Abstract
Maxwell-like field relations which describe spatially-averaged kinematic behaviour of electrons and atomic nuclei (modelled as point charges) are obtained at any prescribed scale using weighting function methodology. Upon appeal to the experimental laws of Coulomb and Biot-Savart, and to dimensional considerations, these relations yield the macroscopic Maxwell equations as they pertain to electrostatics and magnetostatics. Generalisation to classical macroscopic electrodynamics is effected by taking account of signal transmission delay and selection of appropriate retardation potentials. Unlike previous derivations, no appeal is made to the microscopic field relations of Lorentz.
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1 Introduction
The foundations of electromagnetic theory were established in the nineteenth century to explain experimental evidence associated with a wide variety of phenomena concerning electric charge, its flow, and links with magnetic effects. In a two-volume treatise Maxwell in 1873 presented a meticulous and comprehensive survey [1] of phenomena and experimentation, and developed a theoretical framework within which results could be interpreted. In a radical departure from the Newtonian concept of action at a distance, Maxwell’s account introduced relations between fields (that is, continuous functions of position and time) holding in the aether, on the basis of which he conjectured the electromagnetic nature of light. In particular Maxwell’s equations included (using modern notation: see Hendry [2], Chap. 6)
and
Here \(\mathbf{D}\) denotes electric displacement, \(\rho\) charge density, \(\mathbf{j}\) electric current density, \(\mathbf{H}\) magnetic displacement, B magnetic field, A magnetic potential, f electromotive force, v velocity (relative to the aether), and \(\psi\) electrostatic potential.
Immediately (1.3) yields
Further, on taking the divergence of relation (1.2) and the time derivative of (1.1) it follows that
If the electric field E is defined via
then the time derivative of (1.3) yields
and relation (1.4) may be written as
The relations which are now considered to be the most fundamental, and which bear Maxwell’s name, are (1.1), (1.2), (1.5) and (1.8).
Attempts to deduce Maxwell’s equations from dynamical principles were restricted by contemporaneous understanding of the nature of matter. In 1902 Lorentz [3, 4] proposed a system of Maxwell-like equations which related to individual small isolated regions of charged matter he termed electrons. Each electron was characterised by a charge density field \(\rho\), which vanished outside the region it occupied, together with fields of velocity v, electric displacement d, and magnetic force h. In stationary aether these were to satisfy (cf. [4], equations I, II, IV, V, VI)
and
where \(c\) denotes the speed of light in the aether. Further, ‘the force, reckoned per unit charge, which the aether exerts on a charged element of volume’, was postulated (cf. [4], equation VII) to be
Here \(\mathbf{d}\) was identified as ‘the electric force that would act on an immovable charge’. The foregoing fields were considered to change rapidly and irregularly, and observable/measurable quantities were to be identified with local spatial averages computed over many charges (cf. [3]). Specifically, the average of any field \(F\), evaluated at location x, is
Here the volume integral is taken over the averaging region \(C(\mathbf{x})\) of volume \(V\). Relations (1.10)–(1.13) were averaged using propertiesFootnote 1
Distinguishing between electrons associated with conduction, polarisation, and magnetisation, Lorentz obtained the macroscopic Maxwell Eqs. (1.1), (1.5), (1.8), and a version of (1.2) of the form
Identification of composite current C with \(\partial \mathbf{D}/\partial t + \mathbf{j}\) yields (1.2) and, via (1.1) and (1.16)1, also (1.6). Further, Lorentz effected decompositions
where P and M represented electric and magnetic polarisation densities.
Many studies have subsequently refined Lorentz’ pioneering work. In particular, once electrons and atomic nuclei were established to be the fundamental and discrete carriers of charge, it became natural to model these entities as point charges, and to attempt to introduce such knowledge into the microscopic relations. Further, averaging procedures, together with modelling assumptions concerning subatomic behaviour, have been clarified. Specifically, averaging may be purely spatial (Van Vleck [6]), jointly in space and time (Rosenfeld [7]), or statistical (via designation of an appropriate ensemble: Mazur and Nijboer [8]). These approaches were reviewed by de Groot [9] who also discussed covariant derivations. A more rigorous approach to spatial averaging via the introduction of a weighting function was employed by Russakoff [10].
The aforementioned works are based upon the microscopic relations (1.10)–(1.13), even if atomicity is introduced via expressions for \(\rho\) and \(\mathbf{j}\, (:= \rho\mathbf{v})\) in terms of sums involving \(\delta\)-functions and instantaneous point charge locations and velocities. Such atomicity is somewhat at variance with the interpretations of d and h, no matter how locally these fields are defined. It is helpful to bear in mind relevant spatial scales. The respective sizes of nuclei and atoms are of orders \(10^{-15}\mbox{ m}\) and \(10^{-10}\mbox{ m}\), and so the Lorentz equations, postulated to model electromagnetic behaviour within atoms, relate to behaviour at scale \(10^{-10}\mbox{ m}\) or less. On the other hand Maxwell’s equations pertain to reproducible macroscopic behaviour at scales often described as ‘physically infinitesimal’ but ‘microscopically large’. Practically speaking, field values must be related to measurements, and hence to spatial and temporal sensitivity of monitoring devices. In any specific physical context it is thus for experimentation to determine the scales of length and time at which Maxwell’s relations provide a valid description.
Here the approach is from the outset entirely atomistic and no appeal is made to postulated microscopic relations. Electrons and atomic nuclei are modelled as point charges. Spatial averaging of the kinetic behaviour of any set of charges is effected in terms of a scale-dependent weighting function \(w\). The starting point in Sect. 2 is the definition of a charge density field \(\rho_{w}\) whose time derivative immediately introduces a current density \(\mathbf{j}_{w}\) which satisfies (1.6). Any solution a to \(\mathrm {div}\, \mathbf{a} = w\) leads directly to relations of forms (1.1) and (1.2), with specific definitions of the electrokinetic fields \(\mathbf{D} _{w}\) and \(\mathbf{H}_{w}\). A natural choice of \(w\) is introduced in Sect. 3 which accords equal weighting to charges within a prescribed distance \(\epsilon\) from any point x at which an average is to be computed, and zero weighting to charges further than \(\epsilon + \delta\) from x, with \(\delta\ll\epsilon\). For distances in the range \([\epsilon, \epsilon+ \delta]\) weighting corresponds to a choice of mollifier which ensures that \(w\) is everywhere smooth. The assumption that the corresponding function a be isotropic leads to natural decompositions \(\mathbf{D}_{w} = \mathbf{P}_{w} + \boldsymbol {\mathcal {E}} _{w}\) and \(\mathbf{H}_{w} = - \mathbf{M}_{w} + \boldsymbol {\mathcal {B}}_{w}\), together with potentials \(\psi_{w}\) and \(\boldsymbol {\mathcal {A}}_{w}\) for which \(\boldsymbol {\mathcal {E}}_{w} = - \nabla\psi_{w}\), \(\boldsymbol {\mathcal {B}}_{w} = \mathrm{curl}\, \boldsymbol {\mathcal {A}}_{w}\) and \(\partial\psi_{w}/\partial t + \mathrm{div}\, \boldsymbol {\mathcal {A}}_{w} = 0\). The contributions to values \(\boldsymbol {\mathcal {E}}_{w} (\mathbf{x})\) and \(\boldsymbol {\mathcal {B}}_{w} (\mathbf{x})\) from any charge \(q_{i}\) distant further than \(\epsilon+ \delta\) from x are shown to be \(-q_{i} \mathbf{u}_{i}/4\pi u^{3}_{i}\) and \(q_{i} \mathbf{u}_{i} \times\mathbf{v}_{i}/4\pi u^{3}_{i}\), respectively. (Here \(\mathbf{u}_{i}\) denotes the displacement of \(q_{i}\) from x, \(u_{i}\) its magnitude, and \(\mathbf{v}_{i}\) its velocity.) Additional time averaging is introduced in Sect. 4 to elucidate the physical interpretations of fields by considering bound and free/diffusive electrons in simple systems. In particular, \(\mathbf{P}_{w}\) is seen to be a density of a measure of time-averaged electron charge distribution about parent nuclei: individual contributions are time-averaged dipole moments which measure orbital asymmetry. In Sect. 5 the electrokinetic fields \(\boldsymbol {\mathcal {E}}_{w}\) and \(\boldsymbol {\mathcal {B}}_{w}\) are linked with the force-related electrostatic and magnetostatic fields \(\mathbf{E}^{s}_{w}\) and \(\mathbf{B}^{s}_{w}\) via the experimental results of Coulomb and Biot-Savart, respectively. Specifically, \(\mathbf{E}^{s}_{w} = \epsilon_{0}^{-1} \boldsymbol {\mathcal {E}} _{w}\) and \(\mathbf{B}^{s}_{w} = \mu_{0} \boldsymbol {\mathcal {B}}_{w}\), where \(\epsilon_{0}\) and \(\mu_{0}\) are determined by experiment, satisfy \(\epsilon_{0} \mu_{0} = c^{-2}\), and serve to ensure dimensional consistency. A formal generalisation to dynamical contexts motivated by (1.7) leads to a complete set of Maxwell relations, consequent upon knowledge of the instantaneous location and velocity of every charge. However, such global instantaneous information can never be known, but requires time to be communicated. This issue is addressed in Sect. 6. It is assumed that information is transferred at the local speed of light and is visualised via the artifice of hypothetical radar signal reflection. The consequent fully-dynamical Maxwell relations correspond to knowledge of the apparent locations and velocities as monitored at any given location and time. Such information has required time to be transmitted and corresponds to an earlier (retarded) time. In Sect. 7 the consequences of force relation (1.9) holding in a general dynamical context are explored. The individual contribution \(\mathbf{B}^{d}_{i}\) of a charge \(P_{i}\) to the dynamic magnetic field at location x is shown to be orthogonal both to the corresponding dynamic electric field contribution \(\mathbf{E}^{d}_{i}\) and to the apparent displacement of \(P_{i}\) from x. The force \(\mathbf{F}_{i}\) on a charge at x due to \(P_{i}\) is expressed as a linear transformation acting on \(\mathbf{E}^{d}_{i}\). A brief summary and concluding remarks are appended in Sect. 8.
The notation employed is direct (that is, free of co-ordinate considerations: cf. [5]) and standard identities involving scalar and vector fields are employed without reference.
2 Spatial Averaging and Weighting Functions
Electromagnetic phenomena derive from the behaviour of electrons and atomic nuclei which are here modelled as point charges. Relations which describe spatially-averaged kinetic behaviour of assemblies of such charges are derived: these are formally identical to Maxwell’s Eqs. (1.6), (1.1) and (1.2).
Here \(m_{i}\), \(q_{i}\) and \(\mathbf{x}_{i} (t)\) denote the mass, charge, and location at time \(t\), of a typical charge \(P_{i}\). For any assembly of charges, the net charge within any region ℛ at time \(t\), divided by the volume \(V\) of \(R\), yields a volumetric average (that is, a density) \(\rho(\mathcal{R}, t)\) at time \(t\). Symbolically,
where the primed sum is taken only over those charges in ℛ at time \(t\). Equivalently,
where the sum is over all assembly charges, and \(w_{i} (t) = 1\) or 0 according to whether or not \(P_{i}\) lies in ℛ at time \(t\).
The weighted sum in (2.2) can be generalised to yield a candidate electric charge density field via
where the displacement of \(P_{i}\) from location x at time \(t\)
Suppressing arguments, (2.3) may be written as
Here weighting function \(w\) is defined on the space \(\mathcal{V}\) of all displacements in three-dimensional Euclidean space, takes real values with physical dimension \(L^{-3}\), and assigns greater values to charges near x than those far therefrom. In order that the integral of \(\rho_{w}\) over all space should yield the total net assembly charge it is sufficient that
Consideration of an assembly consisting of a single charge indicates that normalisation condition (2.6) is also necessary: cf. [5], p. 45.
Remark 2.1
While the physical interpretation of \(\rho_{w}\) depends upon the specific form of \(w\), it is instructive to proceed formally before making a natural, scale-dependent, choice in Sect. 3.
From (2.3), noting that \(\mathbf{v}_{i} := d\mathbf{x}_{i}/dt\) does not depend upon x, and using the chain rule,Footnote 2
Thus (cf. (1.6))
where a candidate current density field is
Remark 2.2
If \(q_{i}\) is replaced by \(m_{i}\), then the mass density field
is similarly seen to satisfy
where
Here \(\mathbf{p}^{m}_{w}\) is a momentum density field. Writing
yields
and \(\mathbf{v}^{m}_{w}\) is a (mass) velocity field. Definition (2.13) makes sense if \(\rho^{m}_{w} \ne0\); that is, wherever and whenever matter is found.
The electrokinetic analogue of (2.13), namely
makes sense only where and when \(\rho_{w} \ne0\), and hence is not a sensible construct for electrically-neutral assemblies. However, for assemblies consisting only of electrons, or only of nuclei, (charge) velocity fields are well-defined in terms of the relevant densities of current and charge. Indeed, for an assembly of identical charges (so \(q_{i} = q\) and \(m_{i} = m\), say) the constancy of ratio \(q_{i}/m_{i}\) (\(= q/m\)) mandates that the mass and charge velocities be the same:
Now suppose that, for some function \(\mathbf{a} : \mathcal{V} \to \mathcal{V}\),
is a solution to
Accordingly,
and so
where (cf. (2.17))
is a candidate electric displacement field.
From (2.17), use of the chain rule yields
Since \(\mathbf{v}_{i}\) is independent of x, from (2.21), (2.17), the chain rule, and (2.22) it follows thatFootnote 3
Further, from (2.9) and (2.18),
Accordingly, (2.23) and (2.24) yield
where
Relation (2.25) and definition (2.26) may be re-written by introducing relevant axial vectors.
Remark 2.3
To any skew-symmetric tensor W corresponds an axial vector \(ax (\mathbf{W}) =: \mathbf{w}\) for which
for any vector k. (Cf., e.g., Murdoch [5], A.15.) In particular,
where
Further, if W denotes a skew-symmetric field and w the corresponding axial vector field, then for any fixed vector k
Accordingly, since k is arbitrary,
It follows from (2.28) that with \(\mathbf{W} = \mathbf{v}_{i} \wedge \mathbf{a}_{i}\),
Remark 2.3 enables (2.25) to be written as
where (cf. (1.2)) candidate magnetic displacement field
While relations (2.25) and (2.33) are equivalent, the former requires no appeal to orientation (that is, to ‘right-’ and ‘left-handedness’) which is necessary both for the definition of a vector product and of the curl operator.
Remark 2.4
Relations (2.8), (2.20) and (2.33) are formally identical with Maxwell relations (1.6), (1.1) and (1.2). However, the physical interpretation of all fields depends upon the choice of weighting function \(w\). From an analytical viewpoint, relations (2.6) and (2.7) require that \(w\) be, respectively, integrable and differentiable. Indeed, the regularity of all fields depends precisely upon that associated with choice \(w\). In the following section a specific natural choice (appropriate to any prescribed length scale) is made. Such choice provides physical insight, and mandates a natural binary decomposition both of \(\mathbf{D}_{w}\) and \(\mathbf{H}_{w}\), thereby introducing electric and magnetic fields which figure in Maxwell equations (1.5) and (1.8).
3 Electrokinetics via a Natural Choice of Weighting Function
The weighting function \(w\) introduced in (2.3) et seq does not depend explicitly upon location x and its values are invariant under translation, consistent with regarding space to be homogeneous. That space also be isotropic requires that
for every displacement u and every proper orthogonal tensor Q. That is, \(w(\mathbf{u})\) is independent of the direction of u, so
where
Thus homogeneity and isotropy of space have mandated that \(w\) values be Galilean invariant and depend only upon separations.
The simplest form of \(w\) is given by
where \(V_{\epsilon}:= 4\pi\epsilon^{3}/3\). This corresponds to spherical averaging regions and a specific choice \(\epsilon\) of length scale.
Remark 3.1
Choice \(w = w_{\epsilon}\) in (2.3) yields \(\rho_{w} (\mathbf{x}, t)\) as the sum of those charges which at time \(t\) lie within that sphere \(S_{\epsilon}(\mathbf{x})\) of radius \(\epsilon\) and centre x, divided by the volume of the sphere. Similar interpretations of \(\rho^{m}_{w} (\mathbf{x}, t)\) and \(\mathbf{p}^{m}_{w} (\mathbf{x}, t)\) apply in respect of mass and momentum: see (2.10) and (2.12). Further, from (2.13) \(\mathbf{v}^{m} _{w} (\mathbf{x}, t)\) is identifiable with the velocity of the mass centre of those charges within \(S_{\epsilon}(\mathbf{x})\) at time \(t\). Notice from (2.16) that, for any assembly of identical charges, \(\mathbf{v}_{w}\) coincides with the corresponding mass centre velocity.
While choice (3.4) delivers precise definitions and physical interpretations, \(w_{\epsilon}\) fails to be differentiable (indeed, suffers jump discontinuities) wherever \(u = \epsilon\), and hence neither \(\partial\rho_{w}/\partial t\) nor \(\mathrm{div}\, \mathbf{j} _{w}\) are defined thereat. This wrinkle may be overcome by mollifying \(w_{\epsilon}\) for \(u \in[\epsilon, \epsilon+ \delta]\) with \(\delta(> 0)\) arbitrarily small. Specifically,
Here \(k\) is a constant determined by normalisation (2.6), and \(\varphi\) is a monotonic decreasing function on \([0, 1]\) of class \(C^{n}\) for which \(\varphi(0) = 1\), \(\varphi(1) = 0\), and all derivatives up to order \(n\) vanish (one-sidedly) at \(\lambda= 0\) and \(\lambda= 1\).
Remark 3.2
It follows that any fields defined in terms of \(w_{\epsilon, \delta}\) weighted sums inherit class \(C^{n}\) spatial regularity and also (via multiple use of the chain rule) class \(C^{r}\) temporal regularity if trajectories \(\mathbf{x}_{i} (t)\) are of class \(C^{s}\) and \(r = \mathrm{min} (n, s)\).
Normalisation condition (2.6) with (3.2) yield, on employing spherical polar co-ordinates,
It follows from (3.5) that (cf. [5], §4.3.5)
The simplest mollifier sufficient to render \(w_{\epsilon, \delta}\) of class \(C^{1}\) is
From (3.5) and (3.6) such choice implies
Straightforward integration yields
where
It follows (cf. (3.7)) that
Interpretation of \(\mathbf{D}_{w}\) and \(\mathbf{H}_{w}\) (cf. (2.21) and (2.34)) requires specification of a (cf. (2.17) and (2.18)), and hence depends upon both \(\epsilon\) and mollifier \(\varphi\). In view of spatial isotropy we assume that
where \(a\) is scalar-valued.
Theorem 3.1
where (cf. (3.11) and (3.5)2) \(\lambda= (u - \epsilon)/\delta \) and
Proof
The divergence theorem, (2.17) and (2.18) yield
Here \(S_{R} (\mathbf{x}_{i})\) denotes that spherical ball of radius \(R\) centred at \(\mathbf{x}_{i}\) and \(\partial S_{R} (\mathbf{x}_{i})\) its boundary. The outward unit normal at point \(\mathbf{x} \in\partial S_{R} (\mathbf{x}_{i})\) is
Thus from (3.13) and (3.19), noting \(u_{i} = R\) on \(\partial S_{R}( \mathbf{x}_{i})\),
If \(R < \epsilon\), then from (3.5)1 and (3.21),
and hence if \(u < \epsilon\), then \(a(u) = -k/3\).
If \(R > \epsilon+ \delta\), then normalisation (2.6), (3.5)3 and (3.21) yield
so
If \(\epsilon\le R \le\epsilon+ \delta\), then (3.5)1,2 and (3.21) yield
where
Thus
where
with \(F\) given by (3.17). Hence (3.26) yields
Writing \(R = u\) yields (3.16). □
Remark 3.3
Theorem 3.1 highlights a marked difference in the nature of \(a(u)\) for \(u < \epsilon\) as compared with \(u > \epsilon+ \delta\). In particular, for \(0 \le u < \epsilon\),
If
then
Accordingly,
The binary decomposition (3.32) of \(a\) (and hence, via (3.13), of a) motivates corresponding decompositions of fields \(\mathbf{D}_{w}\) and \(\mathbf{H}_{w}\). Specifically (cf. (1.17)),
Here, noting (3.31) and suppressing subscripts \(\epsilon\) and \(\delta\),
Remark 3.4
Fields \(\mathbf{P}_{w}\) and \(\mathbf{M}_{w}\) are local spatial densities (cf. (2.5) and (2.9)) of electrokinetic variables \(-q_{i} \mathbf{u}_{i}/3\) and \(-q_{i} \mathbf{u}_{i} \times \mathbf{v}_{i}/3\). In particular, values \(\mathbf{P}_{w} (\mathbf{x}, t)\) and \(\mathbf{M}_{w} (\mathbf{x}, t)\) derive (cf. (3.36)1, (3.38)1 and (3.34)1) only from charges within a distance \(\epsilon+ \delta\) from x at time \(t\). In contrast, values \(\boldsymbol {\mathcal {E}}_{w} (\mathbf{x}, t)\) and \(\boldsymbol {\mathcal {B}}_{w} (\mathbf{x}, t)\) derive (cf. (3.37)1, (3.39)1 and (3.33)2) only from charges further than \(\epsilon\) from x at time \(t\).
Remark 3.5
In view of (2.20) and (3.35)1 it is natural to examine the separate contributions of \(\mathrm{div}\, \boldsymbol {\mathcal {E}} _{w}\) and \(\mathrm{div}\, \mathbf{P}_{w}\) to \(\rho_{w}\). Now
Since
and
so
Accordingly, (3.40) with (3.42) and (3.43) yield, via the chain rule,
Since \(w'(u_{i})\) vanishes for \(u_{i} < \epsilon\) and \(u_{i} > \epsilon+ \delta\) (cf. (3.5)), the only contributions to \(\mathrm{div}\, \boldsymbol {\mathcal {E}}_{w}\) evaluated at x come from charges for which \(\epsilon\le u_{i} \le\epsilon+ \delta\). It follows that if the contribution to \(\boldsymbol {\mathcal {E}}_{w}\) from charge \(q_{i}\) is labelled \(\boldsymbol {\mathcal {E}}_{i}\), and if ℛ is a region for which \(S_{\epsilon+ \delta} (\mathbf{x}_{i}) \subset \mathcal{R}\) (cf. (3.18)), then (cf. (3.5)2 and (3.11))
However, notice normalisation condition (3.9) may be written as
Accordingly (3.45) becomes
Writing
it follows from (3.47), (2.20) and (3.36), that
Accordingly, for any assembly of charges \(P_{i}\) each of which satisfies \(S_{\epsilon+ \delta} (\mathbf{x}_{i}) \subset\mathcal{R}\),
It should be noted that this does not imply that
since this would require (3.50) to hold for arbitrarily small regions (and hence condition \(S_{\epsilon+ \delta} (\mathbf{x}_{i}) \subset \mathcal{R}\) would fail to be satisfied). Indeed, in general (3.50) is approximate if there is any charge for which \(S_{\epsilon+ \delta} ( \mathbf{x}_{i}) \cap\mathcal{R} \ne S_{\epsilon+ \delta} ( \mathbf{x}_{i})\) (in such case part of the mollifying region would lie outside ℛ).
Fields \(\boldsymbol {\mathcal {E}}_{w}\) and \(\boldsymbol {\mathcal {B}}_{w}\) are expressible in terms of potential functions:
Theorem 3.2
If \(f\) satisfies
then
where
Proof
From the chain rule, (3.43) and (3.51),
and (3.52)1 follows from (3.37)1. Further, noting that \(\mathbf{x}_{i}\) does not depend upon x so \(\mathrm{curl}\, \mathbf{v}_{i} = \mathbf{0}\),
Thus (3.52)2 follows via (3.39)1. □
Corollary 3.2.1
Corollary 3.2.2
Proof
From (3.53)1,
where, from (3.41)2,
so (cf. (3.43)2)
Accordingly,
From (3.53)2, (3.54) and (3.43), noting \(\mathrm{div}\, \mathbf{v}_{i} = 0\),
Adding (3.60) and (3.61) yields (3.57). □
Remark 3.6
From (3.33)2 and (3.34)2, \(a_{2} (u _{i}) = 0\) if \(u_{i} < \epsilon\) and \(a_{2} (u_{i}) = -1/4\pi u^{3} _{i}\) if \(u> \epsilon+ \delta\). Thus, from (3.51), \(f'(u_{i}) = 0\) if \(u_{i} < \epsilon\) and \(f'(u_{i}) = -1/4\pi u^{2}_{i}\) if \(u_{i} > \epsilon+ \delta\). Noting that the constant of integration plays no part in determining \(\boldsymbol {\mathcal {E}}_{w}\) and \(\boldsymbol {\mathcal {B}}_{w}\), without loss of generality
Remark 3.7
Selection of natural choice (3.4) of scale-dependent weighting function \(w\) (suitably mollified in (3.5)), and associated displacement function a (delineated via (2.17), (2.18) and (3.13)) have resulted in relations formally identical to equations in classical electromagnetic theory. In particular, (2.20) and (2.33) are explicitly scale-dependent versions of Maxwell relations (1.1) and (1.2). Additionally, decompositions (3.35), together with the existence of potential functions given in (3.53) and which satisfy (3.57), take standard forms. However, all relations derived here are of purely electrokinetic character. While this is indeed the actual physical nature of spatial densities \(\rho_{w}\), \(\mathbf{j}_{w}\), \(\mathbf{P}_{w}\) and \(\mathbf{M}_{w}\), and of \(\mathbf{D}_{w}\) and \(\mathbf{H}_{w}\), the fields E and B which appear in Maxwell relations (1.5) and (1.8) are intimately related to the transmission of force, via (1.9). Evidently the physical dimensions of \(\boldsymbol {\mathcal {E}}_{w}\) and \(\boldsymbol {\mathcal {B}}_{w}\) are not those of E and B, respectively. Indeed, from (1.9) E and \(\mathbf{v} \times\mathbf{B}\) have the dimensions of force per unit charge. Accordingly,
On the other hand, noting \(w\), \(a\), \(a_{1}\) and \(a_{2}\) have dimension \(L^{-3}\) (cf. (3.30) and (3.31)), from (3.37) and (3.39)
This issue is resolved in Sect. 5 by appeal to the experimentally-based laws of Coulomb and Biot-Savart, after examining the interpretation of fields for simple atomic systems via time averaging.
Remark 3.8
Mollifier \(\varphi\) in (3.8) was the simplest choice. Further regularity may be ensured via polynomials of higher order: cf. [5], §4.3.6. Different choices change field definitions only over separations \(u_{i}\) in the range \([\epsilon, \epsilon+ \delta]\) and \(k\) satisfies (3.7).
4 Time Averaging and Interpretation of Atomic Kinetics
Any theoretical description of material behaviour must be related to its observation and measurement. Since no measurement can reflect behaviour at a geometrical point nor instant in time, but is limited by both the spatial and temporal sensitivity of the monitoring procedure and apparatus (cf. [5], Sect. 8.2), measurement values represent local averages jointly in space and time. This motivates a further temporal averaging of the spatially-averaged relations obtained so far.
The \(\Delta\)-time average \(f_{\Delta}\) of any integrable function of time is
It follows (cf. [5], p. 78) that if \(f\) is a class \(C^{1}\) field, then
Further, if f is a \(C^{1}\) vector field, then (cf. [5], Remark 8.3.1)
Result (4.3)1 also holds for tensor fields of any order.
Accordingly, time averaging relations (2.8), (2.20), (2.33), (3.35), (3.52), (3.56) and (3.57) yields versions formally identical to their originals but wherein all fields are replaced by their time-averaged counterparts.
Remark 4.1
Consider a simple model of a conductor as a system of atomic nuclei \(P_{j}\) (charge \(Z_{j}\, e\)) each of which has a fixed set of \(N_{j}\) bound electrons \(P_{j_{k}}\), together with a system of free/diffusive electrons \(P_{\ell}\). Both systems give rise to a relation of form (2.8), namely
Here ‘\(b\)’ and ‘\(f\)’ designate the bound and free systems. Of course, for the system as a whole, (2.8) is satisfied with
Suppressing time dependence (cf. (2.3)),
Also (cf. (2.9))
If \(P_{j}\) and its bound charges lie in \(S_{\epsilon}(\mathbf{x})\), then from (3.5)1 and (3.12) their contribution to \(\rho^{b}_{w} (\mathbf{x})\) is essentially \((Z_{j} - N_{j})e/V_{\epsilon}\) and to \(\mathbf{j}^{b}_{w}\) is \(\{ (Z_{j} - N_{j})e\mathbf{v}_{j} + \sum^{N_{j}}_{j_{k} = 1} e(\mathbf{v}_{j} - \mathbf{v} _{j_{k}}) \} /V_{\epsilon}\). If this subsystem remains in \(S_{\epsilon}(\mathbf{x})\) for the time interval \((t - \Delta, t)\), then its contribution to \((\rho^{b}_{w})_{\Delta}(\mathbf{x}, t)\) is \((Z_{j} - N_{j})e/V_{\epsilon}\) and to \((\mathbf{j}^{b}_{w})_{\Delta} (\mathbf{x}, t)\) is \((Z_{j} - N_{j})e(\mathbf{v}_{j})_{\Delta}/V_{ \epsilon}\). Here the factor \((\mathbf{v}_{j} - \mathbf{v}_{j_{k}})_{ \Delta}= (\dot{\mathbf{x}} - \dot{\mathbf{x}}_{j_{k}})_{\Delta}= d/dt \{(\mathbf{x}_{j} - \mathbf{x}_{\mathbf{j}_{k}})_{\Delta}\}\) has been regarded to be negligible as a consequence of the bounded nature of \(P_{j_{k}}\). In a solid macroscopically at rest, nuclei vibrate erratically on time scales of order \(10^{-13}~\mbox{s}\) about fixed locations. Accordingly, if \(\Delta\gg10^{-13}~\mbox{s}\) (say \(\Delta\sim10^{-6}~\mbox{s}\)) then \((\mathbf{v}_{j})_{\Delta}= (\dot{\mathbf{x}}_{j})_{\Delta}= d/dt \{(\mathbf{x}_{j})_{\Delta}\} = \mathbf{0}\). In such case (and neglecting nuclei which cross or straddle the mollifying region \(S_{\epsilon+ \delta} (\mathbf{x}) - S_{\epsilon}(\mathbf{x})\)) \((\mathbf{j}^{b}_{w})_{\Delta}\,(\mathbf{x}, t) = \mathbf{0}\) and \((\rho^{b}_{w})_{\Delta}\,(\mathbf{x}, t)\) does not change with time. Thus both terms in the time-averaged version of (4.4)1 vanish and, from (4.5),
and
for a conductor macroscopically at rest. More generally it is time averages of terms \((Z_{j} - N_{j})e/V_{\epsilon}\) and \((Z_{j} - N _{j})e\mathbf{v}_{j}/V_{\epsilon}\), taken over nuclei instantaneously in \(S_{\epsilon}(\mathbf{x})\), which dominate contributions to \((\rho^{b}_{w})_{\Delta}\, (\mathbf{x}, t)\) and \((\mathbf{j}^{b}_{w})_{ \Delta}\, (\mathbf{x}, t)\). The latter is an ionic current density which might be significant in an electrolytic context.
Remark 4.2
Consider an assembly of neutral atoms. In such case all electrons are bound, so \(N_{j} = Z_{j}\) for all atoms. The contribution to \(\mathbf{P}_{w} (\mathbf{x}, t)\) from such an atom, all of whose charges lie in \(S_{\epsilon}(\mathbf{x})\) at time \(t\), is from (3.36)2 closely approximated (here \(r\) is neglected, so \(k = V^{-1}_{\epsilon}\) from (3.12)) by
Here
Term \(-e\mathbf{r}_{j_{k}}\) represents an instantaneous dipole moment associated with the displacement of electron \(P_{j_{k}}\) from its parent nucleus. Such moment will vary erratically on an atomic timescale \((\sim10^{-13}~\mbox{s})\). However, if \(\Delta\sim10^{-6}~\mbox{s}\), then \(-e(\mathbf{r}_{j_{k}})_{\Delta}\) represents a time-averaged dipole moment which is a measure of time-averaged orbital asymmetry. Correspondingly, \((\mathbf{p}_{j})_{\Delta}\) is a measure of the averaged total charge distribution about nucleus \(P_{j}\), namely the polarisation of this atom. From (4.10) the contribution to \(( \mathbf{P}_{w})_{\Delta}(\mathbf{x}, t)\) from atoms which remain in \(S_{\epsilon}(\mathbf{x})\) during time averaging constitutes a polarisation density associated with these atoms: strictly speaking, \(-1/3\) multiplied by the sum of the time-averaged dipole moments and divided by \(V_{\epsilon}\). In general there will also be contributions both from atoms which migrate into and out of \(S_{\epsilon}( \mathbf{x})\), and atoms which straddle the mollifying region, at any time in the interval \((t-\Delta, t)\). If the system constitutes a solid at macroscopic rest \(((\mathbf{v}_{j})_{\Delta}= \mathbf{0})\), then only ‘straddling’ atoms are involved and, since \(\delta\ll\epsilon \), their contribution may be expected to be negligible.
Remark 4.3
The contribution to \(\mathbf{M}_{w} ( \mathbf{x}, t)\) from a neutral atom within \(S_{\epsilon}(\mathbf{x})\) at time \(t\) is (cf. (3.38)2, (3.5)1 and (3.12)) essentially
where
Term \(\mathbf{m}_{j_{k}}\) is the instantaneous magnetic moment of electron \(P_{j_{k}}\) about its parent nucleus. Erratic variation in \(\mathbf{m}_{j_{k}}\) is smoothed by time averaging, and
is a measure of averaged orbital angular momentum of electrons about their parent nucleus (on noting \(\mathbf{m}_{j_{k}} = - (e/m) \mathbf{r}_{j_{k}} \times m\dot{\mathbf{r}}_{j_{k}}\), where \(m\) denotes electron mass). From (4.12), (4.14) and (3.38), the contribution to \((\mathbf{M}_{w})_{\Delta}(\mathbf{x}, t)\), from atoms which remain in \(S_{\epsilon}(\mathbf{x})\) during time averaging, is a density \((\times1/3)\) of net time-averaged atomic moments \((\mathbf{m}_{j})_{ \Delta}\) together with averages \((\mathbf{u}_{j} \times \dot{\mathbf{p}}_{j} + \mathbf{p}_{j} \times\mathbf{v}_{j})_{\Delta }\). For a solid at macroscopic rest \(\mathbf{M}_{w} (\mathbf{x}, t)\) is characterised by such contributions upon neglecting ‘straddling’ atoms (cf. Remarks 4.1 and 4.2).
Remark 4.4
From (3.37) and (3.33)2, only charges further than \(\epsilon\) from x at time \(t\) contribute to \(\boldsymbol {\mathcal {E}}_{w} (\mathbf{x}, t)\). Any charge further than \(\epsilon+ \delta\) at this time yields (cf. (3.34)2) a contribution \(-q_{i} \mathbf{u}_{i}/4\pi u^{3}_{i}\). The net contribution from a neutral atom outside \(S_{\epsilon+ \delta} ( \mathbf{x})\) is
Now
Contribution (4.15) may thus (on neglect of \(O((r_{j_{k}}/u_{j})^{2})\) terms) be written as
where (cf. (3.43)2)
Fluctuations in the location of an atomic nucleus in a solid at rest are much less than a typical atomic radius of \(10^{-10}\mbox{ m}\). Thus if \(\epsilon > 10^{-6}\mbox{ m}\), then (noting here \(u_{j} > \epsilon+ \delta\)) \((\mathbf{A}_{j}\, \mathbf{p}_{j})_{\Delta}\sim(1/4\pi u^{3}_{j}) ( \hat{\mathbf{A}}_{j}\,\mathbf{p}_{j})_{\Delta}\), where \(\hat{\mathbf{A}}_{j} := (\mathbf{1} - 3\hat{\mathbf{u}}_{i} \otimes \hat{\mathbf{u}}_{i})\).
Remark 4.5
From (3.39) and (3.33)2, only charges further than \(\epsilon\) from x at time \(t\) contribute to \(\boldsymbol {\mathcal {B}}_{w} (\mathbf{x}, t)\). Any charge for which \(u_{i} > \epsilon+ \delta\) at this time yields (cf. (3.34)2) a contribution \(q_{i} \mathbf{u}_{i} \times\mathbf{v}_{i}/4\pi u^{3} _{i}\). Thus the contribution from a neutral atom outside \(S_{\epsilon + \delta} (\mathbf{x})\) is (neglecting terms as in (4.17))
In a solid at macroscopic rest \(\mathbf{u}_{j}\) varies negligibly in \([t-\Delta, t]\) and the contribution to \((\boldsymbol {\mathcal {B}}_{w})_{\Delta }(\mathbf{x}, t)\) is essentially
The foregoing remarks emphasise the rôle of time averaging in interpreting the subatomic contributions to macroscopic measures of atomic kinetics for simple systems. The methodology extends to molecular systems modulo context-dependent considerations: for example, book-keeping appropriate to (valence) electrons which are shared by several nuclei.
5 Consequences of the Coulomb and Biot-Savart Laws
If \(\boldsymbol {\mathcal {E}}_{i}\) denotes the contribution to \(\boldsymbol {\mathcal {E}}_{w}\) of a charge \(q_{i}\) for which \(u_{i} > \epsilon+ \delta\), then (cf. (3.37)1 and (3.34)2) for any charge \(q\)
This may be compared with Coulomb’s law for the force \(\mathbf{f}^{es} _{qq_{i}}\) exerted in vacuo on a stationary charge \(q\) at location x by a stationary charge \(q_{i}\) at \(\mathbf{x}_{i}\), namely (in SI units: cf., e.g., Griffiths [11], (2.1))
Thus, for in vacuo separations in excess of \(\epsilon+ \delta\),
Accordingly, for any set \(\{q_{i}\}\) of stationary charges all of which are distant at least \(\epsilon+ \delta\) from x, the assumption of linear superposition (cf. [11], 2.4; Zangwill [12], (2.18); Jackson [13], 24–26; Elliott [14], Sect. 3.2) yields a static electric field \(\mathbf{E}^{s}_{w}\) at x for which
Hence, for such a collection of charges
Similarly, if \(u_{i} > \epsilon+ \delta\), then the contribution of charge \(q_{i}\) to \(\boldsymbol {\mathcal {B}}_{w}\) is (cf. (3.39)1 and (3.34)2)
Consider a localised set of charges \(P_{i}\) which lie within a sphere of radius \(\epsilon\) centred at point X. If \(\mathbf{R} = \mathbf{X} - \mathbf{x}\) and \(R := \Vert\mathbf{R}\Vert\gg\epsilon \), then \(\mathbf{u}_{i} = (\mathbf{x}_{i} - \mathbf{X}) + (\mathbf{X} - \mathbf{x}) \sim\mathbf{R}\) and
This may be compared with the Biot-Savart law in which the contribution \(\Delta\mathbf{B}^{s}\) to the net magnetic field \(\mathbf{B}^{s}\) at x arising from such a collection of charges is (cf., e.g., [12], (10.15)) essentially
Here \(\mathbf{j}^{s}\) is the current density associated with the charges and is steady (emphasised by superscript ‘\(s\)’). Comparison of (5.7) with (5.8) suggests the natural identification (for the charges considered)
Remark 5.1
Relation (5.5) and identification (5.9) pertain to experimental results in vacuo (otherwise described as ‘free space’) based upon force relation (1.9). Specifically, no charges within \(\epsilon+ \delta\) of x are present. More generally, (5.5) and (5.9) motivate definitions
which add contributions (if any) from charges for which \(\epsilon \le u_{i} \le\epsilon+ \delta\) (cf. (3.37)1, (3.39)1, (3.33)2, (3.31)2 and (3.16)). Since \(\delta\ll \epsilon\) is arbitrarily small, definitions (5.10) in general represent precise book-keeping rather than a physical issue.
Further, from (3.52), (3.53) and (5.10),
where
Relation (3.57) may be written as
Remark 5.2
From (5.10)1, (3.63)1 and (3.64)1,
From (5.10)2, (3.63)2 and (3.64)2,
Hence
\(\epsilon_{0} \mu_{0}\) has the dimension of (speed)−2, and so \(\epsilon_{0} \mu_{0} = \alpha c^{-2}\) for some dimensionless constant \(\alpha\). Experimentation yields \(\alpha= 1\): that is,
From (5.12), (5.14) and (5.18),
and
Remark 5.3
The experimental laws of Coulomb and Biot-Savart relate to macroscopically stationary situations and hold at points in the neighbourhoods of which no charges are to be found. Specifically, the stationary requirement yields (cf. (2.8), (2.33) and (5.14))
while the free space condition (at scale \(\epsilon+ \delta\)) implies that (cf. (2.5), (2.9), (3.36)2 and (3.38)2)
Accordingly, from (2.20), (5.22)1, (5.11)1 and (5.22)3,
while from (2.33), (5.21)2, (5.22)2, (5.11)2 and (5.22)4,
It follows from (5.12)1 and (5.23) that
while (5.24) and (5.12)2 yield
Accordingly, from (5.21)3 and (5.14),
Of course, relations (5.25) and (5.26) are immediately evident on noting that here \(u_{i} > \epsilon+ \delta\) and thus \(f(u_{i}) = 1/4\pi u_{i}\) (cf. (3.62)2).
As they stand, relations (5.12) and (5.13) do not of themselves indicate any restriction to macroscopically-static situations, and admit formal generalisation to dynamical contexts without change. In particular, (2.20) may be written, via (5.11)1, as
where
Further, (2.33) may be expressed, via (5.11)1 and (5.11)2, as
where
Thus
and hence, via (2.8),
While (5.27) and (5.29) are two often-cited versions of Maxwell relations (cf., e.g., Griffiths [11], §10.1.1) the foregoing approach does not yield a corresponding version of (1.8). Guided by (1.7), suppose
Then clearly
whence, from (5.12)2,
While this relation has been an immediate consequence of (5.34), relation (5.29), with \(\mathbf{E}^{s}_{w}\) equated with \(\tilde{\mathbf{E}}_{w} + \partial\mathbf{A}^{s}_{w}/\partial t\), introduces an extra term \(\epsilon_{0} \mu_{0}\ \partial^{2} \mathbf{A}^{s}_{w}/\partial t^{2}\). Rather than adopting this approach, consider the direct consequence of definition (5.34) which is summarised in the following result.
Theorem 5.1
where
Further,
where
and
Proof
Result (5.37)1 was derived above. From (5.12)2, (5.20), (5.12)1, (5.34)1 and (5.38),
From (5.34)1, (5.12)1, (5.20) and (5.40),
From (5.38), (5.20), (5.40) and (5.18),
□
Corollary 5.1.1
and
where
Proof
Relation (5.42) with (5.44)1 follow upon taking the curl of (5.38) and noting (5.12)2. Taking the gradient of (5.40) and noting (5.34),
via the time derivative of (5.38). □
Remark 5.4
The foregoing definitions and manipulations require that \(\tilde{\mathbf{E}}_{w}\) and \(\mathbf{B}^{s}_{w}\) be of class \(C^{2}\) in space and time. In particular, (5.10)1,2 require that \(\boldsymbol {\mathcal {E}}_{w}\) and \(\boldsymbol {\mathcal {B}}_{w}\) be spatially of class \(C^{2}\). Accordingly, from (3.37), (3.39) and (3.31), \(w\) must be of class \(C^{2}\). However, simplest choice (3.8) of \(C^{1}\) mollifier \(\varphi\) implies that \(w\) is only of class \(C^{1}\). The simplest choice of \(C^{2}\) mollifier is (cf. [5], (4.3.59))
The corresponding analysis is straightforward, delivers an appropriate polynomial in place of \(F(\lambda)\) (cf. (3.16) and (3.17)), and affects only separations in the range \(\epsilon< u_{i} < \epsilon+ \delta\).
Remark 5.5
Since \(f(u_{i}) = 0\) if \(u_{i} < \epsilon\), from (5.13), (5.12), (5.34), (5.42) and (5.43) only charges further than \(\epsilon\) from x contribute to the values of \(\psi^{s}_{w}\), \(\mathbf{A}^{s}_{w}\), \(\tilde{\mathbf{E}}_{w}\), \(\mathbf{B}^{s}_{w}\), \(\mathbf{s}_{1}\) and \(\mathbf{s}_{2}\). If all charges are further than a distance \(d\) from x (so that x lies in free space), then one can choose a scale \(\epsilon\) and mollifying interval \(\delta\) such that \(\epsilon+ \delta< d\). In such case relations (5.22) hold at any such scale. Further, (5.42) and (5.43) hold with sources given by (5.44)1,2 in which \(\tilde{\rho}_{w}\) and \(\tilde{\mathbf{J}}_{w}\) satisfy (5.40) and (5.38). Here \(\psi^{s} _{w}\) and \(\mathbf{A}^{s}_{w}\) are defined by (5.13)1,2 with \(f(u_{i}) = 1/4\pi u_{i}\). Specifically, we have
Corollary 5.1.2
If all charges are further than \(\epsilon+ \delta\) from location x, then at x
and
where
and
Further,
and
Proof
If \(u_{i} > \epsilon+ \delta\), then (cf. (3.62)2) \(f(u_{i}) = 1/4\pi u_{i}\). Straightforward calculations yield
Relations (5.46) and (5.47) follow from (5.40), (5.38) and definitions (5.13)1,2.
Fields \(\tilde{\mathbf{E}}_{w}\) and \(\mathbf{B}^{s}_{w}\) are given by (5.34) and (5.12)2 with \(\psi^{s}_{w}\) and \(\mathbf {A}^{s}_{w}\) defined by (5.13)1,2. Relations (5.50) and (5.51) follow on noting that
and
□
Remark 5.6
At this point a complete set of Maxwell equations has been obtained, at any spatial scale, from a Newtonian perspective. While relations (2.8), (2.20) and (2.33) are entirely general (and of purely kinematic character) relations (5.37) are associated with forces in free space, via the results of Coulomb and Biot-Savart, and hence have been shown to hold in vacuo. The analytic key to the theory is function \(f\): this determines \(\psi^{s}_{w}\) and \(\mathbf{A}^{s}_{w}\) (cf. (5.13)) and thereby \(\tilde{\mathbf{E}}_{w}\) and \(\mathbf{B}^{s}_{w}\) (cf. (5.12) and (5.34)). The form of \(f\) is determined by choice of scale \(\epsilon\) and mollifier \(\varphi\). Departure from Newtonian dynamics stems from consideration of the argument \(u_{i}\) of \(f\) in respect of any charge. Determination of \(u_{i} (\mathbf{x}, t)\) requires knowledge of the location of \(P_{i}\) at time \(t\). In a dynamic context this cannot be known at x since information transfer is not instantaneous. Indeed, the inhomogeneous wave equations (5.42) and (5.43) indicate that the transfer speed in free space is \(c\). Information available at x at time \(t\) consists of the apparent displacement \(\mathbf{u}^{a}_{i} (\mathbf{x}, t)\) of \(P_{i}\) from x at time \(t\), and time derivatives thereof: in particular, the apparent velocity \(\mathbf{v}^{a}_{i} := \partial\mathbf{u}^{a}_{i}/\partial t\). In generalising the discussion to take account of information delay (‘retardation’) the key is to find analogues of potentials \(\psi^{s} _{w}\) and \(\mathbf{A}^{s}_{w}\) which depend upon available knowledge, namely \(\mathbf{u}^{a}_{i}\) and \(\mathbf{v}^{a}_{i}\).
6 Signal Transmission Times, Retardation and Classical Maxwellian Electrodynamics
Consider the behaviour of a moving point charge \(P_{i}\) as monitored by an observer \(O\) located at a point x in an inertial frame ℱ. Information available to \(O\) at time \(t\) can involve only data that has reached \(O\) at, or before, this time. Since transmission of information is not instantaneous, it is necessary to examine the consequences of transmission time delay. Examined here are the relationship between the apparent location of \(P_{i}\) at time \(t\), the delay such datum of information takes to reach \(O\), and the actual trajectory of \(P_{i}\).
It is instructive to consider how, at least in principle, information about the motion of \(P_{i}\) could be obtained. Suppose that \(O\) has a radar device capable of detecting a reflected segment (from \(P_{i}\)) of any signal that it has transmitted. Suppose further that any such segment, both during its outward and inward paths, travels in a straight line at constant speed \(c\), and is instantaneously reflected by \(P_{i}\). Such a signal, emitted from x at time \(t'\) and received back at time \(t\), will have travelled a total distance \(c(t-t')\) and been reflected at time \(t' + (t-t')/2 = (t + t')/2\) when at a distance \(c (t-t')/2\) from x. If \(\mathbf{x}_{i} ( \tau)\) denotes the location of \(P_{i}\) at time \(\tau\), then the distance travelled by the signal between its reflection and reception is
Changing notation, if \(\tau_{i} (\mathbf{x}, t)\) denotes the time at which the signal reaching x at time \(t\) was reflected from \(P_{i}\), then
a detectable quantity. Accordingly, (6.1) may be written as
Assuming that the device can detect the direction of the incoming signal, such information, together with (6.3), determines location \(\mathbf{x}_{i} (\tau_{i}(\mathbf{x}, t))\). This is the apparent location of \(P_{i}\) at time \(t\) as monitored at location \(\mathbf{x}, \mathbf{x}^{a}_{i} (\mathbf{x}, t)\) say. That is,
Remark 6.1
In any motion of \(P_{i}\), apparent location \(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\) is unique. Indeed, suppose that a signal emitted from x gives rise to segments which are reflected by \(P_{i}\) at times \(\tau\) and \(\tau'\), and are both received at x at time \(t\). Thus \(\tau\) and \(\tau'\) are two values of \(\tau_{i} (\mathbf{x}, t)\), and \(\mathbf{x}_{i} (\tau)\) and \(\mathbf{x}_{i} (\tau')\) are both candidates for \(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\). From (6.3) it follows thatFootnote 5
Hence the average speed of \(P_{i}\) over a time interval of duration \(|\tau- \tau'|\) is at least \(c\), a physical impossibility. Thus the hypothesis of two times \(\tau\) and \(\tau'\) is incorrect, \(\tau_{i} ( \mathbf{x}, t)\) is unique, and so \(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\) is unique.
The apparent displacement of \(P_{i}\) from x at time \(t\) is
Writing
relation (6.3) becomes
Time \(\tau_{i}\), apparent location \(\mathbf{x}^{a}_{i}\), and apparent displacement \(\mathbf{u}^{a}_{i}\) are functions of x and \(t\), and are thus fields; \(\tau_{i}\) is termed the retarded time field. In particular, \(\tau_{i} (\mathbf{x}, t)\) and \(\mathbf{x}^{a}_{i} (\mathbf{x}, t)\) constitute the basic information about any motion of \(P_{i}\) which is available at x at time \(t\). Such information is necessary in computation of velocities. Here a distinction must be made between the actual velocity of \(P_{i}\) at any time \(\tau\), namely
and the apparent velocity
corresponding to \(P_{i}\) as monitored at x and time \(t\). In particular, \(\mathbf{v}_{i}\) is a vector-valued function of time, whereas \(\mathbf{v}^{a}_{i}\) is a vector-valued field. From (6.4) and the chain rule,
That is,
To determine \(\partial\tau_{i}/\partial t\), note that from (6.6) and (6.8)
Differentiation with respect to \(t\) yields, via (6.10)2 and (6.8),
Accordingly,
where
Thus \(\hat{\mathbf{u}}^{a}_{i} (\mathbf{x}, t)\) denotes a unit vector in the direction of \(P_{i}\) from x at the latest time \(\tau_{i} (\mathbf{x}, t)\) that information about \(P_{i}\) reaches \(\mathbf{x}_{i}\) at time \(t\). From (6.12) and (6.15),
Equivalently,
This relation delivers the actual velocity at time \(\tau_{i} ( \mathbf{x}, t)\) in terms of information available at x and time \(t\).
The spatial derivative (or ‘gradient’) of (6.13) yields, with (6.8)
From (6.6), the chain rule and (6.9),
whence
Hence (6.19) may be written, noting (6.16), as
and so
Since, from (6.17) and (6.15)2,
it follows from (6.23) that
From (6.20)3, (6.25) and (6.18),
Further, from (6.26) and (6.15)2,
In order to generalise the discussion of Sect. 5 to dynamic situations, taking account of signal transmission delay, it suffices to consider the potential functions: cf. Remark 5.6. A natural choice \(\mathbf{A}^{d}_{w}\) for the dynamic counterpart of \(\mathbf{A}^{s}_{w}\) (cf. (5.13)2) would seem to be
Choice \(\mathbf{A}^{d}_{w} (\mathbf{x}, t)\) takes account of the latest information concerning charge displacements and velocities that is available at x and time \(t\), has the form of (5.13)2, and would coincide with \(\mathbf{A}^{s}_{w} (\mathbf{x}, t)\) were transmission times to be negligible. A choice of the corresponding dynamic electric potential might be
However, this field does not satisfy the counterpart of the Lorenz gauge relation (cf., e.g., Jackson [13], p. 240), namely
To obtain an appropriate candidate \(\psi^{d}_{w}\), consider (cf. (6.29))
From (6.27),
while, from (6.28),
Accordingly, (6.32), (6.33) and (6.34) imply
Hence, from (6.35) and (6.36),
and thus
satisfies (6.31).
The results of Sect. 5 can now be generalised by essentially replacing superscript ‘\(s\)’ by ‘\(d\)’. Specifically, function \(f\) is known once weighting function \(w\) is chosen: this follows from Theorem 3.1, consequent upon definition (3.5), (3.31)2 and (3.51). In particular (cf. Remark 3.6), \(f(u) = 0\) if \(u < \epsilon\) and \(f(u) = 1/4\pi u\) if \(u > \epsilon+ \delta\). Knowledge of \(f\), and the latest information concerning displacements and velocities (namely \(\mathbf{u}^{a}_{i}\) and \(\mathbf{v}^{a}_{i}\)), yield fields \(\mathbf{A}^{d}_{w}\) and \(\psi^{d}_{w}\) via (6.29) and (6.38). The corresponding dynamic electric field \(\mathbf{E}^{d}_{w}\) and magnetic field \(\mathbf{B}^{d}_{w}\) are
Theorem 6.1
where
and
Proof
Relations (6.40)2 and (6.41)1 are immediate consequences of relations (6.39). Proofs of the remaining relations are precise analogues of those in Theorem 5.1, on invoking (6.31) in place of (5.14), writing \(\mathbf{E}^{d}_{w}\) in place of \(\tilde{\mathbf{E}}_{w}\), and changing superscript ‘\(s\)’ to ‘\(d\)’. □
Remark 6.2
Relations (6.40) and (6.41) constitute the general macroscopic form of Maxwell’s equations in free space at scale \(\epsilon\) in classical electrodynamics when supplemented by the purely kinematic relations (2.8), (2.20) and (2.33), with \(w\) given by (3.5). These relations take exactly the same form when additional time averaging is implemented in the manner of Sect. 4, specifically via invocation of properties (4.2) and (4.3). However, although all relations are thus form invariant under changes of length and time scales, the phenomena they represent may not be so obliging. Indeed, fields \(\rho^{d}_{w}\) and \(\mathbf{J}^{d}_{w}\), which are the dynamic analogues of charge and current densities \(\rho_{w}\) and \(\mathbf{j} _{w}\), are to be expected to depend upon the spatial and temporal scales associated with measurements of the phenomena of interest. Furthermore, fields \(\rho^{d}_{w}\) and \(\mathbf{J}^{d}_{w}\) (or their time-averaged counterparts) can be considered to drive associated electromagnetic phenomena: their prescription gives rise to the inhomogeneous d’Alembert equations (6.42) for the corresponding potentials \(\psi ^{d}_{w}\) and \(\mathbf{A}^{d}_{w}\) which in turn (via (6.39)) generate the relevant fields \(\mathbf{E}^{d}_{w}\) and \(\mathbf{B}^{d}_{w}\) which satisfy (6.40) and (6.41). In particular, suppose the \(\Delta \)-time averages of \(\rho^{d}_{w}\) and \(\mathbf{J}^{d}_{w}\) vanish in some region. Then in this region the corresponding fields \(\mathbf{E}^{d}_{w}\) and \(\mathbf{B}^{d}_{w}\) satisfy (cf. (6.40)1 and (6.41)2),
Accordingly, from (6.45)1, (6.45)2 and (6.44)1,
and, similarly,
That is, both the electric and magnetic fields constitute wave disturbances travelling at speed \(c\) wherever and whenever \(\rho^{d} _{w}\) and \(\mathbf{J}^{d}_{w}\) both vanish.
Remark 6.3
Fields \(\psi^{d}_{w}\) and \(\mathbf{A}^{d}_{w}\) with \(f(u^{a}_{i}) = 1/4\pi u^{a}_{i}\) are the Liénard-Wiechert potentials. These were derived in connection with solving inhomogeneous d’Alembert equations of form (6.42) using Green’s functions (cf., e.g., [13], §6.4 and Jones [15], §3.1).
Remark 6.4
If \(u^{a}_{i} > \epsilon+\delta\), then (cf. (6.38)) \(\beta_{i}\, f(u^{a}_{i}) = \beta_{i}/4\pi u^{a}_{i}\). To discover the physical interpretation of \(\beta_{i}/u^{a}_{i}\), note that
With \(\tau= \tau_{i}(\mathbf{x}, t)\) and recalling (6.6), (6.9), (6.8) and (6.17),
Suppressing arguments x and \(t\), it follows that
via (6.15)2 and neglecting terms of order \(O((v^{a}_{i}/c)^{2})\) and \(o(u^{a}_{i}/c)\). Accordingly, if \(u^{a}_{i} > \epsilon+ \delta \), then
upon neglect of terms as in (6.50).
Remark 6.5
Macroscopic steady-state situations were discussed in Sect. 5 and generalised to dynamic contexts via definition (5.34) of a dynamic electric field \(\tilde{\mathbf{E}}_{w}\) together with the consequences exhibited in Theorem 5.1. In order to take account of non-instantaneous information transfer the artifice of hypothetical radar signalling was introduced. However, such monitoring would not only be impracticable but also intrusive. Indeed, radar signals would interact with electrons and nuclei, and constitute a measurement process requiring a quantum mechanical description. Nevertheless, the discussion captures precisely how information concerning a moving charge is communicated: this is transmitted precisely as the signal after reflection.
7 The Force on a Moving Charge
Consider the consequences of assuming that the Lorentz force relation (1.9) has general validity. In such case the force acting on a point charge \(q\) moving with velocity v in an inertial frame would be (cf. (6.39))
Here field values are calculated at the instantaneous location x of \(q\) at time \(t\). If \(u^{a}_{i} (\mathbf{x}, t) > \epsilon + \delta\) for all charges \(P_{i}\), then \(f(u^{a}_{i}) = 1/4\pi u^{a} _{i}\) (cf. Remark 3.6) and the contribution from \(P_{i}\) to F is
where (cf. (6.39)1, (6.38) and (6.29))
and
Recalling (6.18) and writing
yield (via (5.18))
and
Lemma 7.1
For brevity proofs of these results are omitted, as are those of the following theorem.
Theorem 7.1
and
where
Corollary 7.1
This is an immediate consequence of (7.12) and (7.13).
Remark 7.1
These results duplicate those given in Griffiths [11]. In particular, (7.12) is expressed (cf. op. cit. (10.65)) as
where
Thus (cf. (6.18), (6.15)2 and (7.6))
Remark 7.2
Equivalently,
Remark 7.3
The results of Theorem 7.1 furnish individual contributions to the macroscopic fields which satisfy the Maxwell relations (6.40). There were two underlying assumptions involved: (i) separations \(u^{a}_{i}\) were to exceed \(\epsilon+ \delta\), where \(\epsilon\) was associated with any scale at which the Biot-Savart law is valid (cf. identifications (5.7) and (5.8)), and (ii) the potentials given in (5.13) were assumed to have dynamic generalisations (6.38) and (6.29).
8 Summary and Concluding Remarks
The microscopic basis of classical macroscopic electromagnetic relations has been investigated via recognition of the rôle played by spatial and temporal averaging, here implemented in terms of weighting functions. Modelling electrons and nuclei as point charges, any choice \(w\) of weighting function yielded definitions \(\rho_{w}\) and \(\mathbf{j}_{w}\) of charge and current densities which satisfy (1.6). Any solution a to \(\mathrm{div}\, \mathbf{a} = w\) gave rise to purely electrokinetic fields \(\mathbf{D}_{w}\) and \(\mathbf{H}_{w}\) which satisfy (1.1) and (1.2). At any prescribed scale \(\epsilon\), choice of a specific and natural weighting function resulted in binary decompositions (3.35) of \(\mathbf{D}_{w}\) and \(\mathbf{H}_{w}\). These decompositions introduced electric and magnetic polarisation densities together with electrokinetic fields \(\boldsymbol {\mathcal {E}}_{w}\) and \(\boldsymbol {\mathcal {B}} _{w}\) that were expressible in terms of scalar and vector potentials \(\psi_{w}\) and \(\boldsymbol {\mathcal {A}}_{w}\), respectively. The experimental laws of Coulomb and Biot-Savart enabled the macroscopic time-independent electric and magnetic fields \(\mathbf{E}^{s}_{w}\) and \(\mathbf{B}^{s} _{w}\) in free space to be expressed as scalar multiples of \(\boldsymbol {\mathcal {E}} _{w}\) and \(\boldsymbol {\mathcal {B}}^{s}_{w}\) and hence in terms of potential functions \(\psi^{s}_{w}\) and \(\mathbf{A}^{s}_{w}\). Fields \(\mathbf{B} ^{s}_{w}\) and \(\mathbf{A}^{s}_{w}\) satisfy (1.3) and (1.5), and \(\mathbf{E}^{s}_{w}\) satisfies (1.7) and (1.8) in this macroscopically stationary context. Generalisation to macroscopically dynamic situations was effected in two stages. In the first stage the dynamic electric field was formally defined via (1.7), with \(\psi\) and A identified with \(\psi^{s}_{w}\) and \(\mathbf{A}^{s}_{w}\), and the consequences exhibited in Theorem 5.1. The second stage addressed the consequences of non-instantaneous transmission of information: this involved selection of appropriate modified versions \(\psi^{d}_{w}\) and \(\mathbf{A}^{d}_{w}\) of \(\psi^{s}_{w}\) and \(\mathbf{A}^{s}_{w}\), and resulted in the final form of the relations under investigation, displayed in Theorem 6.1. The individual contribution of any given charge to the force on another moving charge, as predicted by the Lorentz relation (1.9), was computed as an exercise and for completeness.
Remark 8.1
Relations (2.8), (2.20) and (2.33) are completely general and electrokinetic in nature. However, while Maxwell’s equations (6.40)1,2 and (6.41)1,2 hold in vacuo, their counterparts in regions occupied by matter depend upon the matter in question, and thus must incorporate constitutive relations. The simplest, for homogeneous isotropic media, are
where \(\chi_{e}\) and \(\nu_{m}\) are dimensionless constants. Accordingly, from (3.35)1,2,
If the Coulomb and Biot-Savart laws hold within a medium with unchanged values of \(\epsilon_{0}\) and \(\mu_{0}\), then from (5.5) and (5.9)
where, writing \(\chi_{m} := \nu_{m}/(1-\nu_{m})\),
Dimensionless constants \(\chi_{e}\) and \(\chi_{m}\) are termed the electric and magnetic susceptibilities of the medium (cf. [14], p. 354 and p. 412). In such case relations (2.20) and (2.33) yield
Remark 8.2
Consideration of information transmission in Sect. 6 involved invariance of signal speed, the fundamental phenomenon addressed by special relativity. The final results are accordingly consistent with this theory, but have been derived independently of other relativistic considerations (cf. [9, 14]) and are thus described as classical. Of course, the quantum mechanical notion of electron spin and its macroscopic consequences lie outwith a classical formulation.
Remark 8.3
The ubiquitous nature of electromagnetism requires that all charges in the universe be a priori considered to contribute to local values of electric and magnetic fields.Footnote 6 In examining specific localised behaviour those charges in the region of interest may be distinguished from all others, resulting in internal or localised fields together with their complements, the corresponding external or ‘far’ fields.
Remark 8.4
As indicated in Remark 6.2, when time-averaged the relations of Theorem 6.1 are form-invariant with respect to changes in the choices of length and time scales associated with the spacetime averaged fields which appear therein. However, if local field values are associated with measurements, then the sensitivity of the measuring devices introduces associated and specific scales of length and time.
Notes
Derivation of these results can be made rigorous by implementing cellular averaging in terms of a weighting function: cf. Murdoch [5], p. 61 and pp. 197–199.
\(\nabla_{\mathbf{u}}w\) and \(\nabla_{\mathbf{x}} w\) denote derivatives with respect to arguments u and x, respectively. Direct (that is, co-ordinate-free) notation is employed: cf., e.g., Murdoch [5], in particular Lemma 4.2.1.
The tensor (or dyadic) product \(\mathbf{a} \otimes \mathbf{b}\) of vectors a and b is that linear transformation which maps any vector k into \((\mathbf{b}\,. \,\mathbf{k})\mathbf{a}\): cf. [5], p. 317. For any order 2 tensor field L, \(\mathrm{div}\, \mathbf{L}\) is the vector field for which \((\mathrm{div}\, \mathbf{L}).\,\mathbf{k} = \mathrm{div} (\mathbf{L} ^{T}\mathbf{k})\) for any fixed vector k: cf. [5], p. 390.
For any pair of vectors v and \(\mathbf{v}'\), \(\Vert\mathbf{v} - \mathbf{v}' \Vert\ge|\, \Vert\mathbf{v}\Vert- \Vert\mathbf{v}'\Vert\,|\).
For example, the terrestrial magnetic field and its effect upon compass needles, or the effect of solar flares upon long range radio communications.
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Murdoch, A.I. On Spatially-averaged Electrokinetics of Point Charges and Maxwell’s Equations. J Elast 131, 75–109 (2018). https://doi.org/10.1007/s10659-017-9647-0
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DOI: https://doi.org/10.1007/s10659-017-9647-0