How to obtain the covariant form of Maxwell's equations from the continuity equation
aa r X i v : . [ phy s i c s . c l a ss - ph ] D ec How to obtain the covariant form of Maxwell’s equationsfrom the continuity equation
Jos´e A. Heras
Departamento de Ciencias B´asicas,Universidad Aut´onoma Metropolitana, Unidad Azcapotzalco,Av. San Pablo No. 180, Col. Reynosa, 02200,M´exico D. F. M´exico and Departamento de F´ısica y Matem´aticas,Universidad Iberoamericana, Prolongaci´on Paseode la Reforma 880, M´exico D. F. 01210, M´exico
Abstract
The covariant Maxwell equations are derived from the continuity equation for the electric charge.This result provides an axiomatic approach to Maxwell’s equations in which charge conservationis emphasized as the fundamental axiom underlying these equations. . Introduction In classical electrodynamics the conservation of electric charge, which is expressed throughthe continuity equation ∇ · J + ∂ρ∂t = 0 , (1)is not usually considered to be an independent assumption but a formal consequence ofMaxwell’s equations. The familiar argument is that the divergence of the Ampere-Maxwelllaw together with the Gauss law yield equation (1). Regarding equation (1) Griffiths haswritten [1]: “This is, of course, the continuity equation —the precise mathematical statementof local conservation of charge. As I indicated earlier, it can be derived from Maxwell’sequations. Conservation of charge is not an independent assumption, but a consequence ofthe laws of electrodynamics.” The argument is certainly more elegant in spacetime wherethe covariant form of the continuity equation reads ∂ ν J ν = 0. By applying the operator ∂ ν to the inhomogeneous Maxwell equations (in Gaussian units) ∂ µ F µν = (4 π/c ) J ν we obtain ∂ ν J ν = 0 because of the identity ∂ ν ∂ µ F µν ≡ ∇ × B = (4 π/c ) J to the Ampere-Maxwell law: ∇ × B − (1 /c ) ∂ E /∂t =(4 π/c ) J . Accordingly, charge conservation should be interpreted as an axiom of Maxwell’sequations rather than a consequence of them.We have also demonstrated [2] an existence theorem for two retarded fields, which saysthat given localized time-dependent scalar and vector sources satisfying the continuity equa-tion there exist two retarded vector fields that satisfy four field equations. When the sourcesare identified with the usual electromagnetic charge and current densities and appropriatechoice of constants made, the retarded vector fields become the generalized Coulomb andBiot-Savart laws [3,4] and their associated field equations become Maxwell’s equations. Wehave concluded that charge conservation is the fundamental axiom underlying Maxwell’s2quations. Our derivation of Maxwell’s equations has been commented on by Jefimenko[5,6] and Kapu`scik [7,8]. Colussi and Wickramasekara [9] have stressed the result thatMaxwell’s equations, which are Lorentz-invariant, have been obtained from the continuityequation, which is Galilei-invariant according to these authors.An advantage of the derivation of Maxwell’s equations presented in reference 2 is that itnaturally introduces the time-dependent extensions of the Coulomb and Biot-Savart laws inthe form given by Jefimenko [3,4]. However, a practical disadvantage of such a derivation isthat it involves unfamiliar and large identities involving derivatives of retarded quantities.In this paper we extend our three-dimensional derivation of Maxwell’s equations [2] to thefour-dimensional spacetime. We show how the covariant form of Maxwell’s equations can beobtained from the continuity equation for the electric charge. The existence theorem for tworetarded fields demonstrated in reference 2 is now formulated for a retarded antisymmetricfield tensor in spacetime. The existence theorem in spacetime is then applied to the usualelectromagnetic four-current and thus the electromagnetic field together with the covariantMaxwell equations are obtained. This four-dimensional derivation of Maxwell’s equationsis considerably shorter than that formulated in the three-dimensional space. Readers wellversed in the mathematics of the spacetime of special relativity could find more simpleand elegant the four-dimensional derivation because it does not involve large operationson explicit retarded quantities but instead it involves simplified tensor operations in whichretardation is implicit in the retarded Green function of the free-space wave equation.
2. The covariant Maxwell equations
Greek indices µ, ν, κ . . . run from 0 to 3; Latin indices i, j, k, . . . run from 1 to 3; x = x µ = ( ct, x ) is the field point and x ′ = x ′ µ = ( ct ′ , x ′ ) the source point; the signature ofthe metric of spacetime is (+ , − , − , − ); ε µναβ is the totally antisymmetric four-dimensionaltensor with ε = 1 and ε ijk is the totally antisymmetric three-dimensional tensor with ε = 1. Summation convention on repeated indices is adopted.The covariant form of Maxwell’s equations can be written in Gaussian units as: ∂ µ F µν = 4 πc J ν and ∂ µ ∗ F µν = 0 , (2)where F µν is the electromagnetic field tensor and ∗ F µν = (1 / ε µνκσ F κσ is the dual of F µν .The antisymmetric tensor F µν is defined by its components F i = ( E ) i and F ij = − ε ijk ( B ) k , ∗ F µν are ∗ F i = ( B ) i and ∗ F ij = ε ijk ( E ) k . The source of thetensor F µν is the electric four-current: J ν = ( cρ, J ) , which is a conserved quantity ∂ ν J ν = 0 . (3)As above mentioned, equation (3) seems to be a formal consequence of equations (2) becauseof the identity ∂ ν ∂ µ F µν ≡
0, that is, equations (2) imply equation (3). We will investigatenow the converse implication: To what extent equation (3) implies equations (2)?
3. An existence theorem in spacetime
Let us formulate the following existence theorem: Given the localized four-vector J ν satisfying the continuity equation ∂ ν J ν = 0 , (4)there exists the antisymmetric tensor field F µν = Z d x ′ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) , (5)that satisfies the field equations ∂ µ F µν = J ν and ∂ µ ∗ F µν = 0 , (6)where G ( x, x ′ ) = δ ( t ′ − t + R/c ) / (4 πR ) with R = | x − x ′ | is the retarded Green function ofthe free-space wave equation and ∗ F µν = (1 / ε µναβ F αβ is the dual of F µν . The function G satisfies the free-space wave equation ∂ ′ µ ∂ ′ µ G = δ ( x − x ′ ), where δ ( x − x ′ ) is the four-dimensional delta function. The integral in equation (5) is over all spacetime. We note that G is not an explicit Lorentz invariant object. This objection can be avoided by replacing G by the less-known retarded invariant form D ( x, x ′ ) = θ ( x − x ′ ) δ [( x − x ′ ) ] / (2 π ), where θ isthe theta function [10].The proof of the theorem follows from the tensor identity ∂ µ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) − ∂ ′ µ ( G∂ ′ ν J µ − J ν ∂ ′ µ G ) = J ν δ ( x − x ′ ) . (7)This identity is of general validity for a four-vector J ν that satisfies equation (4) but oth-erwise is arbitrary. In particular, equation (7) is valid when J ν is localized into a finiteregion of spacetime, property that guaranties that surface integrals in spacetime involving4 ν vanish at infinity. Equation (7) can be obtained from Eq. (4) as follows. Assume that J ν satisfies the continuity equation evaluated at the source point: ∂ ′ µ J µ = 0 . (8)We apply the operator − G∂ ′ ν to equation (8) and add the term J ν ∂ ′ µ ∂ ′ µ G to both sides: − G∂ ′ ν ∂ ′ µ J µ + J ν ∂ ′ µ ∂ ′ µ G = J ν ∂ ′ µ ∂ ′ µ G. (9)The left-hand side of equation (9) can be written as − ∂ ′ µ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) − ∂ ′ µ ( G∂ ′ ν J µ − J ν ∂ ′ µ G ) = J ν ∂ ′ µ ∂ ′ µ G. (10)When − ∂ ′ µ G = ∂ µ G is used in the first term of the left-hand side of equation (10) and ∂ ′ µ ∂ ′ µ G = δ ( x − x ′ ) in its right-hand side, we finally obtain equation (7).We integrate equation (7) over all spacetime ∂ µ Z d x ′ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) − Z d x ′ ∂ ′ µ ( G∂ ′ ν J µ − J ν ∂ ′ µ G ) = Z d x ′ J ν δ ( x − x ′ ) , (11)where ∂ µ has been extracted from the first integral of the left-hand side. The second integralon this side can be transformed into a surface integral which vanishes at infinity because of J ν is assumed to be a localized quantity. The term on the right-hand side can be integratedyielding J ν . Thus equation (11) reduces to ∂ µ Z d x ′ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) = J ν . (12)This equation shows the existence of the antisymmetric tensor F µν = Z d x ′ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) , (13)in terms of which equation (12) takes the compact (inhomogeneous) form: ∂ µ F µν = J ν . (14)The dual of F µν can be written as ∗ F µν = ε µνκλ Z d x ′ G∂ ′ κ J λ . (15)We apply the operator ∂ µ to equation (15) and perform an integration by parts to obtain ∂ µ ∗ F µν = ε µνκλ Z d x ′ G∂ ′ µ ∂ ′ κ J λ − Z d x ′ ∂ ′ µ ( ε µνκλ G∂ ′ κ J λ ) . (16)5he first integral vanishes because of the symmetry of the indices µ and κ in ∂ ′ µ ∂ ′ κ andthe antisymmetry of such indices in ε µνκλ . The second integral can be transformed intoa surface integral which vanishes at infinity. Then equation (16) reduces to the compact(homogeneous) form ∂ µ ∗ F µν = 0 . (17)Once equations (13), (14), and (17) have been obtained, the theorem has been demonstrated.Notice that this theorem applies to any conserved four-vector which does not necessarilybelong to the electromagnetic theory.
4. Deriving Maxwell’s equations
Let us apply the existence theorem to the four-vector J µ = βJ µ , where J µ = ( cρ, J ) isthe usual electric four-current with ρ and J being the charge and current densities and β isa constant. Then there exists the tensor field G µν = β Z d x ′ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) , (18)which satisfies the field equations ∂ µ G µν = βJ ν and ∂ µ ∗ G µν = 0 , (19)where ∗ G µν can be written as ∗ G µν = βε µνκλ Z d x ′ G∂ ′ κ J λ . (20)The constant β belongs to the αβγ -system which allows us to write electromagnetic expres-sions in a way that is independent of units [11]. The constants α, β and γ satisfy the relation α = βγc with c being the speed of light in vacuum [11]. If we let α = 1 /ǫ , β = µ , and γ = 1, we obtain expressions in SI units, and if we let α = 4 π, β = 4 π/c , and γ = 1 /c , weobtain expressions in Gaussian units.To verify that equations (19) are already the covariant Maxwell equations, that is, thatthe tensor G µν identifies with the well-known electromagnetic field tensor F µν , we proceed asfollows. Let us label the components of the tensor G µν in equation (18) as G i = ( βc/α )( X ) i and G ij = − ε ijk ( Y ) k , where X and Y are two time-dependent vector fields. The componentsof the dual tensor ∗ G µν can be obtained from those of G µν by making the dual changes6 βc/α )( X ) i → ( Y ) i and ( Y ) k → − [1 / ( γc )]( X ) k . Notice that βc/α = 1 / ( γc ). It follows that ∗ G i = ( Y ) i and ∗ G ij = [1 / ( γc )] ε ijk ( X ) k . Our plan is now to identify the vectors X and Y with the electric and magnetic fields E and B . With this purpose in mind, let us considerthe i X ) i = α π Z d x ′ G ( ∂ ′ i J − ∂ ′ J i ) , = α π Z Z d x ′ dt ′ G (cid:26) − ( ∇ ′ ρ ) i − c (cid:18) ∂ J ∂t ′ (cid:19) i (cid:27) = α π Z d x ′ R (cid:20) − ∇ ′ ρ − c ∂ J ∂t ′ (cid:21) i ret , (21)where the square brackets [ ] ret indicate that the enclosed quantity is to be evaluated at theretarded time t ′ = t − R/c.
The right-hand side of Eq. (21) is the same as the right-handside of the i-component of the retarded electric field( E ) i = α π Z d x ′ R (cid:20) − ∇ ′ ρ − c ∂ J ∂t ′ (cid:21) i ret . (22)Comparison between equations (21) and (22) yields the identification X = E . We considernow the i Y ) i = β π ǫ i αβ Z d x ′ G∂ ′ α J β , = β π Z Z d x ′ dt ′ G { ǫ ijk ∂ ′ j J k } = β π Z d x ′ R [ ∇ ′ × J ] i ret . (23)The right-hand side of equation (23) is the same that the right-hand side of the i-componentof the retarded magnetic field ( B ) i = β π Z d x ′ R [ ∇ ′ × J ] i ret . (24)Comparison between equations (22) and (23) yields the identification Y = B . Then thetensor G µν naturally identifies with the electromagnetic field tensor F µν = β Z d x ′ G ( ∂ ′ µ J ν − ∂ ′ ν J µ ) , (25)which satisfies the Maxwell equations ∂ µ F µν = βJ ν and ∂ µ ∗ F µν = 0 , (26)7here ∗ F µν can be written as ∗ F µν = βε µνκλ Z d x ′ G∂ ′ κ J λ . (27)The components of F µν are given by F i = βcα ( E ) i and F ij = − ε ijk ( B ) k , (28)where ( E ) i and ( B ) k are the components of the electric and magnetic fields. It follows that ∗ F i = ( B ) i and ∗ F ij = 1 γc ε ijk ( E ) k . (29)Therefore, the covariant form of Maxwell’s equations have been obtained as an applicationof the existence theorem formulated here.For completeness, we can verify that Eqs. (26) yield the three-dimensional form ofMaxwell’s equations in αβγ -units. By using equations (28), (29) and ∂ µ = [(1 /c ) ∂/∂t, ∇ ],we can write the four-vectors [11]: ∂ µ F µν = (cid:18) βcα ∇ · E , ∇ × B − βα ∂ E ∂t (cid:19) , (30) ∂ µ ∗ F µν = (cid:18) ∇ · B , − γc ∇ × E − c ∂ B ∂t (cid:19) . (31)From equations (26), (30), (31) and J ν = ( cρ, J ) , we obtain the three-dimensional form ofMaxwell’s equations in αβγ units for sources in vacuum [11]: ∇ · E = αρ, (32) ∇ · B = 0 , (33) ∇ × E + γ ∂ B ∂t = 0 , (34) ∇ × B − βα ∂ E ∂t = β J . (35)In particular, if we let α = 1 /ǫ , β = µ , and γ = 1, then we obtain Maxwell’s equations inSI units,
5. Discussion
In section 4 we have established the identification G µν = F µν on the basis that equations(22) and (24) are already known. But suppose that we do not know these equations but8nly the experimental time-independent Coulomb and Biot-Savart laws. In this case thetime-independent limit of the i X ) i = α π Z d x ′ ( ˆ R ) i ρR , (36)where ˆ R = R /R . From the experimental law f = q E for the force f acting on a charge q we know that there exists the electrostatic field E with components( E ) i = α π Z d x ′ ( ˆ R ) i ρR . (37)Comparison between equations (36) and (37) yields the identification X = E in the time-independent regime. We consider now the time-independent limit of the i Y ) i = β π Z d x ′ (cid:18) J × ˆ R R (cid:19) i . (38)From the experimental law f = γ R d x ′ J × B for the force f acting on a current J weknow that there exists the magnetostatic field B given by( B ) i = β π Z d x ′ (cid:18) J × ˆ R R (cid:19) i . (39)Comparison between equations (38) and (39) leads to the identification Y = B in thetime-independent regime.Therefore, by requiring that the time-independent limit of equations (18) and (20) beconsistent with the experimental Coulomb and Biot-Savart laws, we obtain the identifications X = E and Y = B in the time-independent regime. Such identifications naturally extendto the time-dependent regime of the theory and so we conclude again the identification G µν = F µν . Let us imagine for a moment that there is a physicist that only knows theexperimental Coulomb and Biot-Savart laws. He then can infer the electromagnetic fieldtogether with Maxwell’s equations by assuming the validity of the continuity equation. Thephysicist would correctly conclude that charge conservation is the fundamental postulateunderlying Maxwell’s equations.It can be argued that the derivation of the covariant form of Maxwell’s equations from thecontinuity equation involves other implicit physical postulates like retardation (causality)which is implicit in the use of the retarded Green function G . If instead of the Minkowski9pacetime with metric signature (+ , − , − , − ) together with its associated retarded Greenfunction G = δ { t ′ − t + R/c } / (4 πR ), we consider, for example, an Euclidean four-space[12-14] with metric signature (+ , + , + , +) together with its associated retarded-imaginaryGreen function [14] G I = δ { t ′ − t + R/ ( ic ) } / (4 πR ), then we obtain a set of Maxwell-likefield equations in the Euclidean four-space [12-14]. This means that the continuity equationcan also imply other electromagnetic theories different from that of Maxwell. This is anexpected result because charge conservation is independent of the signature of the metric ofspacetime. The independence of the signature can be verified in the equation ∂ α J α = ∂ α J α = ∇ · J + ∂ρ∂t = ˆ ∂ α ˆ J α = ˆ ∂ α ˆ J α , (40)where ∂ α = [(1 /c ) ∂/∂t, ∇ ] , ∂ α = [(1 /c ) ∂/∂t, − ∇ ] , J α = ( cρ, − J ) and J α = ( cρ, J ) areobjects in the Minkowski spacetime and ˆ ∂ α = [(1 /c ) ∂/∂t, ∇ ] , ˆ ∂ α = [(1 /c ) ∂/∂t, ∇ ] , ˆ J α =( cρ, J ) and ˆ J α = ( cρ, J ) are objects in the Euclidean spacetime (there is no difference be-tween covariance and contravariance in this spacetime). Hehl and Obukhov [15] have pointedout that charge conservation is metric-independent because it is based on a counting proce-dure for elementary charges. Therefore if one first postulates the validity of the continuityequation in any four-space, the particular use of the Minkowski spacetime (together with itsassociated retarded Green function G ) would not really be a new postulate, but only just aparticular application of the initial postulate.
6. The subtle relation between sources and fields: Is this an egg-hen problem?
A referee has pointed out that ∂ρ/∂t and J can only be produced by electric and magneticfields (take Ohm’s law for instance J = σ E ) . Therefore a circular process seems to beunavoidable in electromagnetism: ρ and J imply E and B which in turn imply new ρ and J , and so on. Because of this circular characteristic, it is not clear if E and B (satisfyingMaxwell’s equations) are a consequence of ρ and J (satisfying the continuity equation) or viceversa. According to the referee it seems a matter of taste to say which one is a consequenceof the other. In other words: From referee’s comment we could conclude that the connectionbetween sources and fields is a little bit like the egg-hen problem: Who was first?Let us to briefly discuss the very deep relation between sources and fields in the contextof the electromagnetic theory. It is now a common place to say that charges and currents arethe causes of the electric and magnetic fields, but this was not so in the 19th century [16,17].10wo rival point of views on what is now known as electromagnetic theory coexisted throughmost of the 19th century. On one hand, the field point of view followed by Faraday, Maxwell,Thomson, Fitzgerald, and Larmor who considered the fields (and their “observable” lines offorce) to be the primary objects of the theory. They viewed the charges as manifestationsof the terminal points of lines of force having no independent or substantial existence [17].On the other hand, the charge-interaction point of view followed by Ampere, Neumann andWeber who considered the electromagnetic phenomena as originating in the interaction ofcharges which was regarded by some as mediated by an ethereal continuum and by others asa direct action at a distance. In the charge-interaction view, charges and currents were theprimary objects of the theory and the causes of the forces. Modern field theory of the 20thcentury resulted from a combination of these two point of views. Charges were regardedas sources of the electromagnetic forces (this was the legacy of the charge-interaction pointof view) which were considered as propagating through the Maxwellian field (this was thelegacy of the field point of view).The existence of a causal relation between sources and fields is now universally acceptedin electromagnetism: ρ and J (causes) produce the fields E and B (effects). This relationis clearly expressed by the retarded solutions of Maxwell’s equations in the form of time-dependent extensions of the Coulomb and Biot-Savart laws (in Gaussian units) [3]: E ( x , t ) = Z d x ′ ρ ( x ′ , t − R/c ) ˆ R R + ∂∂t Z d x ′ (cid:18) ρ ( x ′ , t − R/c ) ˆ R Rc − J ( x ′ , t − R/c ) Rc (cid:19) , (41) B ( x , t ) = Z d x ′ J ( x ′ , t − R/c ) × ˆ R R c + ∂∂t Z d x ′ J ( x ′ , t − R/c ) × ˆ R Rc . (42)According to these equations the values of E and B at the instant t are determined by thevalues of ρ and J at the retarded (previous) time t − R/c . This result clearly agrees withthe principle of causality which roughly speaking says that the causes (in this case ρ and J )precede in time to their effects (in this case E and B ). Traditionally we place the causes onthe right-hand side of equations and the effects on their left-hand side.The principle of causality provides then a reasonable argument to elucidate what are thecauses and what are the effects in electromagnetism. Who were first? By looking at Eqs. (41)and (42), our answer is straightforward: ρ and J occurred first and therefore they should beconsidered the causes of E and B . This interpretation would seem incontrovertible —andso the supposed egg-hen problem involved in the source-field relation would be an illusion—11xcept because there exist also the so-called advanced solutions of Maxwell’s equations,which can be written as Z d x ′ ρ ( x ′ , t + R/c ) ˆ R R − ∂∂t Z d x ′ (cid:18) ρ ( x ′ , t + R/c ) ˆ R Rc + J ( x ′ , t + R/c ) Rc (cid:19) = E ( x , t ) , (43) Z d x ′ J ( x ′ , t + R/c ) × ˆ R R c − ∂∂t Z d x ′ J ( x ′ , t + R/c ) × ˆ R Rc = B ( x , t ) . (44)According to these equations the values of E and B at the instant t are related with thevalues of ρ and J at the advanced (posterior) time t + R/c . We again invoke the principle ofcausality to determine what are the causes and the effects in electromagnetism. Who werefirst? By looking at Eqs. (43) and (44) we note that E and B occurred first and thereforethey should be considered the causes of ρ and J . Furthermore, if we introduce the variable τ = t + R/c then Eqs. (43) and (44) can alternatively be written as Z d x ′ ρ ( x ′ , τ ) ˆ R R − ∂∂t Z d x ′ (cid:18) ρ ( x ′ , τ ) ˆ R Rc + J ( x ′ , τ ) Rc (cid:19) = E ( x , τ − R/c ) , (45) Z d x ′ J ( x ′ , τ ) × ˆ R R c − ∂∂t Z d x ′ J ( x ′ , τ ) × ˆ R Rc = B ( x , τ − R/c ) , (46)These equations say that the values of ρ and J at the instant τ are related with the valuesof E and B at the previous time τ − R/c . In the light of the causality principle, the advancedsolutions of Maxwell’s equations are consistent with the idea that E and B are the causesof ρ and J , but the retarded solutions of these same equations are consistent with the ideathat ρ and J are the causes of E and B . As above mentioned, the subtle relation betweensources and fields in electromagnetism is a little bit like the egg-hen problem.It can be argued, however, that the above analysis is purely formal and that other physicalconsiderations allow us to identify the causes and the effects in electromagnetism. Supposewe have an electron at rest which is associated with an electrostatic field. In this case,we have neither retarded nor advanced times and so we cannot clearly distinguish betweencauses and effects. Suppose now that the electron is accelerated by mechanical forces. Theresulting electric current obeys the continuity equation and the reason the charge moveshas nothing to do with electromagnetism. Experience shows that the electron radiates,according to Maxwell’s equations, and that this radiation carries of electromagnetic energy,which must come at the expense of the particle’s mechanical energy. The radiated energydetaches from the electron, propagates off to infinity never coming back and we can physicallydetect this electromagnetic energy. In this dynamical case the radiation fields clearly arise12s a consequence of the motion of charges which was originated by mechanical and notelectromagnetic means. This simple example shows that the continuity equation is validoutside the domain of Maxwell’s equations and therefore it should be regarded as a strongeraxiom.
7. Concluding remarks
R. G. Brown wrote [18]: “... observe that if we take the 4-divergence of both sides ofthe inhomogeneous Maxwell equations: ∂ α ∂ β F βα = µ ∂ α J α = 0 the left hand side vanishesbecause again, a symmetric differential operator is contracted with a completely antisym-metric field strength tensor. Thus ∂ α J α = 0, which, by some strange coincidence, is thecharge-current conservation equation in four dimensions. Do you get the feeling that some-thing very deep is going on? This is what I love about physics. Beautiful things are reallybeautiful!” I think there is no strange coincidence. This is what I love about physics! Thebeautiful equation ∂ α J α = 0 representing charge conservation is the fundamental symmetrybehind the inhomogeneous and homogeneous Maxwell equations. “That something verydeep” is just charge conservation. To consider the continuity equation as the basic axiomto build the set of equations known as Maxwell’s equations is then a very natural idea. Inthis paper we have followed this idea by obtaining the covariant form of Maxwell’s equationsfrom the covariant form of the continuity equation.It can be argued that other physical assumptions like the use of the Minkowski spacetimewith its speed of light c and its associated retarded Green function G are ingredients im-plicit in the covariant derivation of Maxwell’ equations presented here. However, if we firstpostulate the validity of the continuity equation in any four-space, the particular use of theMinkowski spacetime (together with its associated retarded Green function G and speed c )would not really be a new postulate, but merely an application of our initial postulate.We advocate an axiomatic presentation of Maxwell’s equations in undergraduate andgraduate courses of electromagnetism. According to Obradovic [19]: “The axiomaticmethod offers the shortest way to the essence of any theory, enables its more accurateformulation and more profound and complete interpretation.” In the case of Maxwell’sequations the essence is charge conservation which should be considered the fundamentalpostulate for an axiomatic presentation of these equations. In the practice, undergraduateinstructors can find the basis for this presentation in equations (4a) and (4b) of Ref. 2 and13raduate instructors in equation (7) of this paper. Acknowledgment
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