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

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.

1. Introduction

In his famous Treatise [1,2], Maxwell derived his equations of electrodynamics based on the knowledge about the three experimental laws known at the time: Coulomb’s law describing the electric force between charges at rest; Ampere’s law describing the force between current carrying wires, and Faraday’s law of induction. Maxwell equations and Lorentz force were never derived from first principles, i.e., from simpler physical laws than Ampere’s law and Faraday’s law.

In modern vector notation, the four Maxwell’s equations that govern the behavior of electromagnetic fields are written as:

∇·D=ρ

∇·B=0

∇×E=−∂B∂t

∇×B=μJ+1^c2∂E∂t

where the symbolDdenotes electric displacement vector,Eis the electric field vector,Bis a vector called magnetic flux density, vectorJis called current density and scalarρis the charge density. Furthermore, there are two more important equations in electrodynamics that relate magnetic vector potentialAand the scalar potentialϕto the electromagnetic fieldsBandE:

B=∇×A

E=−∇ϕ−∂A∂t

In standard electromagnetic theory, if a point charge_qsis moving with velocity_vs(t)along arbitrary path_rs(t)the scalar potentialϕand vector potentialAcaused by moving charge_qs are described by well known, relativistically correct, Liénard–Wiechert potentials [3,4]:

ϕ=ϕ(r,t)=14πϵ_qs1−_ns(_tr)·_βs(_tr)r−_rs(_tr)

A=A(r,t)=μc4π_{qs βs}(_tr)1−_ns(_tr)·_βs(_tr)r−_rs(_tr)

where_tris retarded time,ris the position vector of observer and vectors_ns(_tr)and_βs(_tr)are:

_βs(_tr)=_vs(_tr)c

_ns(_tr)=r−_rs(_tr)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 velocityvthen the forceFexperienced by the charge q is:

F=qE+v×B

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×Bterm) 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 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:

FC=q1 q24πϵr1−r2r1−r23

FG=−Gm1 m2r1−r2r1−r23

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 transformation 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 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 Appendix A, 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 Figure 1a, 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. 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 Appendix A. There have been several previous attempts in the literature to generalize classical Helmholtz decomposition theorem to time dependent vector fields [21,22,23]. 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 Appendix A and Appendix A.2. As it was shown in Appendix A, the generalization of Helmholtz decomposition theorem for the vector function of space and timeF(r,t)can be written as:

F(r,t)=−∇∫Rdt′ ∫R3 ∇′·F(r′,t′)G(r,t;r′,t′)dV′+1c2∂∂t∫Rdt′ ∫R3 ∂∂t′F(r′,t′)G(r,t;r′,t′)dV′+∇×∫Rdt′ ∫R3 ∇′×F(r′,t′)G(r,t;r′,t′)dV′

where scalar functionG(r,t;^r′,^t′)is the fundamental solution of time-dependent inhomogeneous wave equation given as:

^∇2G(r,t;^r′,^t′)−1^c2∂2∂^t2G(r,t;^r′,^t′)=−δ(r−^r′)δ(t−^t′)

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

G(r,t;^r′,^t′)=δ^t′−t+r−^r′c4πr−^r′

where position vector^r′is the location of the source at time^t′.

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 positionras vector function of previous time^t′and of previous position^r′ . Furthermore, the generalized Helmholtz decomposition theorem (14) comes with additional limitation that it is valid if vector functionF(^r′,^t′)approaches zero faster than1/r−^r′asr−^r′→∞.

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 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 Figure 2, where source charge_qsis 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 delayΔtbetween the past time_tswhen the source charge has stopped and the present time_tpwhen this disturbance has propagated to the test charge:

Δt=_tp−_ts=r−_rs(_ts)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_tshad enough time to propagate to test charge. The Coulomb’s force_Fcexperienced by the test charge q at the present time_tpcan be expressed by the following equation:

_Fc(r,_tp)=q_qs4πϵr−_rs(_ts)^{r−_rs(_ts)3}_tp=_ts+Δt

Let us now consider the time t just one brief moment before the stopping time_ts:

t=_ts−δt

whereδt→0is very small time interval. This time intervalδtis so small that it might even be called infinitesimally small. Then at the moment in time infinitesimally before the present time_tpthe force felt by test charge q is still the Coulomb’s force ifδt→0 . Using these considerations, Equation (18) can be rewritten as:

_Fc(r,_tp−δt)=q_qs4πϵr−_rs(t)^r−_rs(t)3t=_ts−δt;δt→0

Note that Equation (20) is equivalent to Equation (18) whenδt→0. The reason Coulomb’s law is written this way is to permit slight variation of time before stopping time_tsso 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.

Because the right hand side of Equation (20) is now the function of time t and positionr 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 positionr 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:

Fc(r,tp−δt)=qqs4πϵr−rs(t)r−rs(t)3=−∇∫Rdt′ ∫R3 ∇′·qqs4πϵr′−rs(t′)r′−rs(t′)3G(r,t;r′,t′)dV′+1c2∂∂t∫Rdt′ ∫R3 ∂∂t′qqs4πϵr′−rs(t′)r′−rs(t′)3G(r,t;r′,t′)dV′+∇×∫Rdt′ ∫R3 ∇′×qqs4πϵr′−rs(t′)r′−rs(t′)3G(r,t;r′,t′)dV′

To clarify the notation in equation above note thatd^V′=d^x′d^y′d^z′represents the differential volume element of an infinite volume^R3 . As defined in Appendixes Appendix A and Appendix A.1, the primed position vector^r′is written in Cartesian coordinate system as:

^r′=^x′x^+^y′y^+^z′z^

where variables^x′,^y′,^z′∈R. Vectorsx^,y^andz^are orthogonal Cartesian unit basis vectors. Furthermore, in Cartesian coordinates, the primed del operator^∇′ that appears in Equation (21) is defined as:

^∇′=x^∂∂^x′+y^∂∂^y′+z^∂∂^z′

From the definition above, it follows that primed del operator^∇′acts only on functions of variables^x′,^y′,^z′, and consequently, on functions of primed position vector^r′=^x′x^+^y′y^+^z′z^. It does not act on functions of position vector of source charge_rs(^t′)because this position vector is function of variable^t′. Using these definitions one can write the following simple relations:

^r′−_rs(^t′)^{r′−_rs(t′)3}=−^∇′1^r′−_rs(^t′)

^∇′·14π^r′−_rs(^t′)^{r′−_rs(t′)3}=−^∇′214π1^r′−_rs(^t′)=δ^r′−_rs(^t′)

^∇′×^r′−_rs(^t′)^{r′−_rs(t′)3}=−^∇′×^∇′1^r′−_rs(^t′)=0

whereδ^r′−_rs(^t′) is 3D Dirac’s delta function. Inserting Equations (25) and (26) into Equation (21), and eliminating charge q from the equation, yields the following relation:

_qs4πϵr−_rs(t)^r−_rs(t)3=−∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+1^c2∂∂t_∫Rd^t′ _∫^R3 ∂∂^t′_qs4πϵ^r′−_rs(^t′)^{r′−_rs(t′)3}G(r,t;^r′,^t′)d^V′

In Appendix C and Appendix 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:

∂∂^t′_qs4πϵ^r′−_rs(^t′)^{r′−_rs(t′)3}=_qs4πϵ^∇′×^∇′×_vs(^t′)^r′−_rs(^t′)−_qsϵ_vs(^t′)δ^r′−_rs(^t′)

where_vs(^t′)is the velocity of the source charge_qsat time^t′:

_vs(^t′)=∂_rs(^t′)∂^t′

By inserting Equation (28) into Equation (27) it is obtained that:

_qs4πϵr−_rs(t)^r−_rs(t)3=−∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂∂t_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+1^c2∂∂t_∫Rd^t′ _{∫^R3 qs}4πϵ^∇′×^∇′×_vs(^t′)^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

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

_∫Rd^t′ _{∫^R3 qs}4πϵ^∇′×^∇′×_vs(^t′)^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

Replacing the last right hand side integral in Equation (30) with Equation (31) and differentiating the resulting equation with respect to time t yields:

_qs4πϵ∂∂tr−_rs(t)^r−_rs(t)3=−∂∂t∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂2∂^t2_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+1^c2∂2∂^t2∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

In the physical setting shown in Figure 2 the coordinates of the test charge q are fixed, hence, the order in which the operator s∇×∇×and second order time derivative∂∂^t2are applied can be swapped (because operator ∇ does not affect variable t). Furthermore, because variables^t′,^x′,^y′and^z′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:

_qs4πϵ∂∂tr−_rs(t)^r−_rs(t)3=−∂∂t∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂2∂^t2_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)1^c2∂2∂^t2G(r,t;^r′,^t′)d^V′

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

_qs4πϵ∂∂tr−_rs(t)^r−_rs(t)3=−∂∂t∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂2∂^t2_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)δr−^r′δt−^t′d^V′+∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)^∇2G(r,t;^r′,^t′)d^V′

Using sifting the property of Dirac’s delta function allows us to rewrite the third right hand side term of Equation (34) as:

∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)δr−^r′δt−^t′d^V′=∇×∇×_∫Rd^t′_qs4πϵ_vs(^t′)r−_rs(^t′)δt−^t′=∇×∇×_qs4πϵ_vs(t)r−_rs(t)

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

∇×∇×_qs4πϵ_vs(t)r−_rs(t)=_qs4πϵ∇∇·_vs(t)r−_rs(t)−_qs4πϵ^∇2_vs(t)r−_rs(t)=_qs4πϵ∇_vs(t)·∇1r−_rs(t)−_qs4πϵ_vs(t)^∇21r−_rs(t)=−_qs4πϵ∇∂∂t1r−_rs(t)+_qsϵ_vs(t)δr−_rs(t)=_qs4πϵ∂∂tr−_rs(t)^r−_rs(t)3+_qsϵ_vs(t)δr−_rs(t)

Combining Equations (34)–(36), after cancellation of appropriate terms, yields:

0=−∂∂t∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂2∂^t2_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+_qsϵ_vs(t)δr−_rs(t)+∇×∇×_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)^∇2G(r,t;^r′,^t′)d^V′

In Appendix C and Appendix C.3, the following mathematical identity was derived:

_∫Rd^t′ _{∫^R3 qs}4πϵ_vs(^t′)^r′−_rs(^t′)^∇2G(r,t;^r′,^t′)d^V′=−_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

By inserting Equation (38) into Equation (37) it is obtained that:

0=−∂∂t∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂2∂^t2_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+_qsϵ_vs(t)δr−_rs(t)−∇×∇×_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

If now the new constantμ=1^c2ϵ was introduced, then by dividing whole Equation (39) by^c2it is obtained:

0=−1^c2∂∂t∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂2∂^t2_∫Rd^t′ _{∫^R3 qs}μc_vs(^t′)cδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+_qsμ_vs(t)δr−_rs(t)−∇×∇×_∫Rd^t′ _{∫^R3 qs}μc_vs(^t′)cδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

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_∫^R3 δ^r′−_rs(^t′)f(^r′)d^V′=f(_rs(^t′)), wheref(^r′)is function of position vector^r′ , to rewrite the right hand side integrals in Equation (40) as:

_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=_∫RqsϵG(r,t;_rs(^t′),^t′)d^t′

_∫Rd^t′ _{∫^R3 qs}μc_vs(^t′)cδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=_∫Rqsμc_vs(^t′)cG(r,t;_rs(^t′),^t′)d^t′

To evaluate right hand side integrals in equations above, the Green’s functionG(r,t;_rs(^t′),^t′) in these equations is replaced with Equation (16) in order to obtain:

_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=_∫Rqsϵδ^t′−t+r−_rs(^t′)c4πr−_rs(^t′)d^t′

_∫Rd^t′ _{∫^R3 qs}μc_vs(^t′)cδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=_∫Rqsμc_vs(^t′)cδ^t′−t+r−_rs(^t′)c4πr−_rs(^t′)d^t′

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:

δf(u)=δ(u−_u0)_{∂∂uf(u)u=u0}

wheref(u)is real function of real argument u, and_u0is the solution of equationf(_u0)=0 . Using identity (45), the Dirac’s delta function in Equations (43) and (44) can be written as:

δ^t′−t+r−_rs(^t′)c=δ(^t′−_tr)1−1c_vs(^t′)·r−_rs(^t′)r−_rs(^t′)=δ(^t′−_tr)1−1c_vs(^t′)·r−_rs(^t′)r−_rs(^t′)=δ(^t′−_tr)1−_βs(_tr)·_ns(_tr)

where_βs(_tr)and_ns(_tr) are given by Equations (9) and (10), respectively. From Equation (45) it follows that the time_tris the solution to the following equation:

t−_tr−r−_rs(_tr)c=0

Evidently, the time_tris 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 positionrof the test charge at time t. In the electromagnetic literature this time_tris 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^t′can be evaluated as:

_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=14πϵ_qsr−_rs(_tr)1−_βs(_tr)·_ns(_tr)

_∫Rd^t′ _{∫^R3 qs}μc_vs(^t′)cδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′=μc4π_{qs βs}(_tr)r−_rs(_tr)1−_βs(_tr)·_ns(_tr)

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

∇×∇×μc4π_{qs βs}(_tr)r−_rs(_tr)1−_βs(_tr)·_ns(_tr)=_qsμ_vs(t)δr−_rs(t)−1^c2∂∂t∇14πϵ_qsr−_rs(_tr)1−_βs(_tr)·_ns(_tr)−1^c2∂2∂^t2μc4π_{qs βs}(_tr)r−_rs(_tr)1−_βs(_tr)·_ns(_tr)

The first right hand side term of the equation above can be identified as the currentJof the point charge distribution moving with velocity_vs(t)multiplied by constantμ:

μJ=μ_{qs vs}(t)δr−_rs(t)

To proceed, scalar functionϕ(r,t)and vector functionA(r,t)are defined:

ϕ(r,t)=14πϵ_qsr−_rs(_tr)1−_βs(_tr)·_ns(_tr)

A(r,t)=μc4π_{qs βs}(_tr)r−_rs(_tr)1−_βs(_tr)·_ns(_tr)

With the aid of scalar functionϕ(r,t), vector functionA(r,t), and expressionμJ given by Equation (51) the Equation (50) can be written as:

∇×∇×A(r,t)=μJ−1^c2∂∂t−∇ϕ(r,t)−∂∂tA(r,t)

To further simplify Equation (54) two additional vector functionsBandEare defined:

B=∇×A(r,t)

E=−∇ϕ(r,t)−∂∂tQ(r,t)

Using definitions of vector functionsBandE , given by Equations (55) and (56), Equation (54) can be rewritten as:

∇×B=μJ+1^c2∂E∂t

The mathematical properties of vector fieldsBandEshall now be investigated. Note that because for any differentiable vector fieldPwe can write∇·∇×P=0 , from Equation (55) it follows that:

∇·B=0

The curl of the gradient of any differentiable scalar functionψis equal to zero, i.e.,∇×∇ψ=0 . Thus, taking the curl of Equation (56) yields:

∇×E=−∇×∂∂tA(r,t)=−∂∂t∇×A(r,t)=−∂B∂t

Finally, in Appendix C and Appendix C.4, it was shown that the divergence of vector fieldEis:

∇·E=_qsϵδr−_rs(t)=ρ(r,t)ϵ

which completes the derivation of electrodynamic equations from Coulomb’s law. From tables Table 1 and Table 2 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).

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 mathematically 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. 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 Figure 3 where the source charge_qsis moving along arbitrary trajectory_rs(t). Then the source charge_qsstops at some time in the past_ts. In this physical setting, closed contour C is at rest at all times. At present time_tp>_tsall the points inside the sphere of radiusR=c(_tp−_ts)are affected only by 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:

_∮CEc(r,_tp)·dr=_∮Cqs4πϵr−_rs(_ts)^{r−_rs(_ts)3}·dr=0

where_Ecis Coulomb’s electrostatic field,_rs(_ts)is the position vector of source charge when it stopped moving, and vectorris 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.

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=_ts−δt

whereδtis infinitesimally small time interval. If time intervalδtapproaches zero (δt→0 ) the contour integral (61) can be rewritten as the function of time t:

_∮CEc(r,_tp−δt)·dr=_∮Cqs4πϵr−_rs(t)^r−_rs(t)3·dr=0

Because the integrand on the right hand side of Equation (63) is the function of varying time t and position vectorr 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:

_qs4πϵr−_rs(t)^r−_rs(t)3=−∇_∫Rd^t′ _{∫^R3 qs}ϵδ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′−1^c2∂∂t_∫Rd^t′ _{∫^R3 qs}ϵ_vs(^t′)δ^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′+1^c2∂∂t_∫Rd^t′ _{∫^R3 qs}4πϵ^∇′×^∇′×_vs(^t′)^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

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/μϵyields:

_qs4πϵr−_rs(t)^r−_rs(t)3=−∇ϕ(r,t)−∂∂tA(r,t)+K(r,t)

where vector functionK(r,t) is equal to the last right hand side term of Equation (64):

K(r,t)=1^c2∂∂t_∫Rd^t′ _{∫^R3 qs}4πϵ^∇′×^∇′×_vs(^t′)^r′−_rs(^t′)G(r,t;^r′,^t′)d^V′

Replacing the first two terms on the right hand side of Equation (65) with Equation (56) yields:

_qs4πϵr−_rs(t)^r−_rs(t)3=E(r,t)+K(r,t)

Then, by inserting Equation (67) into the right hand side of Equation (63) it is obtained that:

0=_∮CEc(r,_tp−δt)·dr=_∮CE(r,t)+K(r,t)·dr

The space-time integral on the right hand side of Equation (66) is very difficult to evaluate. However, vector fieldK(r,t) can be eliminated from the right hand side of Equation (68) by the application of Stokes’ theorem:

0=_∮CEc(r,_tp−δt)·dr=_∮CE(r,t)·dr+_∫S∇×K(r,t)·dS

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)=∂∂tB(r,t)

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

0=_∮CEc(r,_tp−δt)·dr=_∮CE(r,t)·dr+ddt_∫SB(r,t)·dS

The right hand side of Equation (71) is unknown energy conservation principle valid for varying in time dynamic fieldsE(r,t)andM(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 something else. To see this, consider simply connected volume V bounded by surface∂V as shown in Figure 4. The surface∂Vis the union of two surfaces S and_S1bounded by respective contours C and_C1. Contours C and_C1consist of exactly the same spatial points; however, the Stokes’ orientation of these contours is oppositeC=−_C1. Then, using∇×E(r,t)=−∂∂tB(r,t) the first right hand side contour integral of Equation (71) can be written as:

_∮CE(r,t)·dr=−_∮C1 E(r,t)·dr=−_∫S1 ∇×B(r,t)·dS=ddt_∫S1 B(r,t)·dS

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

0=_∮CEc(r,_tp−δt)·dr=ddt_∮∂VB(r,t)·dS

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 straightforward: it is now assumed that all the points on surface∂V shown in Figure 4 have some definite velocityvsuch that|v|<<c. Then the surface∂Vis the function of time, hence,C=C(t)andS=S(t) . Hence, Equation (73) can be written as the sum of two surface integrals over surfacesS(t)and_S1(t):

ddt_∮∂VB(r,t)·dS=ddt_∮S(t)B(r,t)·dS+ddt_∮S1(t)B(r,t)·dS=0

where∂V=S(t)∪_S1(t) . The Leibniz identity [25] for moving surfaces state that for any differentiable vector fieldPit can be written:

ddt_∫S(t)P·dS=_∫S(t)∂∂tP+∇·Pv·dS−_∮C(t)v×P·dr

Applying the Leibniz identity to the surface integral over surface_S1(t) in Equation (74) and using∇·B(r,t)=0yields:

ddt_∮S(t)B(r,t)·dS+_∮S1(t)∂∂tB(r,t)·dS−_∮C1(t)v×B(r,t)·dr=0

Using the result from previous section, i.e.∇×E(r,t)=−∂∂tB(r,t), and applying the Stokes’ theorem yields:

ddt_∮S(t)B(r,t)·dS−_∮C1(t)E(r,t)·dr−_∮C1(t)v×B(r,t)·dr=0

Note that curves_C1(t)andC(t)comprise of the same points; however, Stokes’ orientation of curves_C1(t)andC(t)is opposite, i.e.,_C1(t)=−C(t) . Hence, Equation (77) can be written as:

_∮C(t)E(r,t)+v×B(r,t)·dr+ddt_∫S(t)B(r,t)·dS=0

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).

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 contourC(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 contourC(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_FDon charge q moving with velocityvalong contour C is:

_FD=qE(r,t)+qv×B(r,t)

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:

(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.

Figure 1. In (a) the source chargeqsis at rest at present timetp . Each point on closed contour C is affected by Coulomb's electrostatic fieldEc. Energy conservation principle at present time is∮CEc·dr=0. In (b) the source chargeqsis moving along arbitrary path and it stops at past timets<tp. Dynamic energy conservation principle valid in the past is assumed to be unknown when source charge was moving.

Figure 2. Source chargeqsis moving along arbitrary trajectoryrs(t)and stops at timets. Becauseqsstops moving at past timets, at present timetp , the stationary test charge q experiences electrostatic Coulomb force.

Figure 3. Source chargeqsis moving along arbitrary trajectoryrs(t)and it stops at past timets . Closed contour C is at rest at all times. All the pointsr on contour C are inside the sphere of radiusR=c(tp-ts). At present timetp>ts all the points on contour C are affected only by Coulomb's electrostatic field.

Figure 4. Closed surface∂V that bounds volume V is union of two surfaces S andS1 . Contour C bounds surface S and contourC1bounds surfaceS1 . Contours C andC1are identical; however they have different Stokes' orientation.

Symbol	Equation	Description
ϕ(r,t)	14πϵqsr−rs(tr)1−βs(tr)·ns(tr)	scalar potential derived from Coulomb’s law
A(r,t)	μc4πqs βs(tr)r−rs(tr)1−βs(tr)·ns(tr)	vector potential derived from Coulomb’s law
B	∇×A(r,t)	vector fieldBderived from Coulomb’s law
E	−∇ϕ(r,t)−∂∂tA(r,t)	vector fieldEderived from Coulomb’s law

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

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 vectorsrand^r'are defined in this paper as:

r=xx^+yy^+zz^

^r'=^x'x^+^y'y^+^z'z^

wherex^,y^andz^are Cartesian, mutually orthogonal, unit basis vectors. Variablesx,y,z∈Rand^x',^y',^z'∈Rare linearly independent variables. Furthermore, note that throughout this paper the position vector_rs(^t')indicates the position of the source charge. This position vector_rs(^t')is defined as:

_rs(^t')=_xs(^t')x^+_ys(^t')y^+_zs(^t')z^

where_xs(^t'),_ys(^t')and_zs(^t')are all functions of real variable^t'∈Rwhich is independent of variablesx,y,z∈Rand^x',^y',^z'∈R. The time derivative of position vector_rs(^t')is velocity_vs(^t')of the source charge:

_vs(^t')=∂_rs(^t')∂^t'

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

∇=x^∂∂x+y^∂∂x+z^∂∂x

^∇'=x^∂∂^x'+y^∂∂^y'+z^∂∂^z'

Operator ∇ acts only on functions of variablesx,y,z, hence, on functions of position vectorr. On the other hand, operator^∇'acts only on functions of variables^x',^y',^z', thus, it acts on functions of position vector^r'. For example, if function f is the function of position vectorr, i.e., iff=f(r), one can generally write:

∇f(r)≠0^∇'f(r)=0

On the other hand, if function f is the function of position vector^r', i.e., iff=f(^r')it can be written:

∇f(^r')=0^∇'f(^r')≠0

Furthermore, because variable^t'is independent of variablesx,y,zand^x',^y',^z'neither operator ∇ nor^∇'acts on position vector_rs(^t')and velocity vector_vs(^t'). Using these considerations it is evident that the following equations are correct:

∇·_rs(^t')=0∇·_vs(^t')=0^∇'·_rs(^t')=0^∇'·_vs(^t')=0

However, both operators ∇ and^∇'act on Green's functionG(r,t;^r',^t') given by Equation (16). 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')∂∂tG(r,t;^r',^t')=-∂∂^t'G(r,t;^r',^t')^∂2∂^t2G(r,t;^r',^t')=^∂2∂^t'2G(r,t;^r',^t')

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:

^∇2G(r,t;^r',^t')-1^c2∂2∂^t2G(r,t;^r',^t')=-δ(r-^r')δ(t-^t')

whereG(r,t;^r',^t')is the function called fundamental solution or Green's function andδis Dirac's delta function. The Green's function for inhomogeneous wave equation is well known and it represents an outgoing diverging spherical wave:

G(r,t;^r',^t')=δt-^t'-r-^r'c4πr-^r'

Let us now suppose that vector fieldFis the function of both spacerand time t, i.e.,F=F(r,t). Using sifting property of Dirac delta function the vector functionF(r,t)can be written as the volume integral over infinite volume^R3and over all the timeRas:

F(r,t)=_∫Rd^t' _∫^R3 F(^r',^t')δ(r-^r')δ(t-^t')d^V'

where differential volume elementd^V'isd^V'=d^x'd^y'd^z'. By replacingδ(r-^r')δ(t-^t') in equation above with left hand side of Equation (A11) it is obtained:

F(r,t)=-_∫Rd^t' _∫^R3 F(^r',^t')^∇2G(r,t;^r',^t')-1^c2∂2∂^t2G(r,t;^r',^t')d^V'

From the discussion presented in Appendix A.1, it is evident that the D'Alambert operator^∇2-1^c2∂2∂^t2does not act on variables^x',^y',^z'and^t'nor does it act on vector functionF(^r',^t'). Hence, D'Alambert operator^∇2-1^c2∂2∂^t2can be written in front of the integral:

F(r,t)=-^∇2-1^c2∂2∂^t2_∫Rd^t' _∫^R3 F(^r',^t')G(r,t;^r',^t')d^V'

Using standard vector calculus identity∇×∇×P=∇(∇·P)-^∇2P Equation (A15) can be rewritten as:

F(r,t)=∇×∇×_∫Rd^t' _∫^R3 F(^r',^t')G(r,t;^r',^t')d^V'-∇∇·_∫Rd^t' _∫^R3 F(^r',^t')G(r,t;^r',^t')d^V'+1^c2∂2∂^t2_∫Rd^t' _∫^R3 F(^r',^t')G(r,t;^r',^t')d^V'

Because operators ∇ and∂∂tdo not act on variables^x',^y',^z'and^t'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∇×Pand∇·(ψP)=∇ψ·P+ψ∇·P, and noting that∇×F(^r',^t')=0and∇·F(^r',^t')=0 Equation (A16) can be rewritten as:

F(r,t)=∇×_∫Rd^t' _∫^R3 ∇G(r,t;^r',^t')×F(^r',^t')d^V'-∇_∫Rd^t' _∫^R3 F(^r',^t')·∇G(r,t;^r',^t')d^V'+1^c2∂∂t_∫Rd^t' _∫^R3 F(^r',^t')∂∂tG(r,t;^r',^t')d^V'

One can now use identities∇G(r,t;^r',^t')=-^∇'G(r,t;^r',^t')and∂∂tG(r,t;^r',^t')=-∂∂^t'G(r,t;^r',^t') to rewrite the right hand side integrals in Equation (A17) as:

F(r,t)=-∇×∫Rdt' ∫R3 ∇'G(r,t;r',t')×F(r',t')dV'+∇∫Rdt' ∫R3 F(r',t')·∇'G(r,t;r',t')dV'-1c2∂∂t∫Rdt' ∫R3 F(r',t')∂∂t'G(r,t;r',t')dV'

Using vector calculus identity∇×(ψP)=∇ψ×P+ψ∇×Pand the the form of divergence theorem_∫V∇×PdV=_∮∂VP×dSthe first right hand side integral over^R3can be written as:

∫R3 ∇'G(r,t;r',t')×F(r',t')dV'=∮∂R3G(r,t;r',t')F(r',t')×dS'-∫R3 G(r,t;r',t')∇'×F(r',t')dV'

Note that the surface∂^R3is an infinite surface that bounds an infinite volume^R3. Furthermore, for the surface integral in the equation above, position vector^r'is located on infinite surface∂^R3, i.e.,^r'∈∂^R3. Hence, if vector functionF(^r',^t')decreases faster than1/r-^r'asr-^r'→∞ the surface integral in Equation (A19) vanishes. In that case, it can be written:

∫R3 ∇'G(r,t;r',t')×F(r',t')dV'=-∫R3 G(r,t;r',t')∇'×F(r',t')dV'

Using similar considerations, vector calculus identity∇·(ψP)=∇ψ·P+ψ∇·Pand standard divergence theorem_∫V∇·PdV=_∮∂VP·dSit is obtained that:

_∫^R3 F(^r',^t')·^∇'G(r,t;^r',^t')d^V'=-_∫^R3 G(r,t;r,^t')^∇'·F(^r',^t')d^V'

To treat the last integral on the right hand side of Equation (A18) the following identity can be used:

F(^r',^t')∂∂^t'G(r,t;r,^t')=∂∂^t'F(^r',^t')G(r,t;r,^t')-G(r,t;r,^t')∂∂^t'F(^r',^t')

Using the identity above and noting that^t'is independent of^x',^y'and^z' the last right hand side integral of Equation (A18) can be written as:

_∫Rd^t' _∫^R3 F(^r',^t')∂∂^t'G(r,t;^r',^t')d^V'=_∫Rd^t'∂∂^t'_∫^R3 F(^r',^t')G(r,t;^r',^t')d^V'-_∫Rd^t' _∫^R3 G(r,t;^r',^t')∂∂^t'F(^r',^t')d^V'

By integrating over^t' , it can be shown that the first right hand side integral in Equation (A23) vanishes:

_∫Rd^t'∂∂^t'_∫^R3 F(^r',^t')G(r,t;^r',^t')d^V'=_{∫^R3 F(^r',^t')G(r,t;^r',^t')d^V't'→-∞^t'→∞}

If^t'→±∞ , and if t is finite, then from Equation (A12) follows thatG(r,t;^r',^t')=0 , thus, the right hand side of Equation (A24) is equal to zero. Inserting this result into Equation (A23) yields:

_∫Rd^t' _∫^R3 F(^r',^t')∂∂^t'G(r,t;^r',^t')d^V'=-_∫Rd^t' _∫^R3 G(r,t;^r',^t')∂∂^t'F(^r',^t')d^V'

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)=-∇∫Rdt' ∫R3 ∇'·F(r',t')G(r,t;r',t')dV'+1c2∂∂t∫Rdt' ∫R3 ∂∂t'F(r',t')G(r,t;r',t')dV'+∇×∫Rdt' ∫R3 ∇'×F(r',t')G(r,t;r',t')dV'

The theorem is valid for functionsF(^r',^t')that decrease faster than1/r-^r'asr-^r'→∞.

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

_∫Vψ^∇2PdV=_∮∂VψdS·∇P-_∫V∇ψ·∇PdV

_∫VP^∇2ψdV=_∮∂VP∇ψ·dS-_∫V∇ψ·∇PdV

wherePis differentiable vector field,ψis differentiable scalar function, volumeV⊂^R3is simply connected volume,∂Vis the bounding surface of volume V anddSis differential surface element of∂Vsuch thatdS=ndS. Vectornis an outward unit normal to the surface∂V. In Cartesian coordinate system, the productψ^∇2Pcan be written in terms of Cartesian components as:

ψ^∇2P=x^ψ^∇2 _Px+y^ψ^∇2 _Py+z^ψ^∇2 _Pz

where_Px,_Pyand_Pzare Cartesian components of vectorPand vectorsx^,y^andz^are Cartesian unit basis vectors. Using standard vector calculus identity∇·fT=∇f·T+f∇·T, valid for some scalar function f and for some vector functionT , Equation (A29) can be rewritten as:

ψ^∇2P=x^∇·ψ∇_Px-x^∇ψ·∇_Px+y^∇·ψ∇_Py-y^∇ψ·∇_Py+z^∇·ψ∇_Pz-z^∇ψ·∇_Pz

To proceed, the identity∇ψ·∇Pis expanded in terms of its Cartesian components as:

∇ψ·∇P=∂ψ∂x∂∂x+∂ψ∂y∂∂y+∂ψ∂z∂∂zx^_Px+y^_Py+z^_Pz=x^∇ψ·∇_Px+y^∇ψ·∇_Py+z^∇ψ·∇_Pz

Inserting Equation (A31) into (A30) it is obtained that:

ψ^∇2P=x^∇·ψ∇_Px+y^∇·ψ∇_Py+z^∇·ψ∇_Pz-∇ψ·∇P

By integrating Equation (A32) over volume V and by application of the divergence theorem_∫V∇·TdV=_∮∂VT·dSit is obtained:

_∫Vψ^∇2PdV=x^_∮∂Vψ∇_Px·dS+y^_∮∂Vψ∇_Py·dS+z^_∮∂Vψ∇_Pz·dS-_∫V∇ψ·∇PdV

The first three right hand side terms of Equation (A33) can be rewritten as:

x^_∮∂Vψ∇_Px·dS+y^_∮∂Vψ∇_Py·dS+z^_∮∂Vψ∇_Pz·dS=_∮∂VψdS·∇P

Inserting Equation (A34) into (A33) yields:

_∫Vψ^∇2PdV=_∮∂VψdS·∇P-_∫V∇ψ·∇PdV

which was intended to be proven. To prove Equation (A28) the expressionP^∇2ψis written in terms of its Cartesian components as:

P^∇2ψ=x^_Px ^∇2ψ+y^_Py ^∇2ψ+z^_Pz ^∇2ψ

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

P^∇2ψ=x^∇·_Px∇ψ-x^∇_Px·∇ψ+y^∇·_Py∇ψ-y^∇_Py·∇ψ+z^∇·_Pz∇ψ-z^∇_Pz·∇ψ

Inserting Equation (A31) into Equation (A37) yields:

P^∇2ψ=x^∇·_Px∇ψ+y^∇·_Py∇ψ+z^∇·_Pz∇ψ-∇ψ·∇P

Integrating Equation (A38) over volume V and applying divergence theorem_∫V∇·TdV=_∮∂VT·dSit is obtained that:

_∫VP^∇2ψdV=_∮∂VP∇ψ·dS-_∫V∇ψ·∇PdV

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:

∂∂^t'_qs4πϵ^r'-_rs(^t')^r'-_rs(t')3=-_qs4πϵ∂∂^t'∇'1^r'-_rs(^t')

Because coordinates^x',^y'and^z'are independent of time^t'the order in which operators^∇'and∂∂^t'are applied can be swapped:

∂∂^t'_qs4πϵ^r'-_rs(^t')^r'-_rs(t')3=-_qs4πϵ^∇'∂∂^t'1^r'-_rs(^t')

Furthermore, because coordinates^x',^y'and^z'are independent of time^t', the time derivative of^r'is equal to zero∂^r'∂^t'=0 . The inner time derivative in Equation (A41) can now be written as:

∂∂^t'1^r'-_rs(^t')=_vs(^t')·^r'-_rs(^t')^r'-_rs(t')3=-^∇'·_vs(^t')^r'-_rs(^t')

where_vs(^t')is the velocity of the source charge_qsat time^t' given by Equation (A4). Inserting Equation (A42) into Equation (A41) yields:

∂∂^t'_qs4πϵ^r'-_rs(^t')^r'-_rs(t')3=_qs4πϵ^∇'∇'·_vs(^t')^r'-_rs(^t')

To proceed with derivation, the standard vector calculus identity∇×∇×P=∇∇·P-^∇2P , is used. This identity allows us to rewrite the Equation (A43) as:

∂∂^t'_qs4πϵ^r'-_rs(^t')^r'-_rs(t')3=_qs4πϵ^∇'×^∇'×_vs(^t')^r'-_rs(^t')+_qs4πϵ^∇'2_vs(^t')^r'-_rs(^t')

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

_qs4πϵ^∇'2_vs(^t')^r'-_rs(^t')=_qsϵ_vs(^t')^∇'214π1^r'-_rs(^t')=-_qsϵ_vs(^t')δ^r'-_rs(^t')

Hence, replacing the last right hand side term of Equation (A44) with Equation (A45) yields:

∂∂^t'_qs4πϵ^r'-_rs(^t')^r'-_rs(t')3=_qs4πϵ^∇'×^∇'×_vs(^t')^r'-_rs(^t')-_qsϵ_vs(^t')δ^r'-_rs(^t')

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, whereψis a scalar function andP is a vector function, to rewrite the integrand in the last right hand side term of Equation (30) as:

^∇'×^∇'×_vs(^t')^r'-_rs(^t')G(r,t;^r',^t')=^∇'×^∇'×_vs(^t')^r'-_rs(^t')G(r,t;^r',^t')-^∇'G(r,t;^r',^t')×^∇'×_vs(^t')^r'-_rs(^t')

Integrating Equation (A47) with respect to variables^x',^y',^z'and^t', and making use of a standard form of divergence theorem_∫V∇×PdV=_∮∂VdS×Pit is obtained that:

∫Rdt' ∫R3 ∇'×∇'×vs(t')r'-rs(t')G(r,t;r',t')dV'=∫Rdt' ∮∂R3dS'×∇'×vs(t')r'-rs(t')G(r,t;r',t')-∫Rdt' ∫R3 ∇'G(r,t;r',t')×∇'×vs(t')r'-rs(t')dV'

wheredV=d^x'd^y'd^z',∂^R3is an infinite surface that bounds^R3, anddSis differential surface element of the surface∂^R3. Because∂^R3is an infinite surface, the first right hand side integral vanishes. To see this, one can use standard vector identity∇×(ψP)=∇ψ×P+ψ∇×Pand using^∇'×_vs(^t')=0to rewrite the first term in the first right hand side integrand as:

^∇'×_vs(^t')^r'-_rs(^t')=-^r'-_rs(^t')^r'-_rs(t')3×_vs(^t')

Because^r'∈∂^R3this means that^r'→∞. Provided that charge_qsis moving with finite velocity_vs(^t') it is clear that right hand side term of Equation (A49) vanishes as^r'→∞ . Thus, Equation (A48) can be written as:

∫Rdt' ∫R3 ∇'×∇'×vs(t')r'-rs(t')G(r,t;r',t')dV'=-∫Rdt' ∫R3 ∇'G(r,t;r',t')×∇'×vs(t')r'-rs(t')dV'

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

^∇'G(r,t;^r',^t')=-∇G(r,t;^r',^t')

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

^∇'G(r,t;^r',^t')×^∇'×_vs(^t')^r'-_rs(^t')=-∇G(r,t;^r',^t')×^∇'×_vs(^t')^r'-_rs(^t')=-∇×G(r,t;^r',^t')^∇'×_vs(^t')^r'-_rs(^t')

Equation above is valid because operator ∇ does not act on velocity vector_vs(^t'), nor does it act on position vectors^r'and_rs(^t'). It only acts on Green's functionG(r,t;^r',^t')because it is a function of position vectorr . Inserting Equation (A52) into Equation (A50) yields:

∫Rdt' ∫R3 ∇'×∇'×vs(t')r'-rs(t')G(r,t;r',t')dV'=∫Rdt' ∫R3 ∇×G(r,t;r',t')∇'×vs(t')r'-rs(t')dV'

Because differential volume element isd^V'=d^x'd^y'd^z'and because operator ∇ does not act on variables^x',^y',^z'and^t'the operator ∇ can be written in front of the integral:

∫Rdt' ∫R3 ∇'×∇'×vs(t')r'-rs(t')G(r,t;r',t')dV'=∇×∫Rdt' ∫R3 G(r,t;r',t')∇'×vs(t')r'-rs(t')dV'

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 not act on variables^x',^y'and^z' one can rewrite the integrand in Equation (A54) as:

G(r,t;^r',^t')^∇'×_vs(^t')^r'-_rs(^t')=^∇'×G(r,t;^r',^t')_vs(^t')^r'-_rs(^t')+∇×G(r,t;^r',^t')_vs(^t')^r'-_rs(^t')

Then, by inserting Equation (A55) into Equation (A54) and using a form of standard divergence theorem_∫V∇×PdV=_∮∂VdS×Pit is obtained that:

∫Rdt' ∫R3 ∇'×∇'×vs(t')r'-rs(t')G(r,t;r',t')dV'=∇×∫Rdt' ∮∂R3dS'×vs(t')r'-rs(t')+∇×∫Rdt' ∫∂R3∇×G(r,t;r',t')vs(t')r'-rs(t')dV'

Because surface∂^R3is an infinite surface the magnitude of position vector^r' is infinite, hence the surface integral in the first right hand side term of Equation (A56) vanishes. Furthermore, because operator ∇ does not act on variables^x',^y',^z'and^t' the operator ∇ can be written in front of the second right hand side space-time integral. Hence, Equation (A56) can be written as:

∫Rdt' ∫R3 ∇'×∇'×vs(t')r'-rs(t')G(r,t;r',t')dV'=∇×∇×∫Rdt' ∫∂R3G(r,t;r',t')vs(t')r'-rs(t')dV'

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 functionG(r,t;^r',t) , one can rewrite the left hand side integral in Equation (38) as:

_∫Rd^t' _{∫^R3 qs}4πϵ_vs(^t')^r'-_rs(^t')^∇2G(r,t;^r',^t')d^V'=_∫Rd^t' _{∫^R3 qs}4πϵ_vs(^t')^r'-_rs(^t')^∇'2G(r,t;^r',^t')d^V'

In Appendix B, two novel vector calculus identities were derived. Subtracting vector identity (A27) from vector identity (A28) yields:

_∫VP^∇2ψdV=_∫Vψ^∇2PdV+_∮∂VP∇ψ·dS-_∮∂VψdS·∇P

Using vector identity (A59) Equation (A58) can be written as:

_∫Rd^t' _{∫^R3 qs}4πϵ_vs(^t')^r'-_rs(^t')^∇2G(r,t;^r',^t')d^V'=_∫Rd^t' _{∫^R3 qs}4πϵG(r,t;^r',^t')^∇'2_vs(^t')^r'-_rs(^t')d^V'+_∫Rd^t' _∮∂^R3qs4πϵ_vs(^t')^r'-_rs(^t')^∇'G(r,t;^r',^t')·d^S'-_∫Rd^t' _∮∂^R3G(r,t;^r',^t')d^S'·^∇'_qs4πϵ_vs(^t')^r'-_rs(^t')

Because surface∂^R3is an infinite surface the magnitude of position vector^r'∈∂^R3has an infinite magnitude,^r'→∞. In that case, both right hand side surface integrals over surface∂^R3 vanish in Equation (A60). Hence, Equation (A60) becomes:

_∫Rd^t' _{∫^R3 qs}4πϵ_vs(^t')^r'-_rs(^t')^∇2G(r,t;^r',^t')d^V'=_∫Rd^t' _{∫^R3 qs}ϵG(r,t;^r',^t')^∇'2_vs(^t')4π^r'-_rs(^t')d^V'

The operator^∇'does not affect vector_vs(^t')which is a function of variable^t'. Thus, the Laplacian in the equation above can be written as:

^∇'2_vs(^t')4π^r'-_rs(^t')=_vs(^t')^∇'214π^r'-_rs(^t')=-_vs(^t')δ^r'-_rs(^t')

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

_∫Rd^t' _{∫^R3 qs}4πϵ_vs(^t')^r'-_rs(^t')^∇2G(r,t;^r',^t')d^V'=-_∫Rd^t' _{∫^R3 qs}ϵ_vs(^t')δ^r'-_rs(^t')G(r,t;^r',^t')d^V'

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:

∇·E(r,t)+∇·K(r,t)=∇·_qs4πϵr-_rs(t)^r-_rs(t)3=_qsϵδr-_rs(t)

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:

∇·K(r,t)=-1^c2∂∂t_∫Rd^t' _{∫^R3 qs}4πϵ^∇'×^∇'×_vs(^t')^r'-_rs(^t')·^∇'G(r,t;^r',^t')d^V'

From here, using standard vector identity∇·(ψP)=∇ψ·P+ψ∇·Pand divergence theorem it is obtained that:

∇·K(r,t)=-1^c2∂∂t_∫Rd^t' _∮∂^R3qs4πϵ^∇'×^∇'×_vs(^t')^r'-_rs(^t')G(r,t;^r',^t')·d^S'=+1^c2∂∂t_∫Rd^t' _{∫^R3 qs}4πϵ^∇'·^∇'×^∇'×_vs(^t')^r'-_rs(^t')G(r,t;^r',^t')d^V'

Clearly, the surface integral on the right hand side of Equation (A66) vanishes as^r'→0. Furthermore, because∇·∇×P=0the second right hand side term vanishes as well. Hence, it can be written:

∇·K(r,t)=0

Inserting Equation (A67) into Equation (A64) yields:

∇·E(r,t)=_qsϵδr-_rs(t)

thus, proving Equation (60).

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