \,(\mathbf{E} \times \mathbf{B}),$$
(19)

where the Hall coefficient RH clearly equates to

$$R_{H} = \frac{1}{ne}.$$
(20)

The most straightforward derivation of the Hall effect formula is, nevertheless, to simply identify correction term EH in (13) with the magnetic term of the Lorentz force density f in (16) (cf. e.g. [64]). The empirical relationship (15) then follows immediately, provided that v, the velocity of electricity, again has the meaning of drift velocity vD. Indeed, let us start with the original form of Ohm’s constitutive relation for electric current density i in terms of generalized force density f (= Elektroskopische Kraft [65]), which assumes the existence of constant proportionality factor γ. It reads simply

$$\mathbf{i} = \gamma \,\mathbf{f}\mathbf{.}$$
(21)

By inserting f from (16), we obtain the formula

$$\mathbf{i} = \gamma \,\{ \mathbf{E} + (\mathbf{v}_{D} \times \mathbf{B})\} ,$$
(22)

which is fully congruent with (13). Somewhat formal as it may be, this approach makes clear the nature of the Hall effect as a local magnetically induced correction to linear transport.

Despite the fact that both derivations above provide us with the “correct” result, they have some serious flaws, which in the current literature are either completely ignored or not discussed properly. Quite incomprehensible is the rather teleological use of drift velocity vD instead of, for example, the actual electron velocity v ≤ vF (Fermi velocity) in formulae (17) and (22), which traditionally accounts for deflection of drifting charge carriers. Granted the electrons are deflected by the magnetic part of the Lorentz force, the path of every moving electron must be adequately described by equations for the cyclotron (Larmor) frequency ω and cyclotron radius r,

$$\omega = \frac{eB}{m} = \frac{v}{r},$$
(23)

where m is the electron mass and v is the component of electron velocity perpendicular to the magnetic field. The radius of the cyclotron orbit at B = 0.2 T corresponding to the typical drift velocity (see Appendix) vD ≈ 2.1 × 10−5 m/s is r ≈ 6.0 × 10−16 m, so that it is even smaller than the classical radius of the electron ≈ 2.8 × 10−15 m [66]. The idea that such an effect can be responsible for some observable deflection of individual electrons is questionable at best. On the other hand, the use of a seemingly more plausible mean electron velocity \(\left\langle {\mathbf{v}} \right\rangle \, \le \,{\mathbf{v}}_{F} \approx {1}.{4}\, \times \,{1}0^{{6}}\) m/s in (16) will result in a non-realistically huge estimate for the Hall field.

To resolve this problem, a slightly modified strategy must be used. The central concept of this approach is relaxation time τ, which is assumed to be independent of the energy of the charge carriers freely moving in the conductor. This quantity, which primarily measures the mean time interval between subsequent collisions of a charge carrier with scattering centers in the conductor (see Appendix), also provides an estimate for the characteristic time of charge redistributions within the conductor. This time scale τ may also be used for independent definition of quantities known from Drude’s transport theory [64], i.e. of mobility μ and drift velocity vD,

$$\mu = \frac{e\,\tau }{m} = - \frac{{\mathbf{v}_{D} }}{\mathbf{E}},$$
(24)

where E is again an electric field causing a coordinated drift of the system of electrons.

Another time scale brought about in the problem by a magnetic field is the inverted Larmor frequency 1/ω (23), which is independent of the velocity with which an electron enters the domain of a magnetic field. The ratio of these two time scales, then, is decisive for the behavior of an electronic system in a magnetic field. Of physical importance is the limiting case where

$$\tau \ll \frac{1}{\omega }.$$
(25)

By using (23) and (24), this condition can also be rewritten in the equivalent form

$$\mu B \ll 1.$$
(26)

Relations (25) and (26) define an analytically treatable weak magnetic field limit and also provide the characteristic condition defining the ordinary Hall effect [67]. For samples having medium mobility μ (in our case is μ = 5.2 × 10−3 m2/Vs, see Appendix), condition (26) is fulfilled practically for all magnetic fields accessible in the laboratory. What is the physical meaning of condition (25)?

Using the classical description, it can be said that between subsequent collisions, the electron travels only a small segment of the cyclotron orbit (see Fig. 3). The angle θ corresponding to the circumscribed part of the cyclotron orbit and simultaneously characterizing the deflection of the electron velocity vector can be expressed as

$$\theta \approx \omega \tau .$$
(27)
Fig. 3
figure 3

Projection of the path passed between two subsequent collisions by an electron having velocity v on a plane perpendicular to the vector of the external magnetic field. Insert depicts the configuration of vectors of the primary and the Hall electric field. θ is the Hall angle

Full size image

Thus, regardless of their actual velocities, the paths of all the electrons are rotated within time τ in the plane perpendicular to the vector of the magnetic field, through about angle θ. Since on average the chaotically moving electrons constitute an ordered drift along current lines, it is evident that the current lines themselves must be simultaneously tilted through the same angle θ. These current lines then inevitably collide with the sample boundaries, instantaneously creating surface charges there. Deposited charges induce electric field EH (i.e. the Hall field), which forces the current to flow in parallel with the edges again. These considerations (among others) are the basis for the very fact that the entire process of deviation of electrons or, equivalently, of current paths by the Hall angle θ, takes place during the relaxation time τ; in the opposite case, namely, would be Eq. (27) inconsistent with observations. We can formulate this important piece of knowledge in the following lemma:

It can be taken as an empirical fact that the Hall field is, after application of an external magnetic field, fully established during the relaxation time τ.

The validity of Eq. (27) also enables one to define the Hall angle in a more convenient way, i.e. directly in terms of the electric fields, or in terms of mobility and the magnetic field, as (cf. Fig. 3)

$$\left| {\frac{{\mathbf{E}_{H} }}{\mathbf{E}}} \right| = \mu B = {\text{tg}} \,\theta \approx \theta ,$$
(28)

where the last approximation takes place in very weak magnetic fields.

The success of the use of collective drift velocity vD in the derivation of formulae (19) or (22) for the Hall field EH may be now explained as follows: when the magnetic field appears, all paths of free electrons, regardless of their velocities, simultaneously rotate through the same angle θ. If the distribution of electron velocities were isotropic, there would be no change in the frequency with which the electrons collide with the sample edges; however, in cases where all the electrons have, on average, a common velocity component, i.e. the drift velocity vD, the collisions with one edge will be more frequent than with the other. Consequently, in the vicinity of the sample edges, thin space charge layers appear bearing opposite net charges, inducing the compensating Hall field EH.

For the sake of the completeness of this section, let us give one more potential explanation, also using the Lorentz force, of the ordinary Hall effect (cf. [48]). It is based on analysis of the so-called gyroscopic effect, which was studied, for example, by Page [68]. Accordingly, a free particle of mass m and charge q, passing across a magnetic field with velocity v, starts to rotate like a mechanical flywheel about vector B on a circular or helical path with a cyclotron radius r. The electrical part of the Lorentz force (16), i.e. the term qE, then causes uniform movement of these circular paths with constant velocity u, which is given by formula

$$\mathbf{u} = \frac{{{\mathbf{E}} \times {\mathbf{B}}}}{{B^{2} }}.$$
(29)

Vector u, being collinear with the corresponding Poynting vector (1), is perpendicular to both vectors E and B, meaning there is no mechanical work carried out by these forces. The side progress of circular orbits thus takes place only at the expense of the original kinetic energy of the particles. Due to such a decrease in energy, the cyclotron radius exposed to the electric field is diminished by ratio

$$\eta = \frac{{\sqrt {v^{2} - 2{\mathbf{uv}} + u^{2} } }}{v}.$$
(30)

Using this scenario, the Hall effect can be explained as follows: free electron orbits start to shift with speed u in the direction of the Poynting vector until they reach the sample edge. The charge deposited there then induces the field EH, which stops the process and reestablishes the original distribution of current lines within the sample. A serious flaw in such an explanation, however, is the interesting fact that vector u has, according to (29), the same direction and value for all charged particles regardless of their masses, charges, and sign of charges. Moreover, this gyroscopic model is seriously disturbed by frequent scattering events corresponding to the weak magnetic field limit (25). For these reasons, the gyroscopic effect does not provide any plausible explanation of the ordinary Hall effect.

6 The Hall effect and the Poynting vector

This section attempts to homologize the general conceptual basis currently used for describing electromagnetic fields (see Sect. 2) with the description of our concrete case, i.e. the ordinary Hall effect. To interpret this effect in terms of the flow of energy or electromagnetic momentum, it is first necessary to separate the components of the Poynting vector belonging to the ohmic transport and the Hall effect, respectively. If the contacts are perfectly conducting equipotential surfaces, Ohm’s component SΩ of the Poynting vector incident to the sample surface can be visualized immediately (see Fig. 4). Near its surface, the primary current I flowing through the sample generates electric field E parallel to the current lines and, simultaneously, proper magnetic field BΩ closely circumventing the sample in the planes, which are perpendicular to these lines.

Fig. 4
figure 4

Schematic view of the ohmic components of the steady-state Poynting energy flows. Notice the inward ohmic energy flow SΩ and total outward thermal electromagnetic radiation flow SR

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Considering general formula (1), we can conclude that the vector SΩ enters the interior of the sample through its whole surface with the exception of the contacts. This component of the Poynting vector, for which obviously div SΩ < 0, provides the continuous energy supply from the voltage source, which within the bulk of the sample is then converted to Joule’s heat. The continuous production of this heat is compensated by enhanced thermal radiation represented by the Poynting vector component SR. In addition to these two components ensuring the dynamical energy balance of the sample, there is another independent one, which we tentatively call the Hall component of the Poynting vector SH. This vector SH can be easily constructed from longitudinal electric field E and external magnetic field B by direct application of formula (1) (see Fig. 5), i.e.

$${\mathbf{S}}_{H} = \frac{1}{{\mu_{0} }}\,(\mathbf{E} \times \mathbf{B}).$$
(31)
Fig. 5
figure 5

Construction of the Hall component of the Poynting vector SH, generated by ohmic electric field E and external magnetic field B

Full size image

Obviously, because of the Hall experimental arrangement, the vector SH enters the sample through one edge and leaves it unchanged through the opposite one (see Fig. 5).

At first glance, a striking congruence between formula (31) for the Poynting vector SH and empirical relation (15) for the Hall field EH is apparent. These formulae, differing only in a certain pre-factor, can be recast into an interesting relationship with the help of (19) and (24):

$${\mathbf{E}}_{H} = (\mu_{0} {\mathbf{S}}_{H} )\mu = \frac{e}{m}\tau \,(\mu_{0} {\mathbf{S}}_{H} ).$$
(32)

Accordingly, the vector of Hall field EH and incident Hall component of the Poynting vector SH are collinear and proportional to one another. This formula also reflects a remarkable fact, i.e., that the Hall field EH provides a direct measure of the relaxation time τ. It should be emphasized here, however, that formula (31) is not exactly valid. Its right side is related to the moment of the turning-on of external fields (i.e. specifically of magnetic field B in the case where a steady-state of ohmic flow through the sample was already established), while its left side corresponds to the delay time τ. For the Hall Poynting vector at time τ after the turning-on of the magnetic field, it is more correct to write

$${\mathbf{S}}_{H} (\tau ) = {\mathbf{S}}_{H} + \Delta {\mathbf{S}}_{H} ,$$
(33)

where the correction ΔSH can be estimated from the Hall correction EH to the overall electric field vector appearing at time τ. Thus, taking into account (31) and (32), we immediately obtain

$$\mu_{0\,} \Delta {\mathbf{S}}_{H} = \mu \,\{ \,(\mathbf{E} \times \mathbf{B}) \times \mathbf{B\} } = \mu \,\{ \,\mathbf{B}\,(\mathbf{BE}) - \mathbf{E}(\mathbf{B}^{2} )\} .$$
(34)

Because throughout the process the vector of the total electric field in the plane remains perpendicular to the vector of the magnetic field (i.e. E⊥B), the first term in the last brackets disappears, and for the change of the Poynting vector we arrive at the following formula:

$$\mu_{0\,} \Delta {\mathbf{S}}_{H} = - \mu \,\mathbf{E}\,(\mathbf{B}^{2} ).$$
(35)

In analogy with Eq. (28), for the ratio of quantities on the right side of Eq. (33) we can write

$$\left| {\frac{{\Delta {\mathbf{S}}_{H} }}{{{\mathbf{S}}_{H} }}} \right| = \left| {\frac{{\mu \,\mathbf{E}\,({\mathbf{B}}^{2} )}}{{\mathbf{E} \times \mathbf{B}}}} \right| = \mu B = {\text{tg}} \,\theta .$$
(36)

This means that during time τ the Hall Poynting vector is also deviated by the Hall angle θ, precisely like primary electric field vector E. There is, however, a difference in their layout. Whereas in the case of the electric fields the angle θ should be measured anticlockwise (cf. Fig. 3), in the case of the Poynting vectors it increases in the clockwise direction; moreover, the vectors E and SH are orthogonal. Since the Poynting vector must continuously rotate from its initial position to the end position at time τ, the Hall electric field EH must also simultaneously appear following Larmor rotation of the electron subsystem. From this point of view, EH is not a consequence of charges deposited on the sample edges but is created locally by the deformation of the Poynting vector induced by the Larmor rotation. Our approach to the Hall effect using the Poynting energy flows thus makes clear one persisting logical paradox. As we have already seen above, in standard treatment of the Hall effect, the charging of the edges generates an electric field, which within time τ compensates the action of the magnetic field, thus stopping further deviation of the electron paths. This consideration is based on the tacit assumption that the information about the charging of the edges is spread throughout the sample practically instantaneously. Nevertheless, taking into account the fact that the maximal speed of any signal is ≤ c, we find that in most cases cτ < w (= sample width, see Appendix), so no such coherent compensation of magnetic action is possible.

The formulae presented in this section allow one to make some interesting estimates or conclusions concerning spatial redistributions of electrons within the system and formulate hypotheses concerning the deposition of electromagnetic momentum as well. First of all, let us evaluate the net mass transfer due to the rise of the Hall field. Working with simple electrostatic considerations, the total ponderable mass of electrons effectively transferred during the relaxation time τ from one edge of the sample to the other is, evidently, (m/e) ε0 EH A (A is oriented edge area). If we compare this quantity with the electromagnetic energy influx SH entering the sample through one edge during the establishment of the Hall field and expressed in mass units, we find that both these quantities are exactly equal, namely,

$$\frac{m}{e}\,\varepsilon_{0} \,{\mathbf{E}}_{H} \,{\mathbf{A}} = {\mathbf{S}}_{H} \,\tau \,\frac{{\mathbf{A}}}{{c^{2} }}.$$
(37)

However, this direct consequence of Eq. (32), which in fact bears a characteristic of tautology, is very difficult to interpret convincingly. One is tempted to consider relationship (37) as a manifestation of the preservation of the center of mass/energy mentioned above. Indeed, considering the left side of (37), the shift δ of the center-of-mass of the sample, which is due to the transfer of ponderable matter (i.e. electrons), can be computed by the following formula:

$${{\varvec{\updelta}}} = \,\frac{{\varepsilon_{0} {\mathbf{E}}_{H} }}{ne}.$$
(38)

Such a shift requires an action of electromagnetic forces characterized by the electromagnetic momentum “consumed” within the sample during time τ. This loss of electromagnetic momentum realized in the vicinity of the enriched edge, however, shifts the center-of-mass/energy back about the length ≈ − μ RH (SH/c2), just opposite to the length given by formula (38). When the stationary state of the Poynting flow through the sample is reestablished after time τ, the center-of-mass/energy is set into its original position. Even though the deeper insight into the problem makes such an explanation seem somewhat naive, the above quantitative estimates supporting this scenario are correct.

Knowing now the rough features of the time evolution of the Poynting vector, we can immediately determine the electromagnetic volume forces active during the establishment of the Hall field. For the time derivative of the Poynting vector, we can use (33) and (35) to obtain estimate

$$\dot{\mathbf{S}} \to \frac{{\Delta {\mathbf{S}}_{H} (\tau )}}{\tau } = - \frac{e}{{m\mu_{0} }}{\mathbf{E}}\,({\mathbf{B}}^{2} ),$$
(39)

which can be inserted into the Lorentz Eq. (2) for the electromagnetic volume force, namely,

$$\mathbf{F} = \frac{e}{m}\int_{V} {\varepsilon_{0} {\mathbf{E}}\,({\mathbf{B}}^{2} )} {\text{ d}}V.$$
(40)

Operating for time τ along the bulk of the sample, this force results in the Hall field, which manifests itself by permanent surface forces seated at the edges of the sample. The electric component of the surface forces, which until now has been ignored and should be added as an extra term to formula (2), can be expressed by an integral [19], i.e.

$${\mathbf{F}}_{SE} = \int_{2A} \, \,\left\{ {\varepsilon_{0} ({\mathbf{n}}\,{\mathbf{E}}_{H} )\,{\mathbf{E}}_{H} - \frac{1}{2}\varepsilon_{0} \,{\mathbf{n}}\,\left| {{\mathbf{E}}_{H} } \right|^{2} } \right\}{\text{d}}A,$$
(41)

taken over the sample edges only; n is the unit normal vector to the sample surface. This formula can be reduced to a simpler relation for the electrostatic stress P (i.e. pressure) imposed on the sample edges by the Hall field, namely

$$P = \frac{1}{2}\varepsilon_{0} ({\mathbf{E}}_{H} )^{2} .$$
(42)

This stress of electrostatic origin is inevitably relaxing into the mechanical strain of the edges, cumulating in this way some potential energy. One can thus be sure that at least one part of the electromagnetic momentum entering the sample is deposited there without affecting the movement of the system. It should be noted that the expression on the right side of (42) is also exactly equal to the energy density of the electrostatic Hall field inside the sample (for the fringing effect see [69]). After the external magnetic field has been turned off, the capacitor consisting of the sample edges will be discharged and the mechanical strain localized at the edges during the assembly period in the form of electromagnetic momentum will be released. While the discharge of said capacitor may, with some proviso, be treated as a reversible process, the mechanical part of the Hall field decay has features typical for temporal reservoirs of hidden momentum, as discussed above. Stress related phenomena of this type thus provide a “traditional” account of the existence of hidden momentum. Importantly, these effects do always exist, and as such they should be carefully identified among other possible sources of hidden momentum.

We have shown the usefulness of interpretation of the ordinary Hall effect in terms of electromagnetic energy flows represented by the Poynting vector. This approach, among others, has revealed some new aspects of this interesting effect and also generated formulae enabling deeper insight into its mechanism.

7 Discussion

As we have already seen, arguments concerning linear hidden momentum are exclusively confined to analysis of thought experiments using abstract models of various kinds. The thought experiment, though, traditionally a mighty tool of theoretical physics, does have its limitations. A similar demonstrative method, one that has been widely used since antiquity, is known for generating many valuable yet also quite misleading ideas. The term thought experiment, which was first coined by Oersted (Gedankenexperiment or Gedankenversuch), was later systematically treated by Mach [70]. Accordingly, we can distinguish two essentially different interpretations of the thought experiment. The first one works with an idealized system and imaginary scenario with the aim of obtaining (through speculations which are not at odds with known laws of nature) answers to some hypothetical questions. The main difficulty of this approach is the arbitrariness of starting assumptions, such as artificial initial and boundary conditions, which are often decisive for the resulting conclusions. The second interpretation, proposed by Mach, views the thought experiment as a necessary precondition for the realization of any physical experiment; it is, in fact, a detailed technical plan, the goal of which is to eliminate weak points in the future execution of the experiment. It may happen, however, that such a thought experiment solves the posed problem even without the actual experiment being performed.

When it comes to problems relating to hidden momentum, clearly, we must work with the first of the two approaches mentioned above. There is no intent to perform the suggested experiments. We dare to claim that most of the authors of such experiments are convinced there is no hope of actually conducting them and that, therefore, there will never be any quantitative results to compare with their expectations. This is why some important or even unrealistic aspects of their conceptual models can be safely overlooked. Since our ultimate aim is to implement the real Hall effect experiment into considerations concerning problems of momentum balance in electromechanical systems, we incline rather to Mach’s (second) conception of the thought experiment. The discussion, then, must begin with an assessment of the range of quantities involved and with detailed specification of the experimental conditions, including the preparation phase.

Firstly, to have some idea of the order of the quantities we are talking about, let us make some quantitative estimates of the forces and effects involved. Using our model of the gold leaf Hall structure (described in full in the Appendix), we determine from Eq. (37) that the excess (or lack) of electron density on the edges is ≈ ± 230 electrons/m2 and the total mass transferred is approximately 2.1 × 10−36 kg. The volume force operating in the bulk of the sample during establishment of the Hall field may then be determined from Eq. (40), F ≈ 1.27 × 10−14 N. For the surface stress, we use (42) to arrive at an estimate of P ≈ 7.73 × 10−23 N/m2, the same as for the volume energy density of the Hall field (≈ 7.73 × 10−23 J/m3). The center-of-mass shift δ ≈ 3.9 × 10−27 m, computed by means of (38), also seems unrealistically small. Obviously, we are dealing with very tiny quantities and extremely subtle effects, though these quantities were derived not from some abstract theoretical considerations but are computed from realistic values currently observed in the laboratory using the Hall measurements. Such measurements, then, are sensitive enough to “weigh” small fractions of electron masses and indicate force densities of the order of μN/m3, so the ordinary Hall effect evidently belongs to the same realm of electromagnetic masses and momenta as those encountered in hypothetical problems relating to hidden momentum and other electromagnetic thought experiments.

Until now, we have investigated the processes taking place in the Hall structure under rather abstract conditions, in particular after the instantaneous application of finite external magnetic field B. The practical laboratory time limit for setting up a typical magnetic field of the order of \(\sim\) 10−1 Tesla is, however, T \(\sim\) 10−2 s and greater. As relaxation time τ in the current materials is much shorter (i.e. τ ≪ T), the rise of the Hall field EH is in fact controlled by relatively slow changes of B; EH follows these changes with a negligible delay equal to τ ≈10−12–10−14 s; therefore, it is necessary to investigate how these facts may influence our considerations above.

Let us assume that the external electric field E and corresponding electric current flowing through the sample are already established and stationary. The following turning-on of a magnetic field can be then satisfactorily modelled by a magnetic field linearly increasing from zero to some required end value. A realistic approximation of such a process is given by formula

$${\mathbf{B}} = {{\varvec{\upbeta}}}\,t,$$
(43)

where the sweep rate β is a constant vector collinear with the magnetic field and t ∈ (0, T) is the time measured from the beginning of application of the field. Keeping in mind that τ <  < T, we can rewrite (32) as

$${\mathbf{E}}_{H} (t) = \frac{e}{m}\,\,\tau \,({\mathbf{E}} \times \,{{\varvec{\upbeta}}}\,t).$$
(44)

This equation describes the progression of generation of the Hall field in real time. Similarly, the time evolution of the corresponding Poynting vector can be characterized by

$$\dot{\mathbf{S}} \to \frac{{{\mathbf{S}}_{H} (t + \tau ) - {\mathbf{S}}_{H} (t)}}{\tau } = - \frac{e}{{m\mu_{0} }}{\mathbf{E}}\,({{\varvec{\upbeta}}}\,t)^{2} .$$
(45)

The time derivative of SH(t), namely Ṡ, then continuously increases to its ultimate value − (e/mμ0) E (β2T2), which is in fact identical to that given by (39). Therefore, curiously enough, the stepwise increasing of the magnetic field for time T should have the same effect as instantaneous application of the magnetic field, B = β T. In light of this fact, one can conclude that our previous estimates remain essentially correct quite independently of the particular way in which the stationary end state of the system was achieved. In this connection, simultaneously, the key significance of relaxation time τ for the establishment of any stationary end state emerges. Indeed, τ seconds after the achievement of a certain magnetic field B, the Hall field EH given by (15) appears and the resulting system starts to be stationary. Notice that without such a prelude of duration τ, no stationary state of the system and no Hall effect at all would be possible. This idea can also be naturally generalized to other similar systems controlled by the relaxation time. We can claim that this parameter, measuring the smallest time in which the transition between two stationary states of the system can be finalized, plays the role of assembly period, cf. [22], which simultaneously represents the unique dynamical aspect of any stationary system.

As we have already seen in Sect. 6 above, in principle the Poynting–Abraham formalism allows one to follow the fate of the energy and momentum fluxes responsible for the establishment of the Hall field within the sample. Inspection of the experimental arrangement and allied conditions, however, clearly reveals another quite serious difference between most of the hypothetical models and real experiments: while hypothetical systems in which the momentum balance is studied are considered to be isolated, i.e. completely decoupled from the rest of the Universe, any real experimental apparatus is inevitably open.

For example, in the real Hall measurement, where the sample is firmly fixed in its position with respect to the sample holder, the momenta and the forces appearing during application of the external fields are completely compensated by the mechanical stresses both inside and outside the sample. The decomposition of momenta of differing natures among the parts of such an open system are obviously far from unambiguous. At the same time, the generated Hall field EH does not depend on the details of the experimental arrangement but only on the quantities explicitly entering into formulae (15), (32), and (37). It is, however, quite a general and essential property of any experimental method striving for a sense of objectivity. It necessarily has a component that (in order to provide reproducible results) is decoupled as far as possible from random external influences; on the other hand, it also has the components controlled mainly by the surrounding. If we provisionally label the component of the experimental setup that is effectively decoupled from the surrounding as milieu-independent, we can conclude that the part of the electromagnetic energy and momentum fluxes needed for the establishment of the Hall field, which are fully converted and stored by the electrons redistributed inside the sample, are the milieu-independent ones. Furthermore, there are inevitably momenta compensating the reaction of the electromagnet during application of the magnetic field and mechanical stresses due to the charging of sample surfaces, which are fully dependent on the concrete configuration of the experiment. Since the sample reveals no overt motion, these latter momenta entering the momentum balance might, in principle, possess a characteristic of hidden momentum. Yet this is inadmissible, because in an open system these momenta cannot be identified and localized unambiguously.

Let us now return to the entity known as an isolated system. It is specified by the property that it cannot exchange either energy or matter with the environment beyond the system boundary. The boundary, which does not belong to the isolated system, can be realized either by an enclosure that is non-penetrable to matter and external forces or by maintaining sufficient (ideally infinite) distance from the other systems. An example of the first approach, frequently used in electrical measurements, is electrostatic shielding [64, 71], where the regions inside and outside of the conductive enclosure are electrically isolated from each other and electric fields from one region cannot penetrate or affect the other. On the other hand, the possibility of using some artificial means to eliminate the influence of long-range gravitational forces is still only hypothetical [72], thus we must rely on the method of maintaining a large distance from other bodies. Clearly these means of isolating are not perfect, and each has its own limitations. As a rule, they are sufficient with required accuracy to realize the milieu-independent system but not the ideally isolated system. Despite the fact that milieu-independent systems have much in common with isolated systems (and are in fact weak versions of such systems), these two concepts are fundamentally different. A real milieu-independent system can be approximated by a certain isolated system, but the success of such an approximation can never be ensured a priori. This circumstance, which is closely related to the non-critical use of infinity in physics [73], can be illustrated by an example mentioned earlier in Sect. 3. The angular electromagnetic momentum of a magnetic monopole plus an electric point charge (independent of their separation) does not converge in infinity; thus, attempts to isolate this system by increasing the relevant distances must totally fail. It becomes apparent, then, that the gap between isolated systems and milieu-independent systems, which are crucial for real experiments, is in principle hardly surmountable. Deciding whether the approximation of an investigated system by a given isolated system is right requires solving difficult convergence-divergence problems, among others. The devil is in the details, and since, as we have seen, any neglect of the smallest detail in system specification can lead to unexpected consequences, we must keep in mind that a single assumption in a thought experiment, e.g. that the system under consideration is isolated, could lead to either incorrect or correct results.

8 Conclusions

In this paper, we have made an attempt to confront the thought experiment methodology traditionally used to identify a solution to the puzzling problem of hidden momentum, addressing momentum balance in electromechanical systems submitted to static electromagnetic fields, with an actual experiment belonging to the same class of phenomena. As we have suggested, the ordinary Hall effect is a good candidate for such an experimentally treatable phenomenon operating in static electromagnetic fields. It is simultaneously sensitive enough to have the potential to effectively visualize some discrepancies persisting in classical electrodynamics. In order to explore this possibility, the phenomenology and standard theory of this effect in terms of the Lorentz force has been critically reviewed; furthermore, it has been shown that the ordinary Hall effect can also be alternatively treated in the framework of the Poynting-Abraham formalism based on electromagnetic energy and momentum flows. This approach provided us with relationships that allowed us to trace with sufficient credibility the evolution of an electronic subsystem and pertaining electromagnetic fields.

Another key result of this work is that it represents a comparison of various features intrinsic either to real experiments or to thought experiments. This allows us to formulate the following important pieces of knowledge; they are seemingly trivial and often ignored, but we are convinced that they are deserving of respect:

  • The relations based on the Poynting-Abraham formalism made it possible to show that the ordinary Hall effect, which is commonly regarded as quite robust, is in fact, as concerns momentum-related effects, also a very tiny effect of the same rank as those studied by thought experiments relating, for example, to the problem of hidden momentum.

  • In a real experiment or any thought experiment, one must take into account that every stationary situation is the result of a certain development. As we have seen in the case of the ordinary Hall effect, the preparational phase may be controlled by a couple of different characteristic time scales, but only the shortest one is relevant for final establishment of the stationary state. Identification of this time scale, the so-called period of system assembly, which represents the dynamical aspect of the studied stationary system, is a crucial condition for correct comprehension of a real experiment as well as for the non-contradictory design of thought experiments.

  • Further important differences between real and thought experiments are firmly rooted in the fact that any real experiment is inevitably made with open systems, while in thought experiments systems are frequently used that are assumed to be strictly isolated. Since the paradoxes are symptoms of non-justified assumptions, it is no wonder that incompetent use of isolated systems in the construction of thought experiments could generate such paradoxes. Therefore, possible consequences of any implementation of an isolated system into the thought experiment should be considered individually and very critically.