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Article Direct Derivation of Liénard–Wiechert Potentials, Maxwell’s Equations and Lorentz Force from Coulomb’s Law

Hrvoje Dodig †

Department of Electrical and Information Technology, Faculty of Maritime Studies, University of Split, 21000 Split, Croatia; [email protected] † Current address: Faculty of Maritime Studies, University of Split, Rudera¯ Boškovi´ca37, 21000 Split, Croatia.

Abstract: In this paper, the solution to long standing problem of deriving Maxwell’s equations and Lorentz force from ﬁrst principles, i.e., from Coulomb’s law, is presented. This problem was studied by many authors throughout history but it was never satisfactorily solved, and it was never solved for charges in arbitrary motion. In this paper, relativistically correct Liénard–Wiechert potentials for charges in arbitrary motion and Maxwell equations are both derived directly from Coulomb’s law by careful mathematical analysis of the moment just before the charge in motion stops. In the second part of this paper, the electrodynamic energy conservation principle is derived directly from Coulomb’s law by using similar approach. From this energy conservation principle the Lorentz force is derived. To make these derivations possible, the generalized Helmholtz theorem was derived along with two novel vector identities. The special relativity was not used in our derivations, and the results show that electromagnetism as a whole is not the consequence of special relativity, but it is rather the consequence of time retardation.

Keywords: coulomb’s law; liénard–wiechert potentials; maxwell equations; lorentz force

Citation: Dodig, H. Direct Derivation 1. Introduction of Liénard–Wiechert Potentials, Maxwell’s Equations and Lorentz In his famous Treatise [1,2], Maxwell derived his equations of electrodynamics based Force from Coulomb’s Law. on the knowledge about the three experimental laws known at the time: Coulomb’s law Mathematics 2021, 9, 237. describing the electric force between charges at rest; Ampere’s law describing the force https://doi.org/10.3390/ between current carrying wires, and Faraday’s law of induction. Maxwell equations and math9030237 Lorentz force were never derived from first principles, i.e., from simpler physical laws than Ampere’s law and Faraday’s law. Received: 5 January 2021 In modern vector notation, the four Maxwell’s equations that govern the behavior of Accepted: 22 January 2021 electromagnetic fields are written as: Published: 26 January 2021 D = ρ (1) Publisher’s Note: MDPI stays neu- ∇ · B = 0 (2) tral with regard to jurisdictional clai- ∇ · ∂B ms in published maps and institutio- E = (3) nal affiliations. ∇ × − ∂t 1 ∂E B = µJ + (4) ∇ × c2 ∂t

Copyright: © 2021 by the authors. Li- where the symbol D denotes electric displacement vector, E is the electric field vector, B is censee MDPI, Basel, Switzerland. a vector called magnetic flux density, vector J is called current density and scalar ρ is the This article is an open access article charge density. Furthermore, there are two more important equations in electrodynamics distributed under the terms and con- that relate magnetic vector potential A and the scalar potential φ to the electromagnetic ditions of the Creative Commons At- fields B and E: tribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Mathematics 2021, 9, 237. https://doi.org/10.3390/math9030237 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 237 2 of 28

B = A (5) ∇ × ∂A E = φ (6) −∇ − ∂t

In standard electromagnetic theory, if a point charge qs is moving with velocity vs(t) along arbitrary path rs(t) the scalar potential φ and vector potential A caused by moving charge qs are described by well known, relativistically correct, Liénard–Wiechert potentials [3,4]:

1 q φ = φ(r, t) = s (7) 4πe (1 ns(tr) β (tr)) r rs(tr) − · s | − | µc qsβ (tr) A = A(r, t) = s (8) 4π (1 ns(tr) β (tr)) r rs(tr) − · s | − |

where tr is retarded time, r is the position vector of observer and vectors ns(tr) and βs(tr) are:

v (t ) β (t ) = s r (9) s r c r rs(tr) ns(tr) = − (10) r rs(tr) | − | These equations were almost simultaneously discovered by Liénard and Wiechert around the 1900s and they represent explicit expressions for time-varying electromagnetic fields caused by the charge in arbitrary motion. Nevertheless, Liénard–Wiechert potentials were derived from retarded potentials, which in turn, are derived from Maxwell equations. Maxwell’s electrodynamic equations provide the complete description of electromagnetic fields; however, these equations say nothing about mechanical forces experienced by the charge moving in the electromagnetic field. If the charge q is moving in electromagnetic field with velocity v then the force F experienced by the charge q is:

F = q(E + v B) (11) × The force described by Equation (11) is the well-known Lorentz force. The discovery of this electrodynamic force is historically credited to H.A. Lorentz [5]; however, the similar expression for electromagnetic force can be found in Maxwell’s Treatise, article 598 [2]. The difference between the two is that Maxwell’s electromotive force acts on moving circuits and Lorentz force acts on moving charges. However, it has not yet been explained what causes the Lorentz force, Ampere’s force law, and Faraday’s law. Maxwell derived his expression for electromotive force along the moving circuit from the knowledge of experimental Faraday’s law. Later, Lorentz extended Maxwell’s reasoning to discover the force acting on charges moving in electromagnetic field [5]. Nevertheless, it would be impossible for Lorentz to derive his force law without the prior knowledge of Maxwell equations [6]. Presently, the magnetism and the Lorentz force (qv B term) is commonly viewed × as a consequence of Einstein’s special relativity. For example, an observer co-moving with source charge would not measure any magnetic field, while on the other hand, the stationary observer would measure the magnetic field caused by moving source charge. However, in this work, it is demonstrated that special relativity is not needed to derive the Lorentz force and Maxwell equations. In fact, Maxwell’s equations and Lorentz force are derived from more fundamental principles: Coulomb’s law and the time retardation. There is another reason the idea to derive Maxwell’s equations and Lorentz force from Coulomb’s law may seem plausible. Because of the mathematical similarity between Coulomb’s law and Newton’s law of gravity, many researchers thought that if Maxwell’s Mathematics 2021, 9, 237 3 of 28

equations and Lorentz force could be derived from Coulomb’s law that this would be helpful in the understanding of gravity. These two inverse-square physical laws are written:

q1q2 r1 r2 FC = − (12) 4πe r r 3 | 1 − 2| r1 r2 FG = Gm1m2 − (13) − r r 3 | 1 − 2| The expressions for Coulomb’s force and Newton’s gravitational force are indeed similar; however, these two forces significantly differ in physical nature. The latter force is always attractive, while the former can be either attractive or repulsive. Nevertheless, several researchers attempted to derive Maxwell’s equations from Coulomb’s law, and most of these attempts rely on Lorentz transformation of space-time coordinates between the rest frame of the moving charge and laboratory frame. The first hint that Maxwell’s equations could be derived from Coulomb’s law and Lorentz transformation can be found in Einstein’s original 1905 paper on special relativity [7]. Einstein suggested that the Lorentz force term (v B) is to be attributed to Lorentz trans- × formation of the electrostatic field from the rest frame of moving charge to the laboratory frame where the charge has constant velocity. Later, in 1912, Leigh Page derived Faraday’s law and Ampere’s law from Coulomb’s law using Lorentz transformation [8]. Frisch and Willets discussed the derivation of Lorentz force from Coulomb’s law using the relativistic transformation of force [9]. A similar route to the derivation of Maxwell’s equations and Lorentz force from Coulomb’s law was taken by Elliott in 1966 [10]. Kobe in 1986 derives Maxwell’s equations as the generalization of Coulomb’s law using special relativity [11]. Lorrain and Corson derive Lorentz force from Coulomb’s law, again, by using Lorentz transformation and special relativity [12]. Field in 2006 derives Lorentz force and magnetic field starting from Coulomb’s law by relating the electric field to electrostatic potential in a manner consistent with special relativity [13]. The most recent attempt comes from Singal [14] who attempted to derive electromagnetic fields of accelerated charge from Coulomb’s law and Lorentz transformation. All of the mentioned attempts have in common that they attempt to derive Maxwell equations from Coulomb’s law by exploiting Lorentz transformation or Einstein’s special theory of relativity. However, historically the Lorentz transformation was derived from Maxwell’s equations [15], thus, the attempt to derive Maxwell’s equations using Lorentz transformation seems to involve circular reasoning [16]. The strongest criticism came from Jackson, who pointed out that it should be immediately obvious that without additional assumptions, it is impossible to derive Maxwell’s equations from Coulomb’s law using the theory of special relativity [17]. Schwartz addresses these additional assumptions and starting from Gauss’ law of electrostatics, and by exploiting the Lorentz invariance and properties of Lorentz transformation, he derives Maxwell’s equations [18]. In addition to the criticism above, it should be emphasized that the derivations of Maxwell’s equations from Coulomb’s law using Lorentz transformation should only be considered valid for the special case of the charge moving along the straight line with constant velocity. This is because the Lorentz transformation is derived under the assumption that electron moves with constant velocity along straight line [15]. For example, if the particle moves with uniform acceleration along a straight line, the transformation of coordinates between the rest frame of the particle and the laboratory frame takes a different mathematical form than that of the Lorentz transformation [19]. If the particle is in uniform circular motion, yet another coordinate transformation from the rest frame to the laboratory frame, called Franklin transform, is valid [20]. None of the above cited papers consider the fact that Lorentz transformation is no longer valid when the charge is not moving along straight line with constant velocity. To circumvent problems with special relativity and Lorentz transformation, in this paper, an entirely different approach is taken to derive Liénard–Wiechert potentials and Maxwell’s equations from Coulomb law. The derivation starts from the analysis of the Mathematics 2021, 9, 237 4 of 28

following hypothetical experiment: consider two charges at rest at the present time, one called the test charge, and the other called the source charge. The source charge was moving in the past, but it is at rest at the present time. Because both charges are at rest at present, the force acting on test charge at present time is Coulomb’s force. However, in the past, when the source charge was moving, it is assumed that the force acting on test charge was not Coulomb’s force. To discover the mathematical form of this “unknown” electrodynamic force acting in the past from the knowledge of known electrostatic force (Coulomb’s law) acting at the present time, the generalized Helmholtz decomposition theorem was applied. This theorem, derived in AppendixA, allows us to relate Coulomb’s force acting at the present time to the positions of source charge at past time. From here, Liénard–Wiechert potentials and Maxwell’s equations were derived by careful mathematical manipulation. However, from Maxwell’s equations, it is very difficult, if not entirely impossible, to derive the Lorentz force without resorting to some form of energy conservation law. As shown in Figure1a, at the present time, the single stationary charge creates Coulomb’s electrostatic field. The known energy conservation law valid at the present time states that the contour integral of Coulomb’s field along closed contour C is equal to zero. However, this electrostatic energy conservation law, valid at present, is not necessarily valid in the past when the source charge was moving. Thus, in the second part of this paper, we derive this "unknown" dynamic energy conservation principle valid in the past from the knowledge of electrostatic energy conservation principle valid at the present time. This was again achieved by the careful application of generalized Helmholtz decomposition theorem which allowed us to transform electrostatic energy conservation law valid at present to dynamic energy conservation law valid in the past. This dynamic energy conservation law states that the work of non-conservative force along closed contour is equal to the time derivative of the flux of certain vector field through the surface bounded by this closed contour. From this dynamic energy conservation law the Lorentz force was finally derived. Thus, in this work, the entire electromagnetic theory for charges in arbitrary motion was derived from two simple postulates: (a) when charges are at rest, the Coulomb’s force acts between the charges, and (b) the disturbances caused by charges in motion propagate with finite velocity. It should be emphasized that the theory of special relativity and Lorentz transformation were not used to derive Maxwell’s equations, hence, this work shows that magnetism and Lorentz force cannot be considered to be the consequence of special relativity. The magnetism and Lorentz force could be considered the consequence of special relativity only for the special case of motion with uniform velocity along straight line. The theory presented in this paper can be considered to be more general than those derived from special relativity because it is valid for charges in arbitrary motion. However, it was derived from time retardation, hence, electromagnetism should be considered to be the consequence of time retardation and not as the consequence of special relativity. Mathematics 2021, 9, 237 5 of 28

C C

SOURCE CHARGE qs IS SOURCE CHARGE qs STOPS STATIONARY AT PRESENT TIME tp MOVING AT PAST TIME ts < tp

AT PRESENT TIME tp ELECTRIC FIELD AT POINT ON CONTOUR r IS Ec r COULOMB’S ELECTROSTATIC FIELD Ec

qs r rs Ec = − 3 4πε r rs qs | − | qs ENERGY CONSERVATION LAW rs AT PRESENT TIME tp : E dr = 0 C c · H (a) present time (b) past time

Figure 1. In (a) the source charge qs is at rest at present time tp. Each point on closed contour C is affected by Coulomb’s electrostatic ﬁeld E . Energy conservation principle at present time is E dr = 0. In (b) the source charge q is moving c C c · s along arbitrary path and it stops at past time t < t . Dynamic energy conservation principle valid in the past is assumed to s p H be unknown when source charge was moving.

2. Generalized Helmholtz Decomposition Theorem Because the generalized Helmholtz decomposition theorem is central for deriving Maxwell equations and Lorentz force from Coulomb’s law, this important theorem is briefly presented in this section while the derivation itself is moved to AppendixA. There have been several previous attempts in the literature to generalize classical Helmholtz decomposition theorem to time dependent vector fields [21–23]. However, in none of the cited articles, the Helmholtz theorem for functions of space and time is presented in the mathematical form usable for the mathematical developments described in this paper. This is probably caused by difficulties in stating such a theorem and this was clearly stated in [22]: “There does not exist any simple generalization of this theorem for time-dependent vector fields”. However, it is shown in this paper that there indeed exists the simple generalization of Helmholtz decomposition theorem for time-dependent vector fields and that it can be derived from time-dependent inhomogeneous wave equation. To improve the clarity of this paper, the complete derivation of Helmholtz decomposition theorem for functions of space and time is moved to AppendixsA and A.2. As it was shown in AppendixA, the generalization of Helmholtz decomposition theorem for the vector function of space and time F(r, t) can be written as:

F(r, t) = dt0 0 F(r0, t0) G(r, t; r0, t0)dV0 (14) − ∇ 3 ∇ · ZR ZR 1 ∂ ∂ + dt0 F(r0, t0) G(r, t; r0, t0)dV0 c2 ∂t 3 ∂t ZR ZR 0

+ dt0 0 F(r0, t0) G(r, t; r0, t0)dV0 ∇ × 3 ∇ × ZR ZR

where scalar function G(r, t; r0, t0) is the fundamental solution of time-dependent inhomogeneous wave equation given as:

2 2 1 ∂ G(r, t; r0, t0) G(r, t; r0, t0) = δ(r r0)δ(t t0) (15) ∇ − c2 ∂t2 − − − Mathematics 2021, 9, 237 6 of 28

In the equation above, δ(r r ) = δ(x x )δ(y y )δ(z z ) is 3D Dirac delta function, − 0 − 0 − 0 − 0 and δ(t t ) is Dirac delta function in one dimension. Fundamental solution G(r, t; r , t ), − 0 0 0 sometimes called Green’s function, represents the retarded in time solution of the inhomogeneous time dependent wave equation and it can be written as:

r r0 δ t0 t + | −c | G(r, t; r0, t0) = − (16) 4π r r | − 0|

where position vector r0 is the location of the source at time t0. From Equation (14) it is evident that the Helmholtz decomposition theorem for functions of space and time can be regarded as a mathematical tool that allows us to rewrite any vector function that is function of present time t and of present position r as vector 0 function of previous time t0 and of previous position r . Furthermore, the generalized Helmholtz decomposition theorem (14) comes with additional limitation that it is valid if vector function F(r , t ) approaches zero faster than 1/ r r as r r ∞. 0 0 | − 0| | − 0| → Very similar theorem was presented in article written by Heras [24]; the difference is that in Heras’ article the time integrals in Equation (14) were a priori evaluated at retarded time t = t r r /c. As such, the generalized Helmholtz theorem presented 0 − | − 0| in [24] is not suitable for the derivation of Maxwell equations and Lorentz force from Coulomb’s law. Reason for this, as it will become evident later in this paper, is that if the time integrals in Equation (14) are immediately evaluated the important information is lost from the equation.

3. Derivation of Maxwell Equations from Coulomb’s Law In this section, Maxwell equations are derived from Coulomb’s law using generalized Helmholtz decomposition theorem represented by Equation (14). To begin the discussion, consider the hypothetical experiment shown in Figure2, where source charge qs is moving along trajectory rs(t) and it stops at some past time ts. The test charge q is stationary at all times. It is assumed that the disturbances caused by moving source charge propagate outwardly from the source charge with ﬁnite velocity c. These disturbances originating from the source charge at past time manifest themselves as the force acting on stationary test charge at present time. This means that there is a time delay ∆t between the past time ts when the source charge has stopped and the present time tp when this disturbance has propagated to the test charge:

r rs(ts) ∆t = t t = | − | (17) p − s c At precise moment in time tp, which is called the present time, the force acting on stationary test charge q is the Coulomb’s force because source charge and test charge are both at rest, and because the effect of source charge stopping at past time ts had enough time to propagate to test charge. The Coulomb’s force Fc experienced by the test charge q at the present time tp can be expressed by the following equation:

qqs r rs(ts) Fc(r, tp) = − 3 tp = ts + ∆t (18) 4πe r rs(ts) | − | Let us now consider the time t just one brief moment before the stopping time ts:

t = t δt (19) s − where δt 0 is very small time interval. This time interval δt is so small that it might → even be called inﬁnitesimally small. Then at the moment in time inﬁnitesimally before the present time t the force felt by test charge q is still the Coulomb’s force if δt 0. Using p → these considerations, Equation (18) can be rewritten as: Mathematics 2021, 9, 237 7 of 28

qqs r rs(t) Fc(r, tp δt) = − 3 t = ts δt; δt 0 (20) − 4πe r rs(t) − → | − | Note that Equation (20) is equivalent to Equation (18) when δt 0. The reason → Coulomb’s law is written this way is to permit slight variation of time before stopping time ts so that generalized Helmholtz decomposition theorem can be exploited in order to derive Maxwell’s equations from Coulomb’s law. Had this not been done, the source charge position vector rs(ts) could simply be considered to be constant vector and generalized Helmholtz decomposition could not be used.

y STATIONARY TEST q CHARGE q FEELS THE COULOMB FORCE AT PRESENT TIME tp > ts

r rs(ts) tp = tr + | − c | r

SOURCE CHARGE qs STOPS AT TIME ts x

r s(ts) qs

Figure 2. Source charge qs is moving along arbitrary trajectory rs(t) and stops at time ts. Because qs stops moving at past time ts, at present time tp, the stationary test charge q experiences electrostatic Coulomb force.

Because the right hand side of Equation (20) is now the function of time t and position r the generalized Helmholtz decomposition theorem can be used to rewrite the right hand side of Equation (20). This is because generalized Helmholtz decomposition theorem states that any vector function of time t and position r can be decomposed as described by this theorem if that function meets certain criteria. Thus, using generalized Helmholtz decomposition theorem, the right hand side of Equation (20) is rewritten as:

qqs r rs(t) Fc(r, tp δt) = − 3 = (21) − 4πe r rs(t) | − | qqs r0 rs(t0) = dt0 0 − G(r, t; r0, t0)dV0 3 3 −∇ R R ∇ · 4πe r rs(t ) Z Z | 0 − 0 | 1 ∂ ∂ qqs r0 rs(t0) + dt0 − G(r, t; r0, t0)dV0 2 3 3 c ∂t R R ∂t0 4πe r rs(t ) ! Z Z | 0 − 0 | qqs r0 rs(t0) + dt0 0 − G(r, t; r0, t0)dV0 3 3 ∇ × R R ∇ × 4πe r rs(t ) Z Z | 0 − 0 | Mathematics 2021, 9, 237 8 of 28

To clarify the notation in equation above note that dV0 = dx0dy0dz0 represents the differential volume element of an inﬁnite volume R3. As deﬁned in AppendixesA and A.1, the primed position vector r0 is written in Cartesian coordinate system as:

r0 = x0 ˆx + y0 ˆy + z0 ˆz (22)

where variables x0, y0, z0 R. Vectors ˆx, ˆy and ˆz are orthogonal Cartesian unit basis ∈ vectors. Furthermore, in Cartesian coordinates, the primed del operator that appears in ∇0 Equation (21) is defined as: ∂ ∂ ∂ 0 = ˆx + ˆy + ˆz (23) ∇ ∂x0 ∂y0 ∂z0 From the definition above, it follows that primed del operator acts only on functions ∇0 of variables x0, y0, z0, and consequently, on functions of primed position vector r0 = x0 ˆx + y0 ˆy + z0 ˆz. It does not act on functions of position vector of source charge rs(t0) because this position vector is function of variable t0. Using these definitions one can write the following simple relations:

r0 rs(t0) 1 − 3 = 0 (24) r rs(t ) −∇ r0 rs(t0) | 0 − 0 | | − | 1 r0 rs(t0) 2 1 1 0 0 − = 0 = δ r r (t0) (25) 4π 3 4π r r (t ) s ∇ · r0 rs(t0) −∇ 0 s 0 − | − | | − | r0 rs(t0) 1 0 − 3 = 0 0 = 0 (26) ∇ × r rs(t ) −∇ × ∇ r0 rs(t0) | 0 − 0 | | − | where δ(r0 r (t )) is 3D Dirac’s delta function. Inserting Equations (25) and (26) into − s 0 Equation (21), and eliminating charge q from the equation, yields the following relation:

qs r rs(t) qs 0 − = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (27) πe 3 3 e 4 r rs(t) − ∇ ZR ZR − | − | 1 ∂ ∂ qs r0 rs(t0) + dt0 − G(r, t; r0, t0)dV0 2 3 3 c ∂t R R ∂t0 4πe r rs(t ) ! Z Z | 0 − 0 | In AppendixsC and C.1, we have shown that the time derivative that appears in the second right hand side integral of Equation (27) can be written as:

∂ qs r0 rs(t0) qs vs(t0) qs − = 0 0 v (t0)δ r0 r (t0) (28) ∂t 4πe 3 4πe r r (t ) e s s 0 r0 rs(t0) ∇ × ∇ × 0 s 0 − − | − | | − | where vs(t0) is the velocity of the source charge qs at time t0:

∂rs(t0) vs(t0) = (29) ∂t0 By inserting Equation (28) into Equation (27) it is obtained that:

qs r rs(t) qs 0 − = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (30) πe 3 3 e 4 r rs(t) − ∇ ZR ZR − | − | 1 ∂ qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 c2 ∂t 3 e − ZR ZR − 1 ∂ qs vs(t0) + 2 dt0 0 0 G(r, t; r0, t0)dV0 c ∂t 3 4πe ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | Mathematics 2021, 9, 237 9 of 28

To rewrite the last right hand side term of Equation (30) the following identity, derived in AppendixsC and C.2, is used:

qs vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (31) 3 4πe ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | qs vs(t0) = dt0 G(r, t; r0, t0)dV0 ∇ × ∇ × 3 4πe r rs(t ) ZR ZR | 0 − 0 | Replacing the last right hand side integral in Equation (30) with Equation (31) and differen- tiating the resulting equation with respect to time t yields:

qs ∂ r rs(t) ∂ qs 0 − = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (32) 3 3 4πe ∂t r rs(t) − ∂t ∇ R R e − | − | Z Z 2 1 ∂ qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 c2 t2 3 e − ∂ ZR ZR − 2 1 ∂ qs vs(t0) + 2 2 dt0 G(r, t; r0, t0)dV0 c ∂t ∇ × ∇ × 3 4πe r rs(t ) ZR ZR | 0 − 0 | In the physical setting shown in Figure2 the coordinates of the test charge q are ﬁxed, hence, the order in which the operator s and second order time derivative ∂ ∇ × ∇× ∂t2 are applied can be swapped (because operator does not affect variable t). Furthermore, ∇ because variables t0, x0, y0 and z0 are independent of time t the double time derivative can be moved under the integral sign in the last right hand side integral of the above equation:

qs ∂ r rs(t) ∂ qs 0 − = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (33) 3 3 4πe ∂t r rs(t) − ∂t ∇ R R e − | − | Z Z 2 1 ∂ qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 c2 t2 3 e − ∂ ZR ZR − 2 qs vs(t0) 1 ∂ + dt0 2 2 G(r, t; r0, t0)dV0 ∇ × ∇ × 3 4πe r rs(t ) c ∂t ZR ZR | 0 − 0 |

The second order time derivative of G(r, t; r0, t0) in the last term of Equation (33) can be replaced with Equation (15) to obtain:

qs ∂ r rs(t) ∂ qs 0 − = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (34) 3 3 4πe ∂t r rs(t) − ∂t ∇ R R e − | − | Z Z 2 1 ∂ qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 c2 t2 3 e − ∂ ZR ZR − qs vs(t0) + dt0 δ r r0 δ t t0 dV0 3 πe r r (t ) ∇ × ∇ × ZR ZR 4 0 s 0 − − | − | qs vs(t0) 2 + dt0 G(r, t; r0, t0)dV0 ∇ × ∇ × 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | Using sifting the property of Dirac’s delta function allows us to rewrite the third right hand side term of Equation (34) as:

qs vs(t0) dt0 δ r r0 δ t t0 dV0 (35) 3 πe r r (t ) ∇ × ∇ × ZR ZR 4 0 s 0 − − | − | qs vs(t0) = dt0 δ t t0 ∇ × ∇ × 4πe r rs(t ) − ZR | − 0 | q v (t) = s s ∇ × ∇ × 4πe r rs(t) | − | Mathematics 2021, 9, 237 10 of 28

To continue the derivation of Maxwell’s equations from Coulomb’s law it should be noted that operator does not affect vector v (t) because v (t) is a function of variable t. ∇ s s Hence, the application of standard vector calculus identity P = ( P) 2P ∇ × ∇ × ∇ ∇ · − ∇ yields:

q v (t) q v (t) q v (t) s s = s s s 2 s ∇ × ∇ × 4πe r rs(t) 4πe ∇ ∇ · r rs(t) − 4πe ∇ r rs(t) | − | | − | | − | qs 1 qs 2 1 = vs(t) vs(t) (36) 4πe ∇ · ∇ r rs(t) − 4πe ∇ r rs(t) | − | | − | qs ∂ 1 qs = + vs(t)δ(r rs(t)) − 4πe ∇ ∂t r rs(t) e − | − | qs ∂ r rs(t) qs = − 3 + vs(t)δ(r rs(t)) 4πe ∂t r rs(t) e − | − | Combining Equations (34)–(36), after cancellation of appropriate terms, yields:

∂ qs 0 0 = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (37) ∂t 3 e − ∇ ZR ZR − 2 1 ∂ qs 2 2 dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 − c ∂t R R3 e − q Z Z + s v (t)δ(r r (t)) e s − s qs vs(t0) 2 + dt0 G(r, t; r0, t0)dV0 ∇ × ∇ × 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | In AppendixsC and C.3, the following mathematical identity was derived:

qs vs(t0) 2 dt0 G(r, t; r0, t0)dV0 (38) 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | qs = dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 3 e − ZR ZR − By inserting Equation (38) into Equation (37) it is obtained that:

∂ qs 0 0 = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (39) ∂t 3 e − ∇ ZR ZR − 2 1 ∂ qs 2 2 dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 − c ∂t R R3 e − q Z Z + s v (t)δ(r r (t)) e s − s qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 3 e − ∇ × ∇ × ZR ZR − = 1 If now the new constant µ c2e was introduced, then by dividing whole Equation (39) by c2 it is obtained:

1 ∂ qs 0 0 = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (40) c2 ∂t 3 e − ∇ ZR ZR − 2 1 ∂ vs(t0) dt0 qsµc δ r0 rs(t0) G(r, t; r0, t0)dV0 c2 t2 3 c − ∂ ZR ZR − + q µv (t)δ(r r (t)) s s − s vs(t0) dt0 qsµc δ r0 rs(t0) G(r, t; r0, t0)dV0 3 c − ∇ × ∇ × ZR ZR − Mathematics 2021, 9, 237 11 of 28

Although it is perhaps not yet apparent, Equation (39) is Maxwell-Ampere equation given in introductory part of this paper as Equation (4). To evaluate right hand side integrals in Equation (40) one can use sifting property of Dirac’s delta function δ(r r (t ))f(r )dV = R3 0 − s 0 0 0 f(r (t )), where f (r ) is function of position vector r , to rewrite the right hand side integrals s 0 0 0 R in Equation (40) as:

qs qs dt0 δ r0 rs(t0) G(r, t; r0, t0)dV0 = G(r, t; rs(t0), t0)dt0 (41) 3 e e ZR ZR − ZR vs(t0) vs(t0) dt0 qsµc δ r0 rs(t0) G(r, t; r0, t0)dV0 = qsµc G(r, t; rs(t0), t0)dt0 (42) 3 c c ZR ZR − ZR To evaluate right hand side integrals in equations above, the Green’s function G(r, t; rs(t0), t0) in these equations is replaced with Equation (16) in order to obtain:

qs dt0 δ r0 rs(t0) G(r, t; r0, t0)dV0 (43) 3 e ZR ZR − r rs(t0) δ t t + | − | qs 0 c = − dt0 e 4π r rs(t ) ZR | − 0 | vs(t0) dt0 qsµc δ r0 rs(t0) G(r, t; r0, t0)dV0 (44) 3 c ZR ZR − r rs(t0) δ t t + | − | vs(t0) 0 − c = qsµc dt0 c 4π r rs(t ) ZR | − 0 | The right hand side integrals in Equations (43) and (44) can be evaluated by making use of the following standard mathematical identity involving Dirac’s delta function:

δ(u u ) δ( f (u)) = − 0 (45) ∂ ∂u f (u) u=u0

where f (u) is real function of real argument u, and u 0 is the solution of equation f (u0) = 0. Using identity (45), the Dirac’s delta function in Equations (43) and (44) can be written as:

r rs(t0) δ(t0 tr) δ(t0 tr) δ t0 t + | − | = − = − (46) 1 vs(t ) (r rs(t )) 1 vs(t ) (r rs(t )) − c 1 0 · − 0 1 0 · − 0 c r rs(t ) c r rs(t ) − | − 0 | − | − 0 | δ(t t ) = 0 − r 1 β (t ) n (t ) − s r · s r

where βs(tr) and ns(tr) are given by Equations (9) and (10), respectively. From Equation (45) it follows that the time tr is the solution to the following equation:

r rs(tr) t t | − | = 0 (47) − r − c

Evidently, the time tr is the time when the disturbance created by moving source charge at the position in space rs(tr) was created. This disturbance moves through the space with ﬁnite velocity c and reaches the position r of the test charge at time t. In the electromagnetic literature this time tr is commonly known as retarded time. To proceed with derivation of Maxwell equations, Equation (46) is inserted into Equations (43) and (44), and then by using the sifting property of Dirac’s delta integrals over t0 can be evaluated as:

qs 1 qs dt0 δ r0 rs(t0) G(r, t; r0, t0)dV0 = (48) 3 e πe r r (t ) ( β (t ) n (t )) ZR ZR − 4 s r 1 s r s r | − | − · vs(t0) µc qsβs(tr) dt0 qsµc δ r0 rs(t0) G(r, t; r0, t0)dV0 = (49) 3 c π r r (t ) ( β (t ) n (t )) ZR ZR − 4 s r 1 s r s r | − | − · Mathematics 2021, 9, 237 12 of 28

By inserting Equations (48) and (49) into Equation (40), and rearranging, it is obtained:

µc qsβs(tr) =qsµvs(t)δ(r rs(t)) (50) ∇ × ∇ × 4π r rs(tr) (1 β (tr) ns(tr)) − | − | − s · 1 ∂ 1 qs 2 − c ∂t ∇ 4πe r rs(tr) (1 β (tr) ns(tr)) | − | − s · 2 1 ∂ µc qsβs(tr) 2 2 − c ∂t 4π r rs(tr) (1 β (tr) ns(tr)) | − | − s · The ﬁrst right hand side term of the equation above can be identiﬁed as the current J of the point charge distribution moving with velocity vs(t) multiplied by constant µ:

µJ = µq v (t)δ(r r (t)) (51) s s − s To proceed, scalar function φ(r, t) and vector function A(r, t) are deﬁned:

1 q φ(r, t) = s (52) 4πe r rs(tr) (1 β (tr) ns(tr)) | − | − s · µc qsβ (tr) A(r, t) = s (53) 4π r rs(tr) (1 β (tr) ns(tr)) | − | − s · With the aid of scalar function φ(r, t), vector function A(r, t), and expression µJ given by Equation (51) the Equation (50) can be written as:

1 ∂ ∂ A(r, t) = µJ φ(r, t) A(r, t) (54) ∇ × ∇ × − c2 ∂t −∇ − ∂t To further simplify Equation (54) two additional vector functions B and E are deﬁned:

B = A(r, t) (55) ∇ × ∂ E = φ(r, t) Q(r, t) (56) −∇ − ∂t Using deﬁnitions of vector functions B and E, given by Equations (55) and (56), Equation (54) can be rewritten as:

1 ∂E B = µJ + (57) ∇ × c2 ∂t The mathematical properties of vector fields B and E shall now be investigated. Note that because for any differentiable vector field P we can write P = 0, from ∇ · ∇ × Equation (55) it follows that: B = 0 (58) ∇ · The curl of the gradient of any differentiable scalar function ψ is equal to zero, i.e., ∇ × ψ = 0. Thus, taking the curl of Equation (56) yields: ∇ ∂ ∂ ∂B E = A(r, t) = A(r, t) = (59) ∇ × −∇ × ∂t − ∂t ∇ × − ∂t Finally, in AppendixC and C.4, it was shown that the divergence of vector field E is:

q ρ(r, t) E = s δ(r r (t)) = (60) ∇ · e − s e which completes the derivation of electrodynamic equations from Coulomb’s law. From tables Tables1 and2 it is clear that potentials and differential equations derived directly from Coulomb’s law are identical to potentials and differential equations given in the introduction, i.e., to Equations (1)–(10). Mathematics 2021, 9, 237 13 of 28

Thus, it should be evident by now that we have derived Maxwell equations and Liénard–Wiechert potentials directly from Coulomb’s law. This was achieved by mathe- matically relating known electrostatic Coulomb’s law acting on test charge at present time to “unknown” electrodynamic ﬁelds acting at the past. The mathematical link between the static case in the present and dynamic case in the past was provided by generalized Helmholtz theorem. The derived equations are valid for arbitrarily moving source charge, and these equations are not conﬁned to motions along straight line. Furthermore, it should be noted that the Maxwell equations and Liénard–Wiechert potentials were derived directly from Coulomb’s law without resorting to special relativity or Lorentz transformation.

Table 1. Potentials and vector ﬁelds derived from Coulomb’s law.

Symbol Equation Description 1 qs φ(r, t) scalar potential derived from Coulomb’s law 4πe r rs(tr) (1 β (tr) ns(tr)) | − | − s · µc qsβ (tr) A(r, t) s vector potential derived from Coulomb’s law 4π r rs(tr) (1 β (tr) ns(tr)) | − | − s · B A(r, t) vector ﬁeld B derived from Coulomb’s law ∇ × E φ(r, t) ∂ A(r, t) vector ﬁeld E derived from Coulomb’s law −∇ − ∂t

Table 2. Maxwell equations for electromagnetic ﬁelds E and B compared to differential equations governing the vector ﬁelds B and E derived from Coulomb’s law.

Maxwell Equation Description Equations Derived from Coulomb’s Law 1 ∂E 1 ∂E B = µJ + Maxwell-Ampere equation B = µJ + 2 ∇ × c2 ∂t ∇ × c ∂t E = ∂B Faraday’s law E = ∂B ∇ × − ∂t ∇ × − ∂t E = ρ Gauss’ law for electric ﬁeld E = ρ ∇ · e ∇ · e B = 0 Gauss’ law for magnetic ﬁeld B = 0 ∇ · ∇ ·

4. Derivation of Electrodynamic Energy Conservation Law and Lorentz Force To derive the electrodynamic energy conservation law from Coulomb’s law, we first consider the hypothetical physical setting shown in Figure3 where the source charge qs is moving along arbitrary trajectory rs(t). Then the source charge qs stops at some time in the past ts. In this physical setting, closed contour C is at rest at all times. At present time t > t all the points inside the sphere of radius R = c(t t ) are affected only by p s p − s electrostatic Coulomb’s field. The known energy conservation law valid at present dictates that contour integral of electrostatic field along any closed contour immersed inside the sphere of radius R equals zero:

qs r rs(ts) Ec(r, tp) dr = − 3 dr = 0 (61) C · C 4πe r rs(ts) · I I | − | where Ec is Coulomb’s electrostatic ﬁeld, rs(ts) is the position vector of source charge when it stopped moving, and vector r is the position vector of the point on contour C. This electrostatic energy conservation law, valid at present time tp, states that no net work is done in transporting the unit charge along any closed contour immersed in the electrostatic ﬁeld. Mathematics 2021, 9, 237 14 of 28

CLOSED CONTOUR C

C SPHERE OF MOVING SOURCE CHARGE qs RADIUS R r STOPS AT TIME ts x

r s(ts) qs c(tp ts) R = −

Figure 3. Source charge qs is moving along arbitrary trajectory rs(t) and it stops at past time ts. Closed contour C is at rest at all times. All the points r on contour C are inside the sphere of radius R = c(t t ). At present time t > t all the points on contour C are affected only by Coulomb’s p − s p s electrostatic ﬁeld.

To proceed, it is assumed that in the past, when the source charge was moving, the energy conservation law is unknown. However, the generalized Helmholtz decomposition theorem allows us to derive this "unknown" electrodynamic energy conservation law valid in the past from the knowledge of electrostatic energy conservation law valid at present. To derive this unknown electrodynamic conservation law we consider the contour integral (61) at the moment t inﬁnitesimally before the time when the source charge stopped:

t = t δt (62) s − where δt is inﬁnitesimally small time interval. If time interval δt approaches zero (δt 0) → the contour integral (61) can be rewritten as the function of time t:

qs r rs(t) Ec(r, tp δt) dr = − 3 dr = 0 (63) C − · C 4πe r rs(t) · I I | − | Because the integrand on the right hand side of Equation (63) is the function of varying time t and position vector r the generalized Helmholtz decomposition theorem can be applied to rewrite this integrand as the function of past positions and velocities of the source charge. In fact, such expression is already derived in previous section as Equation (33), repeated here for clarity:

qs r rs(t) qs 0 − = dt0 δ r rs(t0) G(r, t; r0, t0)dV0 (64) πe 3 3 e 4 r rs(t) − ∇ ZR ZR − | − | 1 ∂ qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 c2 ∂t 3 e − ZR ZR − 1 ∂ qs vs(t0) + 2 dt0 0 0 G(r, t; r0, t0)dV0 c ∂t 3 4πe ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | Mathematics 2021, 9, 237 15 of 28

Substituting the ﬁrst two right hand side terms of Equation (64) with Equations (48) and (49) and combining the result with Equations (52) and (53), and using c2 = 1/µe yields:

qs r rs(t) ∂ − 3 = φ(r, t) A(r, t) + K(r, t) (65) 4πe r rs(t) −∇ − ∂t | − | where vector function K(r, t) is equal to the last right hand side term of Equation (64):

1 ∂ qs vs(t0) K(r, t) = 2 dt0 0 0 G(r, t; r0, t0)dV0 (66) c ∂t 3 4πe ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | Replacing the first two terms on the right hand side of Equation (65) with Equation (56) yields:

qs r rs(t) − 3 = E(r, t) + K(r, t) (67) 4πe r rs(t) | − | Then, by inserting Equation (67) into the right hand side of Equation (63) it is obtained that:

0 = Ec(r, tp δt) dr = (E(r, t) + K(r, t)) dr (68) IC − · IC · The space-time integral on the right hand side of Equation (66) is very difﬁcult to evaluate. However, vector ﬁeld K(r, t) can be eliminated from the right hand side of Equation (68) by the application of Stokes’ theorem:

0 = Ec(r, tp δt) dr = E(r, t) dr + K(r, t) dS (69) IC − · IC · ZS ∇ × · From here, by taking the curl of both sides of Equation (67) and by combining with Equation (59) it is obtained that:

∂ K(r, t) = E(r, t) = B(r, t) (70) ∇ × −∇ × ∂t

∂ Because surface S and contour C are stationary it can be written that ∂t B(r, t) = d dt B(r, t). Inserting Equation (70) into Equation (69) and taking into account that surface S and contour C are not moving yields:

d 0 = E (r, t δt) dr = E(r, t) dr + B(r, t) dS (71) c p dt IC − · IC · ZS · The right hand side of Equation (71) is unknown energy conservation principle valid for varying in time dynamic fields E(r, t) and M(r, t) and it is derived from electrostatic energy conservation principle valid at present time. Clearly, we have just obtained the physical law known in electrodynamics as Faraday’s law. From Equation (71) the conclusion can be drawn about the nature of Faraday’s law. It represents the energy conservation principle valid for non-conservative dynamic fields, and it is the dynamic equivalent of the electrostatic energy conservation principle valid for Coulomb’s electrostatic field. However, even Faraday’s law itself can be considered to be a consequence of some- thing else. To see this, consider simply connected volume V bounded by surface ∂V as shown in Figure4. The surface ∂V is the union of two surfaces S and S1 bounded by respective contours C and C1. Contours C and C1 consist of exactly the same spatial points; however, the Stokes’ orientation of these contours is opposite C = C . Then, using − 1 E(r, t) = ∂ B(r, t) the first right hand side contour integral of Equation (71) can be ∇ × − ∂t written as:

d E(r, t) dr = E(r, t) dr = B(r, t) dS = B(r, t) dS (72) dt IC · − IC1 · − ZS1 ∇ × · ZS1 · Mathematics 2021, 9, 237 16 of 28

Replacing the ﬁrst right hand side term of Equation (71) with Equation (72) yields different form of dynamic energy conservation law:

d 0 = E (r, t δt) dr = B(r, t) dS (73) c p dt IC − · I∂V · Evidently, the right hand side of Equation (73) is the time derivative of Gauss’ law for magnetic fields. The standard interpretation of Gauss’ law for magnetic fields is that magnetic monopoles do not exist. However, from Equation (73) it can be concluded that an alternative interpretation of this law is that its time derivative represents the dynamic energy conservation law. From the derivations presented, it might be even said that Faraday’s law is the consequence of Gauss’ law for magnetic fields. It should be noted that these energy-conservation equations were all derived from simple electrostatic Coulomb’s law. From dynamic energy conservation law, the derivation of Lorentz force is straightfor- ward: it is now assumed that all the points on surface ∂V shown in Figure4 have some definite velocity v such that v << c. Then the surface ∂V is the function of time, hence, | | C = C(t) and S = S(t). Hence, Equation (73) can be written as the sum of two surface integrals over surfaces S(t) and S1(t): d d d B(r, t) dS = B(r, t) dS + B(r, t) dS = 0 (74) dt dt dt I∂V · IS(t) · IS1(t) · where ∂V = S(t) S (t). The Leibniz identity [25] for moving surfaces state that for any ∪ 1 differentiable vector field P it can be written:

d ∂ P dS = P + ( P)v dS v P dr (75) dt S(t) · S(t) ∂t ∇ · · − C(t) × · Z Z I Applying the Leibniz identity to the surface integral over surface S1(t) in Equation (74) and using B(r, t) = 0 yields: ∇ · d ∂ B(r, t) dS + B(r, t) dS v B(r, t) dr = 0 (76) dt ∂t IS(t) · IS1(t) · − IC1(t) × · Using the result from previous section, i.e. E(r, t) = ∂ B(r, t), and applying the ∇ × − ∂t Stokes’ theorem yields:

d B(r, t) dS E(r, t) dr v B(r, t) dr = 0 (77) dt IS(t) · − IC1(t) · − IC1(t) × · Note that curves C1(t) and C(t) comprise of the same points; however, Stokes’ orientation of curves C (t) and C(t) is opposite, i.e., C (t) = C(t). Hence, Equation (77) can be 1 1 − written as:

d [E(r, t) + v B(r, t)] dr + B(r, t) dS = 0 (78) dt IC(t) × · ZS(t) · Note that Equation (78) could not be derived from the right hand side of Equation (71), i.e., from Faraday’s law, even with Leibniz rule. For that reason, it might be considered that the energy conservation law on the right hand side of Equation (73) is perhaps more general than the one given by Equation (71). Mathematics 2021, 9, 237 17 of 28

∂V = S S ∪ 1 = S U

C C S1

Figure 4. Closed surface ∂V that bounds volume V is union of two surfaces S and S1. Contour C bounds surface S and contour C1 bounds surface S1. Contours C and C1 are identical; however they have different Stokes’ orientation.

Furthermore, note that the time derivative of the surface integral in Equation (78) does not represent the work of any force. However, from Equation (73), it is evident that the terms in Equation (78) have dimensions of the work done by the electrodynamic force in moving the unit charge along contour C(t). Because the ﬁrst term in Equation (78) is contour integral of the the vector ﬁeld, it can be concluded that this term represents the non-zero work done by the non-conservative electrodynamic force in transporting the unit charge along contour C(t). Hence, just as the left hand side of Equation (73) represents the work done by the conservative electrostatic force in transporting the unit charge along contour C, the contour integral on the left of Equation (78) represents the work done by the non-conservative electrodynamic force in transporting the unit charge along the same contour. The purpose of the surface integral on the left hand side of Equation (78) is to balance non-zero work of non-conservative electrodynamic force along contour C. Thus, it can be concluded that the electrodynamic force FD on charge q moving with velocity v along contour C is:

F = qE(r, t) + qv B(r, t) (79) D × which is an expression for well known Lorentz force. It was derived theoretically from the knowledge of electrostatic energy conservation law, which in turn, can be derived from Coulomb’s law. Thus, it might be said that we have just derived the Lorentz force from simple electrostatic Coulomb’s law.

5. Conclusions In this paper, the theoretical framework that explains Maxwell equations and the Lorentz force is presented on a more fundamental level than it was previously done. Maxwell equations and the Lorentz force were derived in this paper directly from Coulomb’s law. In contrast, Maxwell derived Maxwell equations from experimental Ampere’s force law and experimental Faraday’s law, and Lorentz continued work on Maxwell’s theory to discover the Lorentz force. Except for the theory presented in this paper, in the last 150 years, no successful theory was presented that derives Maxwell’s equations and Lorentz force from more fundamental laws than Ampere’s force law and Faraday’s law. In contrast to frequently criticized previous attempts to derive Maxwell’s equations from Coulomb’s law using special relativity and Lorentz transformation, the Lorentz transformation was not used in derivations presented in this paper nor the theory of special relativity. In fact, it was shown that Einstein’s theory of special relativity is completely unnecessary to derive Maxwell’s equations and Lorentz force. Effectively, this shows that Maxwell’s equations, magnetism, and Lorentz force are not the consequence of special relativity. Rather than from special relativity, in this work, dynamic Liénard–Wiechert potentials, Maxwell equations, and Lorentz force were derived from Coulomb’s law using the following two simple postulates: Mathematics 2021, 9, 237 18 of 28

(a) when charges are at rest the Coulomb’s law describes the force acting between charges (b) disturbances caused by moving charges propagate outwardly from moving charge with ﬁnite velocity Then with the aid of generalized Helmholtz decomposition theorem, also derived in this paper, Liénard–Wiechert potentials, Maxwell equations, and Lorentz force were derived. In contrast to previous attempts to derive Maxwell equations from special relativity and Coulomb’s law, which can only be considered valid for motion along straight line, the Maxwell equations and Lorentz force derived from these two postulates are valid for charges in arbitrary motion. Thus, in effect, it was shown that magnetism, Maxwell equations, and Lorentz force are not the consequence of Einstein’s special relativity. The only case possible when Maxwell’s equations, magnetism, and Lorentz force could be considered to be the consequence of special relativity is the special case of the charge moving along straight line. On the other hand, the theory presented in this paper is valid for arbitrary motion, and it is the consequence of time retardation. Hence, electromagnetism as a whole should be considered the consequence of time retardation and not of special relativity.

Funding: The APC was funded by University of Split. Data Availability Statement: Data sharing not applicable. Conﬂicts of Interest: The author declare no conﬂict of interest. Sample Availability: Data is available from corresponding author upon reasonable request.

Appendix A. Derivation of Generalized Helmholtz Decomposition Theorem In this appendix, the generalized Helmholtz decomposition theorem for vector functions of space and time is derived. However, in an effort to enhance the readability of this work, some basic identities presented in Appendix A.1 are derived ﬁrst.

Appendix A.1. Preliminary Considerations

To clarify notation used throughout this paper, note that position vectors r and r0 are deﬁned in this paper as:

r = xˆx + yˆy + zˆz (A1)

r0 = x0 ˆx + y0 ˆy + z0 ˆz (A2)

where ˆx, ˆy and ˆz are Cartesian, mutually orthogonal, unit basis vectors. Variables x, y, z R ∈ and x0, y0, z0 R are linearly independent variables. Furthermore, note that throughout this ∈ paper the position vector rs(t0) indicates the position of the source charge. This position vector rs(t0) is deﬁned as:

rs(t0) = xs(t0)ˆx + ys(t0)ˆy + zs(t0)ˆz (A3)

where xs(t0), ys(t0) and zs(t0) are all functions of real variable t0 R which is independent ∈ of variables x, y, z R and x0, y0, z0 R. The time derivative of position vector rs(t0) is ∈ ∈ velocity vs(t0) of the source charge:

∂rs(t0) vs(t0) = (A4) ∂t0 Mathematics 2021, 9, 237 19 of 28

On many occasions in this paper, the differential operators and were used. These ∇ ∇0 operators are deﬁned as:

∂ ∂ ∂ = ˆx + ˆy + ˆz (A5) ∇ ∂x ∂x ∂x ∂ ∂ ∂ 0 = ˆx + ˆy + ˆz (A6) ∇ ∂x0 ∂y0 ∂z0 Operator acts only on functions of variables x, y, z, hence, on functions of position vector ∇ r. On the other hand, operator acts only on functions of variables x , y , z , thus, it acts ∇0 0 0 0 on functions of position vector r0. For example, if function f is the function of position vector r, i.e., if f = f (r), one can generally write:

f (r) = 0 0 f (r) = 0 (A7) ∇ 6 ∇

On the other hand, if function f is the function of position vector r0, i.e., if f = f (r0) it can be written: f (r0) = 0 0 f (r0) = 0 (A8) ∇ ∇ 6 Furthermore, because variable t0 is independent of variables x, y, z and x0, y0, z0 neither operator nor acts on position vector r (t ) and velocity vector v (t ). Using these ∇ ∇0 s 0 s 0 considerations it is evident that the following equations are correct:

r (t ) = 0 v (t ) = 0 ∇ · s 0 ∇ · s 0 (A9) r (t ) = 0 v (t ) = 0 ∇0 · s 0 ∇0 · s 0 However, both operators and act on Green’s function G(r, t; r , t ) given by Equation (16). ∇ ∇0 0 0 In fact, one can easily verify that the following equations hold:

G(r, t; r , t ) = G(r, t; r , t ) 2G(r, t; r , t ) = 2G(r, t; r , t ) ∇ 0 0 −∇0 0 0 ∇ 0 0 ∇0 0 0 ∂ ∂ ∂2 ∂2 (A10) G(r, t; r0, t0) = G(r, t; r0, t0) 2 G(r, t; r0, t0) = 2 G(r, t; r0, t0) ∂t − ∂t0 ∂t ∂t0 Appendix A.2. Generalized Helmholtz Decomposition Theorem To start deriving generalized Helmholtz decomposition theorem for vector functions of space and time, consider the inhomogeneous transient wave equation:

2 2 1 ∂ G(r, t; r0, t0) G(r, t; r0, t0) = δ(r r0)δ(t t0) (A11) ∇ − c2 ∂t2 − − −

where G(r, t; r0, t0) is the function called fundamental solution or Green’s function and δ is Dirac’s delta function. The Green’s function for inhomogeneous wave equation is well known and it represents an outgoing diverging spherical wave:

r r0 δ t t0 | −c | G(r, t; r0, t0) = − − (A12) 4π r r | − 0| Let us now suppose that vector ﬁeld F is the function of both space r and time t, i.e., F = F(r, t). Using sifting property of Dirac delta function the vector function F(r, t) can be written as the volume integral over inﬁnite volume R3 and over all the time R as:

F(r, t) = dt0 F(r0, t0)δ(r r0)δ(t t0)dV0 (A13) 3 ZR ZR − − Mathematics 2021, 9, 237 20 of 28

where differential volume element dV is dV = dx dy dz . By replacing δ(r r )δ(t t ) in 0 0 0 0 0 − 0 − 0 equation above with left hand side of Equation (A11) it is obtained:

2 2 1 ∂ F(r, t) = dt0 F(r0, t0) G(r, t; r0, t0) G(r, t; r0, t0) dV0 (A14) − 3 ∇ − c2 ∂t2 ZR ZR From the discussion presented in Appendix A.1, it is evident that the D’Alambert 2 2 1 ∂ x y z t operator c2 ∂t2 does not act on variables 0, 0, 0 and 0 nor does it act on vector function ∇ − 2 F(r , t ). Hence, D’Alambert operator 2 1 ∂ can be written in front of the integral: 0 0 ∇ − c2 ∂t2 2 2 1 ∂ F(r, t) = dt0 F(r0, t0)G(r, t; r0, t0)dV0 (A15) − ∇ − c2 ∂t2 3 ZR ZR Using standard vector calculus identity P = ( P) 2P Equation (A15) can ∇ × ∇ × ∇ ∇ · − ∇ be rewritten as:

F(r, t) = dt0 F(r0, t0)G(r, t; r0, t0)dV0 (A16) 3 ∇ × ∇ × ZR ZR dt0 F(r0, t0)G(r, t; r0, t0)dV0 − ∇ ∇ · 3 ZR ZR 1 ∂2 + dt0 F(r0, t0)G(r, t; r0, t0)dV0 c2 t2 3 ∂ ZR ZR ∂ Because operators and ∂t do not act on variables x0, y0, z0 and t0 the operator and ∂ ∇ ∇ partial derivative ∂t can be moved under the right hand side integrals in Equation (A16). Then using standard vector calculus identities (ψP) = ψ P + ψ P and ∇ × ∇ × ∇ × ∇ · (ψP) = ψ P + ψ P, and noting that F(r , t ) = 0 and F(r , t ) = 0 Equation (A16) ∇ · ∇ · ∇ × 0 0 ∇ · 0 0 can be rewritten as:

F(r, t) = dt0 G(r, t; r0, t0) F(r0, t0)dV0 (A17) 3 ∇ × ZR ZR ∇ × dt0 F(r0, t0) G(r, t; r0, t0)dV0 3 − ∇ ZR ZR · ∇ 1 ∂ ∂ + dt0 F(r0, t0) G(r, t; r0, t0)dV0 c2 ∂t 3 ∂t ZR ZR ∂ ∂ One can now use identities G(r, t; r0, t0) = 0G(r, t; r0, t0) and G(r, t; r0, t0) = G(r, t; r0, t0) ∇ −∇ ∂t − ∂t0 to rewrite the right hand side integrals in Equation (A17) as:

F(r, t) = dt0 0G(r, t; r0, t0) F(r0, t0)dV0 (A18) 3 − ∇ × ZR ZR ∇ × + dt0 F(r0, t0) 0G(r, t; r0, t0)dV0 3 ∇ ZR ZR · ∇ 1 ∂ ∂ dt0 F(r0, t0) G(r, t; r0, t0)dV0 c2 ∂t 3 ∂t − ZR ZR 0 Using vector calculus identity (ψP) = ψ P + ψ P and the the form of diver- ∇ × ∇ × ∇ × gence theorem PdV = P dS the ﬁrst right hand side integral over R3 can be V ∇ × ∂V × written as: R H

0G(r, t; r0, t0) F(r0, t0)dV0 = G(r, t; r0, t0)F(r0, t0) dS0 (A19) 3 3 ZR ∇ × I∂R × G(r, t; r0, t0) 0 F(r0, t0)dV0 3 − ZR ∇ × Note that the surface ∂R3 is an inﬁnite surface that bounds an inﬁnite volume R3. Furthermore, for the surface integral in the equation above, position vector r0 is located on Mathematics 2021, 9, 237 21 of 28

3 3 inﬁnite surface ∂R , i.e., r0 ∂R . Hence, if vector function F(r0, t0) decreases faster than ∈ 1/ r r as r r ∞ the surface integral in Equation (A19) vanishes. In that case, it | − 0| | − 0| → can be written:

0G(r, t; r0, t0) F(r0, t0)dV0 = G(r, t; r0, t0) 0 F(r0, t0)dV0 (A20) 3 3 ZR ∇ × − ZR ∇ × Using similar considerations, vector calculus identity (ψP) = ψ P + ψ P and ∇ · ∇ · ∇ · standard divergence theorem PdV = P dS it is obtained that: V ∇ · ∂V · R H F(r0, t0) 0G(r, t; r0, t0)dV0 = G(r, t; r, t0) 0 F(r0, t0)dV0 (A21) 3 3 ZR · ∇ − ZR ∇ · To treat the last integral on the right hand side of Equation (A18) the following identity can be used: ∂ ∂ ∂ F(r0, t0) G(r, t; r, t0) = F(r0, t0)G(r, t; r, t0) G(r, t; r, t0) F(r0, t0) (A22) ∂t0 ∂t0 − ∂t0 Using the identity above and noting that t0 is independent of x0, y0 and z0 the last right hand side integral of Equation (A18) can be written as:

∂ ∂ dt0 F(r0, t0) G(r, t; r0, t0)dV0 = dt0 F(r0, t0)G(r, t; r0, t0)dV0 (A23) 3 ∂t ∂t 3 ZR ZR 0 ZR 0 ZR ∂ dt0 G(r, t; r0, t0) F(r0, t0)dV0 3 ∂t − ZR ZR 0

By integrating over t0, it can be shown that the first right hand side integral in Equation (A23) vanishes:

t ∞ ∂ 0→ dt0 F(r0, t0)G(r, t; r0, t0)dV0 = F(r0, t0)G(r, t; r0, t0)dV0 (A24) 3 3 R ∂t0 R R t ∞ Z Z Z 0→− If t ∞, and if t is ﬁnite, then from Equation (A12) follows that G(r, t; r , t ) = 0, 0 → ± 0 0 thus, the right hand side of Equation (A24) is equal to zero. Inserting this result into Equation (A23) yields:

∂ ∂ dt0 F(r0, t0) G(r, t; r0, t0)dV0 = dt0 G(r, t; r0, t0) F(r0, t0)dV0 (A25) 3 ∂t 3 ∂t ZR ZR 0 − ZR ZR 0 By inserting Equations (A20), (A21) and (A25) into Equation (A18) the generalized Helmholtz theorem for vector functions of space and time is obtained:

F(r, t) = dt0 0 F(r0, t0) G(r, t; r0, t0)dV0 (A26) − ∇ 3 ∇ · ZR ZR 1 ∂ ∂ + dt0 F(r0, t0) G(r, t; r0, t0)dV0 c2 ∂t 3 ∂t ZR ZR 0

+ dt0 0 F(r0, t0) G(r, t; r0, t0)dV0 ∇ × 3 ∇ × ZR ZR The theorem is valid for functions F(r , t ) that decrease faster than 1/ r r as r r ∞. 0 0 | − 0| | − 0| → Appendix B. Novel Vector Calculus Identities In this appendix, two novel vector calculus identities are proven, without which it would be very difﬁcult, perhaps even not possible, to derive Maxwell’s equations from Mathematics 2021, 9, 237 22 of 28

Coulomb’s law. These two novel vector calculus identities are given by the following two equations:

ψ 2PdV = ψ(dS )P ( ψ )PdV (A27) ZV ∇ I∂V · ∇ − ZV ∇ · ∇ P 2ψdV = P( ψ dS) ( ψ )PdV (A28) ZV ∇ I∂V ∇ · − ZV ∇ · ∇ where P is differentiable vector ﬁeld, ψ is differentiable scalar function, volume V R3 is ⊂ simply connected volume, ∂V is the bounding surface of volume V and dS is differential surface element of ∂V such that dS = ndS. Vector n is an outward unit normal to the surface ∂V. In Cartesian coordinate system, the product ψ 2P can be written in terms of ∇ Cartesian components as:

ψ 2P = xˆψ 2P + yˆψ 2P + zˆψ 2P (A29) ∇ ∇ x ∇ y ∇ z

where Px, Py and Pz are Cartesian components of vector P and vectors ˆx, ˆy and ˆz are Cartesian unit basis vectors. Using standard vector calculus identity f T = f T + ∇ · ∇ · f T, valid for some scalar function f and for some vector function T, Equation (A29) can ∇ · be rewritten as:

ψ 2P = xˆ (ψ P ) xˆ ψ P (A30) ∇ ∇ · ∇ x − ∇ · ∇ x + yˆ ψ P yˆ ψ P ∇ · ∇ y − ∇ · ∇ y + zˆ (ψ P ) zˆ ψ P ∇ · ∇ z − ∇ · ∇ z To proceed, the identity ( ψ )P is expanded in terms of its Cartesian components as: ∇ · ∇ ∂ψ ∂ ∂ψ ∂ ∂ψ ∂ ( ψ )P = + + ˆxP + ˆyP + ˆzP (A31) ∇ · ∇ ∂x ∂x ∂y ∂y ∂z ∂z x y z = ˆx ψ P + ˆy ψ P + ˆz ψ P ∇ · ∇ x ∇ · ∇ y ∇ · ∇ z Inserting Equation (A31) into (A30) it is obtained that:

ψ 2P = xˆ (ψ P ) + yˆ ψ P + zˆ (ψ P ) ( ψ )P (A32) ∇ ∇ · ∇ x ∇ · ∇ y ∇ · ∇ z − ∇ · ∇ By integrating Equation (A32) over volume V and by application of the divergence theorem TdV = T dS it is obtained: V ∇ · ∂V ·

R 2 H ψ PdV =xˆ ψ Px dS + yˆ ψ Py dS + zˆ ψ Pz dS (A33) ZV ∇ I∂V ∇ · I∂V ∇ · I∂V ∇ · ( ψ )PdV − ZV ∇ · ∇ The ﬁrst three right hand side terms of Equation (A33) can be rewritten as:

xˆ ψ Px dS + yˆ ψ Py dS + zˆ ψ Pz dS = ψ(dS )P (A34) I∂V ∇ · I∂V ∇ · I∂V ∇ · I∂V · ∇ Inserting Equation (A34) into (A33) yields:

ψ 2PdV = ψ(dS )P ( ψ )PdV (A35) ZV ∇ I∂V · ∇ − ZV ∇ · ∇ which was intended to be proven. To prove Equation (A28) the expression P 2ψ is written ∇ in terms of its Cartesian components as:

P 2ψ = ˆxP 2ψ + ˆyP 2ψ + ˆzP 2ψ (A36) ∇ x∇ y∇ z∇ Mathematics 2021, 9, 237 23 of 28

By using standard differential calculus identity f 2h = ( f h) f h, where ∇ ∇ · ∇ − ∇ · ∇ f and h are differentiable scalar functions, Equation (A36) can be written as:

P 2ψ =ˆx (P ψ) ˆx P ψ+ (A37) ∇ ∇ · x∇ − ∇ x · ∇ ˆy P ψ ˆy P ψ+ ∇ · y∇ − ∇ y · ∇ ˆz (P ψ) ˆz P ψ ∇ · z∇ − ∇ z · ∇ Inserting Equation (A31) into Equation (A37) yields:

P 2ψ = ˆx (P ψ) + ˆy P ψ + ˆz (P ψ) ( ψ )P (A38) ∇ ∇ · x∇ ∇ · y∇ ∇ · z∇ − ∇ · ∇ Integrating Equation (A38) over volume Vand applying divergence theorem TdV = V ∇ · T dS it is obtained that: ∂V · R H P 2ψdV = P( ψ dS) ( ψ )PdV (A39) ZV ∇ I∂V ∇ · − ZV ∇ · ∇ which was intended to be proven.

Appendix C. Derivation of Auxiliary Mathematical Identities In this appendix some auxiliary mathematical identities are derived which are helpful for the derivation of Maxwell equations from Coulomb’s law.

Appendix C.1. Derivation of Equation (28) Equation (24) allows us to rewrite the time derivative in the second right hand side integral in Equation (27) as:

∂ qs r0 rs(t0) qs ∂ 1 − 3 = 0 (A40) ∂t0 4πe r rs(t ) − 4πe ∂t0 ∇ r0 rs(t0) | 0 − 0 | | − |

Because coordinates x0, y0 and z0 are independent of time t0 the order in which operators ∂ 0 and are applied can be swapped: ∇ ∂t0

∂ qs r0 rs(t0) qs ∂ 1 − 3 = 0 (A41) ∂t0 4πe r rs(t ) − 4πe ∇ ∂t0 r0 rs(t0) | 0 − 0 | | − |

Furthermore, because coordinates x0, y0 and z0 are independent of time t0, the time ∂r derivative of r0 is equal to zero 0 = 0. The inner time derivative in Equation (A41) can ∂t0 now be written as:

∂ 1 vs(t0) (r0 rs(t0)) vs(t0) = · − 3 = 0 (A42) ∂t0 r0 rs(t0) r rs(t ) −∇ · r0 rs(t0) | − | | 0 − 0 | | − |

where vs(t0) is the velocity of the source charge qs at time t0 given by Equation (A4). Inserting Equation (A42) into Equation (A41) yields:

∂ qs r0 rs(t0) qs vs(t0) − 3 = 0 0 (A43) ∂t0 4πe r rs(t ) 4πe ∇ ∇ · r0 rs(t0) | 0 − 0 | | − | To proceed with derivation, the standard vector calculus identity P = ∇ × ∇ × ( P) 2P, is used. This identity allows us to rewrite the Equation (A43) as: ∇ ∇ · − ∇

∂ qs r0 rs(t0) qs vs(t0) qs 2 vs(t0) − 3 = 0 0 + 0 (A44) ∂t0 4πe r rs(t ) 4πe ∇ × ∇ × r0 rs(t0) 4πe ∇ r0 rs(t0) | 0 − 0 | | − | | − | Mathematics 2021, 9, 237 24 of 28

Since Laplacian operator 2 does not have effect on velocity vector v (t ) the last right ∇0 s 0 hand side term in Equation (A44) can be written using 3D Dirac’s delta function as:

qs 2 vs(t0) qs 2 1 1 qs 0 = vs(t0) 0 = vs(t0)δ r0 rs(t0) (A45) 4πe ∇ r0 rs(t0) e ∇ 4π r0 rs(t0) − e − | − | | − | Hence, replacing the last right hand side term of Equation (A44) with Equation (A45) yields:

∂ qs r0 rs(t0) qs vs(t0) qs − = 0 0 v (t0)δ r0 r (t0) (A46) ∂t 4πe 3 4πe r r (t ) e s s 0 r0 rs(t0) ∇ × ∇ × 0 s 0 − − | − | | − | which proves Equation (28).

Appendix C.2. Derivation of Equation (31) To derive Equation (31) we make use of standard vector calculus identity (ψP) = ∇ × ψ P + ψ P, where ψ is a scalar function and P is a vector function, to rewrite the ∇ × ∇ × integrand in the last right hand side term of Equation (30) as:

vs(t0) 0 0 G(r, t; r0, t0) (A47) ∇ × ∇ × r rs(t ) | 0 − 0 | vs(t0) vs(t0) = 0 0 G(r, t; r0, t0) 0G(r, t; r0, t0) 0 ∇ × ∇ × r rs(t ) − ∇ × ∇ × r rs(t ) | 0 − 0 | | 0 − 0 |

Integrating Equation (A47) with respect to variables x0, y0, z0 and t0, and making use of a standard form of divergence theorem PdV = dS P it is obtained that: V ∇ × ∂V × R H vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (A48) 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | vs(t0) = dt0 dS0 0 G(r, t; r0, t0) ∂ 3 × ∇ × r rs(t ) ZR I R | 0 − 0 | vs(t0) dt0 0G(r, t; r0, t0) 0 dV0 − 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | 3 3 where dV = dx0dy0dz0, ∂R is an infinite surface that bounds R , and dS is differential surface element of the surface ∂R3. Because ∂R3 is an infinite surface, the first right hand side integral vanishes. To see this, one can use standard vector identity (ψP) = ∇ × ψ P + ψ P and using v (t ) = 0 to rewrite the first term in the first right hand ∇ × ∇ × ∇0 × s 0 side integrand as: vs(t0) r0 rs(t0) 0 = − 3 vs(t0) (A49) ∇ × r0 rs(t0) − r rs(t ) × | − | | 0 − 0 | 3 0 Because r0 ∂R this means that r ∞. Provided that charge qs is moving with ∈ | | → finite velocity vs(t0) it is clear that right hand side term of Equation (A49) vanishes as r0 ∞. Thus, Equation (A48) can be written as: | | →

vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (A50) 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | vs(t0) = dt0 0G(r, t; r0, t0) 0 dV0 − 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 |

There is another useful property of Green’s function G(r, t; r0, t0) which enables us to proceed with the derivation of Equation (31). This property can be written as follows:

0G(r, t; r0, t0) = G(r, t; r0, t0) (A51) ∇ −∇ Mathematics 2021, 9, 237 25 of 28

Using this property and standard vector calculus identity (ψP) = ψ P + ψ P ∇ × ∇ × ∇ × allows us to rewrite the integrand in Equation (A48) as:

vs(t0) 0G(r, t; r0, t0) 0 (A52) ∇ × ∇ × r rs(t ) | 0 − 0 | vs(t0) = G(r, t; r0, t0) 0 −∇ × ∇ × r rs(t ) | 0 − 0 | vs(t0) = G(r, t; r0, t0) 0 −∇ × ∇ × r rs(t ) | 0 − 0 | Equation above is valid because operator does not act on velocity vector v (t ), nor ∇ s 0 does it act on position vectors r0 and rs(t0). It only acts on Green’s function G(r, t; r0, t0) because it is a function of position vector r. Inserting Equation (A52) into Equation (A50) yields:

vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (A53) 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | vs(t0) = dt0 G(r, t; r0, t0) 0 dV0 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | Because differential volume element is dV = dx dy dz and because operator does not 0 0 0 0 ∇ act on variables x , y , z and t the operator can be written in front of the integral: 0 0 0 0 ∇

vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (A54) 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | vs(t0) = dt0 G(r, t; r0, t0) 0 dV0 ∇ × 3 ∇ × r rs(t ) ZR ZR | 0 − 0 | Using the same trick again, i.e., by using standard vector calculus identity (ψP) = ∇ × ψ P + ψ P, using G(r, t; r , t ) = G(r, t; r , t ) and noting that operator does ∇ × ∇ × ∇0 0 0 −∇ 0 0 ∇ not act on variables x0, y0 and z0 one can rewrite the integrand in Equation (A54) as:

vs(t0) G(r, t; r0, t0) 0 (A55) ∇ × r rs(t ) | 0 − 0 | vs(t0) vs(t0) = 0 G(r, t; r0, t0) + G(r, t; r0, t0) ∇ × r rs(t ) ∇ × r rs(t ) | 0 − 0 | | 0 − 0 | Then, by inserting Equation (A55) into Equation (A54) and using a form of standard divergence theorem PdV = dS P it is obtained that: V ∇ × ∂V × R H vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (A56) 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | vs(t0) = dt0 dS0 ∇ × ∂ 3 × r rs(t ) ZR I R | 0 − 0 | vs(t0) + dt0 G(r, t; r0, t0) dV0 ∇ × ∂ 3 ∇ × r rs(t ) ZR Z R | 0 − 0 | 3 Because surface ∂R is an infinite surface the magnitude of position vector r0 is infinite, hence the surface integral in the first right hand side term of Equation (A56) vanishes. Fur- thermore, because operator does not act on variables x , y , z and t the operator can ∇ 0 0 0 0 ∇ Mathematics 2021, 9, 237 26 of 28

be written in front of the second right hand side space-time integral. Hence, Equation (A56) can be written as:

vs(t0) dt0 0 0 G(r, t; r0, t0)dV0 (A57) 3 ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | vs(t0) = dt0 G(r, t; r0, t0) dV0 ∇ × ∇ × ∂ 3 r rs(t ) ZR Z R | 0 − 0 | thus, proving Equation (31).

Appendix C.3. Derivation of Equation (38) Using the mathematical identity 2G(r, t; r , t) = 2G(r, t; r , t), valid for Green’s ∇ 0 ∇0 0 function G(r, t; r0, t), one can rewrite the left hand side integral in Equation (38) as:

qs vs(t0) 2 dt0 G(r, t; r0, t0)dV0 (A58) 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | qs vs(t0) 2 = dt0 0 G(r, t; r0, t0)dV0 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | In AppendixB, two novel vector calculus identities were derived. Subtracting vector identity (A27) from vector identity (A28) yields:

P 2ψdV = ψ 2PdV + P( ψ dS) ψ(dS )P (A59) ZV ∇ ZV ∇ I∂V ∇ · − I∂V · ∇ Using vector identity (A59) Equation (A58) can be written as:

qs vs(t0) 2 dt0 G(r, t; r0, t0)dV0 (A60) 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | qs 2 vs(t0) = dt0 G(r, t; r0, t0) 0 dV0 3 4πe ∇ r rs(t ) ZR ZR | 0 − 0 | qs vs(t0) + dt0 0G(r, t; r0, t0) dS0 3 πe r r (t ) ZR I∂R 4 0 s 0 ∇ · | − | qs vs(t0) dt0 G(r, t; r0, t0) dS0 0 3 πe r r (t ) − ZR I∂R · ∇ 4 0 s 0 | − | 3 3 Because surface ∂R is an inﬁnite surface the magnitude of position vector r0 ∂R has ∈ an inﬁnite magnitude, r ∞. In that case, both right hand side surface integrals over | 0| → surface ∂R3 vanish in Equation (A60). Hence, Equation (A60) becomes:

qs vs(t0) 2 dt0 G(r, t; r0, t0)dV0 (A61) 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | qs 2 vs(t0) = dt0 G(r, t; r0, t0) 0 dV0 3 e ∇ 4π r rs(t ) ZR ZR | 0 − 0 | The operator does not affect vector v (t ) which is a function of variable t . Thus, the ∇0 s 0 0 Laplacian in the equation above can be written as:

2 vs(t0) 2 1 0 = vs(t0) 0 = vs(t0)δ r0 rs(t0) (A62) ∇ 4π r0 rs(t0) ∇ 4π r0 rs(t0) − − | − | | − | Mathematics 2021, 9, 237 27 of 28

Inserting Equation (A62) into right hand side of Equation (A61) yields:

qs vs(t0) 2 dt0 G(r, t; r0, t0)dV0 = (A63) 3 4πe r rs(t ) ∇ ZR ZR | 0 − 0 | qs dt0 vs(t0)δ r0 rs(t0) G(r, t; r0, t0)dV0 3 e − ZR ZR − which proves Equation (38).

Appendix C.4. Derivation of Equation (60) Equation (60) can be derived from Equation (67) by taking the divergence of both sides of this equation to obtain:

qs r rs(t) qs E(r, t) + K(r, t) = − 3 = δ(r rs(t)) (A64) ∇ · ∇ · ∇ · 4πe r rs(t) e − | − | To ﬁnd K(r, t) the operator can be written under the right hand side integral of ∇ · ∇ Equation (66) and then apply identity G(r, t; r , t) = G(r, t; r , t) in order to obtain: ∇ 0 −∇0 0

1 ∂ qs vs(t0) K(r, t) = 2 dt0 0 0 0G(r, t; r0, t0)dV0 (A65) ∇ · − c ∂t 3 4πe ∇ × ∇ × r rs(t ) · ∇ ZR ZR | 0 − 0 | From here, using standard vector identity (ψP) = ψ P + ψ P and divergence ∇ · ∇ · ∇ · theorem it is obtained that:

K(r, t) (A66) ∇ · 1 ∂ qs vs(t0) = 2 dt0 0 0 G(r, t; r0, t0) dS0 − c ∂t ∂ 3 4πe ∇ × ∇ × r rs(t ) · ZR I R | 0 − 0 | 1 ∂ qs vs(t0) + 2 dt0 0 0 0 G(r, t; r0, t0)dV0 c ∂t 3 4πe ∇ · ∇ × ∇ × r rs(t ) ZR ZR | 0 − 0 | Clearly, the surface integral on the right hand side of Equation (A66) vanishes as r 0. | 0| → Furthermore, because P = 0 the second right hand side term vanishes as well. ∇ · ∇ × Hence, it can be written: K(r, t) = 0 (A67) ∇ · Inserting Equation (A67) into Equation (A64) yields: q E(r, t) = s δ(r r (t)) (A68) ∇ · e − s thus, proving Equation (60).

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