Helmholtz's theorem for two retarded fields and its application to Maxwell's equations
HHelmholtz’s theorem for two retarded fieldsand its application to Maxwell’s equations
Jos´e A. Heras and Ricardo Heras Instituto de Geof´ısica, Universidad Nacional Aut´onoma de M´exico, Ciudad de M´exico 04510,M´exico. E-mail: [email protected] Department of Physics and Astronomy, University College London, London WC1E 6BT, UK.E-mail: [email protected]
Abstract.
An extension of the Helmholtz theorem is proved, which states that two retardedvector fields F and F satisfying appropriate initial and boundary conditions are uniquelydetermined by specifying their divergences ∇ · F and ∇ · F and their coupled curls −∇ × F − ∂ F /∂ t and ∇ × F − (1 / c ) ∂ F /∂ t , where c is the propagation speed of the fields.When a corollary of this theorem is applied to Maxwell’s equations, the retarded electric andmagnetic fields are directly obtained. The proof of the theorem relies on a novel demonstrationof the uniqueness of the solutions of the vector wave equation.
1. Introduction
The mathematical foundations of electrostatics and magnetostatics relies on the Helmholtztheorem of vector analysis. Among the several equivalent formulations of this theorem presentedin standard textbooks [1, 2, 3, 4], let us consider the formulation given by Gri ffi ths [1]. Thetheorem states that if the divergence ∇ · F = D ( r ) and the curl ∇ × F = C ( r ) of a vector function F ( r ) are specified, and if they both go to zero faster than 1 / r as r → ∞ , and if F ( r ) goes tozero as r → ∞ , then F is uniquely given by F = −∇ U + ∇ × W , where U ( r ) ≡ π (cid:90) D ( r (cid:48) ) R d r (cid:48) , W ( r ) ≡ π (cid:90) C ( r (cid:48) ) R d r (cid:48) . (1)Here r is the field point and r = | r | is its magnitude, r (cid:48) is the source point and R = | r − r (cid:48) | . Theintegrals are over all space and d r (cid:48) is the volume element. This theorem has a useful corollary:Any vector function F ( r ) that goes to zero faster than 1 / r as r → ∞ can be expressed as F ( r ) = −∇ (cid:18) π (cid:90) ∇ (cid:48) · F ( r (cid:48) ) R d r (cid:48) (cid:19) + ∇ × (cid:18) π (cid:90) ∇ (cid:48) × F ( r (cid:48) ) R d r (cid:48) (cid:19) . (2)In a first application, we write F = E and use the electrostatic equations: ∇ · E = ρ/(cid:15) and ∇ × E = E ( r ) produced by the charge density ρ ( r ): E ( r ) = −∇ (cid:18) π(cid:15) (cid:90) ρ ( r (cid:48) ) R d r (cid:48) (cid:19) , (3) a r X i v : . [ phy s i c s . c l a ss - ph ] J un A Heras and R Heras 2or equivalently E = −∇ Φ , where Φ is the scalar potential: Φ ( r ) = π(cid:15) (cid:90) ρ ( r (cid:48) ) R d r (cid:48) . (4)In a second application we make F = B and use the magnetostatic equations: ∇ · B = ∇ × B = µ J to obtain the magnetostatic field B ( r ) produced by the current density J ( r ): B ( r ) = ∇ × (cid:18) µ π (cid:90) J ( r (cid:48) ) R d r (cid:48) (cid:19) , (5)or equivalently B = ∇ × A , where A is the vector potential: A ( r ) = µ π (cid:90) J ( r (cid:48) ) R d r (cid:48) . (6)Here (cid:15) and µ are the permittivity and permeability of vacuum which satisfy (cid:15) µ = / c , with c being the speed of light in vacuum.The Helmholtz theorem is also applicable to the time-dependent regime of Maxwell’sequations [5, 6]. The reason is simple: the derivation of (2) does not involve time and thereforeit can be applied to a time-dependent vector field F ( r , t ). We simply make the replacements F ( r ) → F ( r , t ) and F ( r (cid:48) ) → F ( r (cid:48) , t ) in (2) and obtain an instantaneous form of the theorem [6]: F ( r , t ) = −∇ (cid:18) π (cid:90) ∇ (cid:48) · F ( r (cid:48) , t ) R d r (cid:48) (cid:19) + ∇ × (cid:18) π (cid:90) ∇ (cid:48) × F ( r (cid:48) , t ) R d r (cid:48) (cid:19) . (7)However, this form of the theorem is of limited practical utility in the time-dependent regime ofMaxwell’s equations because the Faraday law and the Amp`ere-Maxwell law in this regime donot specify the curls of the fields E and B as such in terms of sources, rather they specify the hybrid quantities ∇× E + ∂ B /∂ t and ∇× B − (1 / c ) ∂ E /∂ t , which couple space and time variationsof both fields. In the more general case in which there are magnetic monopoles, the Faradayand Amp`ere-Maxwell laws read −∇ × E − ∂ B /∂ t = µ J g and ∇ × B − (1 / c ) ∂ E /∂ t = µ J e ,where J g is the magnetic current density and J e is the electric current density. As may be seen,these laws connect the hybrid quantities −∇ × E − ∂ B /∂ t and ∇ × B − (1 / c ) ∂ E /∂ t with theirrespective electric and magnetic currents. It is clear that the time derivatives of the fields coupletheir corresponding curls. Let us call the first hybrid quantity the coupled curl of E and thesecond quantity the coupled curl of B .It would be desirable to have an extension of the Helmholtz theorem which allows us todirectly find the retarded electric and magnetic fields in terms of the retarded scalar and vectorpotentials, in the same form that the standard Helmholtz theorem allows us to directly findthe electrostatic and magnetostatic fields in terms of their respective static scalar and vectorpotentials. Expectably, this appropriate generalisation of the theorem should be formulated fortwo retarded vector fields and in terms of their respective divergences and coupled curls.In this paper we formulate a generalisation of the Helmholtz theorem for two retardedvector functions. We show how a corollary of this theorem allows us to find the retarded electricand magnetic fields and introduce the corresponding retarded potentials in three specific cases:for Maxwell equations with the electric charge and current densities (the standard case), whenthese equations additionally have polarisation and magnetisation densities and finally when A Heras and R Heras 3they additionally have magnetic charge and current densities. The proof of the theorem relieson the uniqueness of the solutions of the vector wave equation. A novel demonstration ofthis uniqueness is presented in the Appendix A. Although more complicated than the standardHelmholtz theorem, the extension of this theorem for two retarded vector fields formulated heremay be presented in an advanced undergraduate course of electrodynamics.
2. The Helmholtz theorem for two retarded fields
We begin by defining a retarded function as one function of space and time whose sources areevaluated at the retarded time. For example, the vector F ( r , t ) = (cid:82) g ( r (cid:48) , t − R / c ) d r (cid:48) is a retardedfield because its vector source g is evaluated at the retarded time t − R / c . In order to simplifythe notation, we will use the retardation brackets [ ] to indicate that the enclosed quantity is tobe evaluated at the source point r (cid:48) and at the retarded time t − R / c , that is, [ F ] = F ( r (cid:48) , t − R / c ).For example, F = (cid:82) [ g ] d r (cid:48) denotes a retarded quantity.Suppose we are told that the divergences ∇ · F and ∇ · F of two retarded vector functions F = F ( r , t ) and F = F ( r , t ) are specified by the time-dependent scalar functions D = D ( r , t )and D = D ( r , t ): ∇ · F = D , ∇ · F = D , (8)and that the coupled curls −∇ × F − ∂ F /∂ t and ∇ × F − (1 / c ) ∂ F /∂ t are specified by thetime-dependent vector functions C = C ( r , t ) and C = C ( r , t ): −∇ × F − ∂ F ∂ t = C , ∇ × F − c ∂ F ∂ t = C , (9)where c is the propagation speed of the fields F and F . † For consistence, C , C , D and D must satisfy the “continuity” equations: ∇ · C + c ∂ D ∂ t = , ∇ · C + ∂ D ∂ t = , (10)which are implied by (8) and (9). Question : Can we, on the basis of this information, uniquelydetermine F and F ? If D , D , C and C go to zero su ffi ciently rapidly at infinity, the answeris yes , as we will show by explicit construction. ‡ We claim that the solution of (8) and (9) is F = −∇ U − ∇ × W − ∂ W ∂ t , F = −∇ U + ∇ × W − c ∂ W ∂ t , (11) † Here the speed c does not necessarily have to represent the speed of light. However, if c is identified with thespeed of light in vacuum and the following identifications: F = E , F = B , D = ρ/(cid:15) , D = , C = µ J and C = (cid:15) µ = / c are made, then (8) and (9) become Maxwell’s equations in SI units. Analogously, if theidentification F = c E , F = B , D = π c ρ, D = , C = (4 π/ c ) J and C = ‡ The answer is also yes for the case in which the quantities D , D , C and C are localised in space, that is,when they are zero outside a finite region of space. This frequently occurs in practical applications because thesequantities usually play the role of sources, which are physically localised in space. A Heras and R Heras 4where U = π (cid:90) [ D ] R d r (cid:48) , U = π (cid:90) [ D ] R ] d r (cid:48) , (12)and W = π (cid:90) [ C ] R d r (cid:48) , W = π (cid:90) [ C ] R d r (cid:48) . (13)The integrals are over all space. Our demonstration requires two additional sets of equations.The first set is formed by the “Lorenz” conditions: ∇ · W + ∂ U ∂ t = , ∇ · W + c ∂ U ∂ t = . (14)We will prove the first condition using the result [2]: ∇ · ([ F ] / R ) + ∇ (cid:48) · ([ F ] / R ) = [ ∇ (cid:48) · F ] / R , ∇ · W = π (cid:90) ∇ · (cid:18) [ C ] R d r (cid:48) (cid:19) = π (cid:90) [ ∇ (cid:48) · C ] R d r (cid:48) − π (cid:90) ∇ (cid:48) · (cid:18) [ C ] R (cid:19) d r (cid:48) = − ∂∂ t (cid:18) π (cid:90) [ D ] R d r (cid:48) (cid:19) − π (cid:73) ˆ n · [ C ] R dS = − ∂ U ∂ t , (15)where we have used the second equation given in (10), the property [7]: [ ∂ F /∂ t ] = ∂ [ F ] /∂ t ,the Gauss theorem to transform the second volume integral of the second line into a surfaceintegral ( dS is the surface element), which is seen to vanish at infinity by assuming the boundarycondition that C goes to zero faster than 1 / r as r → ∞ , and finally considering the secondequation in (12). § Following a similar procedure, we can prove the second “Lorenz” conditiongiven in (14). The required second set of equations is formed by the wave equations (cid:3) U = − D , (cid:3) U = − D , (cid:3) W = − C , (cid:3) W = − C , (16)where (cid:3) ≡ ∇ − (1 / c ) ∂ /∂ t is the d’Alembertian operator. Using the result [7]: (cid:3) (cid:18) [ X ] R (cid:19) = − π [ X ] δ ( r − r (cid:48) ) , (17)where X represents a scalar or vector function, (cid:107) we can show the first wave equation in (16), (cid:3) U = π (cid:90) (cid:3) (cid:18) [ D ] R (cid:19) d r (cid:48) = − (cid:90) [ D ] δ ( r − r (cid:48) ) d r (cid:48) = − D . (18)By an entirely similar procedure, we can show the remaining wave equations given in (16).Considering (11), (14) and (16), we can now prove that F and F satisfy (8) and (9). We take § In physical applications, the vector C usually represents a localised quantity. This means that C is di ff erentfrom zero only within a limited region of space. However, the surface of integration S in the surface integral onthe third line of (15) encloses all space, and therefore S is outside the region where C is di ff erent from zero. Itfollows that C is zero everywhere on S and then the surface integral vanishes. (cid:107) This identity is true for functions X such that the quantities [ X ] / R have not the form [ X ] / R = f ( R )[ F ]with f ( R ) being a polynomial function. If for example f ( R ) = R then [ X ] / R = R [ F ]. It follows that (cid:3) ( R [ F ]) = − π R [ F ] δ ( r − r (cid:48) ) = R δ ( r − r (cid:48) ) vanishes for r (cid:44) r (cid:48) because of the delta function andalso for r = r (cid:48) because this equality implies R = A Heras and R Heras 5the divergences to F and F and obtain ∇ · F = −∇ U − ∂∂ t ∇ · W = −∇ U + c ∂ U ∂ t = − (cid:3) U = D , (19) ∇ · F = −∇ U − c ∂∂ t ∇ · W = −∇ U + c ∂ U ∂ t = − (cid:3) U = D . (20)We now calculate the coupled curls of F and F and obtain −∇ × F − ∂ F ∂ t = ∇ ( ∇ · W ) − ∇ W + ∂∂ t ∇ × W + ∇ (cid:18) ∂ U ∂ t (cid:19) − ∇ × ∂ W ∂ t + c ∂ W ∂ t = ∇ (cid:18) ∇ · W + ∂ U ∂ t (cid:19) − (cid:3) W = − (cid:3) W = C , (21) ∇ × F − ∂ F ∂ t = ∇ ( ∇ · W ) − ∇ W + ∂∂ t ∇ × W + ∇ (cid:18) ∂ U ∂ t (cid:19) − ∇ × ∂ W ∂ t + c ∂ W ∂ t = ∇ (cid:18) ∇ · W + ∂ U ∂ t (cid:19) − (cid:3) W = − (cid:3) W = C . (22)Therefore the functions F and F given in (11) constitute a solution of (8) and (9).Now, assuming that the conditions on D , D , C and C are met, is the solution in (11) unique ? The answer is clearly no , for we can add to F the function H and F the function H , with the divergences and coupled curls of H and H being zero, and the result stillhas divergences D and D and coupled curls C and C . However, it so happens that if F , F , ∂ F /∂ t and ∂ F /∂ t vanish at t =
0, and if ∇ F , ∇ F , ∂ F /∂ t and ∂ F /∂ t go to zero fasterthan 1 / r as r → ∞ , and if F and F themselves go to zero as r → ∞ , then H and H are zeroand therefore the solution (11) is unique. In the notation we are using a generic second-ordertensor ∇ F denotes the gradient of the vector F . The explicit proof that H and H are zerounder their specified initial and boundary conditions is presented in Appendix A. Notice that theconditions that ∇ F , ∇ F , ∂ F /∂ t and ∂ F /∂ t go to zero faster than 1 / r as r → ∞ automatically imply that D , D , C and C go to zero faster than 1 / r as r → ∞ .Now we can state the Helmholtz theorem for two retarded fields more rigorously: Theorem.
If the divergences D and D and the coupled curls C and C of two retardedvector functions F and F are specified, and if F , F , ∂ F /∂ t and ∂ F /∂ t vanish at t =
0, and if ∇ F , ∇ F , ∂ F /∂ t and ∂ F /∂ t go to zero faster than 1 / r as r → ∞ , and if F and F themselvesgo to zero as r → ∞ , then F and F are uniquely given by (11).This theorem has the following corollary: The fields F and F can be expressed as F = − ∇ (cid:26) (cid:90) [ ∇ (cid:48) · F ]4 π R d r (cid:48) (cid:27) − ∇ × (cid:26) (cid:90) [ −∇ (cid:48) × F − ∂ F /∂ t ]4 π R d r (cid:48) (cid:27) − ∂∂ t (cid:26) (cid:90) [ ∇ (cid:48) × F − (1 / c ) ∂ F /∂ t ]4 π R d r (cid:48) (cid:27) , (23) A Heras and R Heras 6 F = − ∇ (cid:26) (cid:90) [ ∇ (cid:48) · F ]4 π R d r (cid:48) (cid:27) + ∇ × (cid:26) (cid:90) [ ∇ (cid:48) × F − (1 / c ) ∂ F /∂ t ]4 π R d r (cid:48) (cid:27) − ∂∂ t (cid:26) (cid:90) [ −∇ (cid:48) × F − ∂ F /∂ t ]4 π R d r (cid:48) (cid:27) . (24)These expressions of the Helmholtz theorem for two retarded vector functions are considerablymore complicated than the expression of the Helmholtz theorem (2) for a static vector function.However, the practical advantage (23) and (24) is that they allows us to directly find the retardedfields of Maxwell’s equations, as we will see in the next section.
3. Applications
In a first application of the corollary expressed in (23) and (24), we make the identifications: F = E and F = B in (23) and (24) and subsequently use the Maxwell equations in SI units ∇ · E = (cid:15) ρ, ∇ · B = , (25) ∇ × E + ∂ B ∂ t = , ∇ × B − c ∂ E ∂ t = µ J , (26)to directly obtain the familiar retarded fields E = − ∇ (cid:18) π(cid:15) (cid:90) [ ρ ] R d r (cid:48) (cid:19) − ∂∂ t (cid:18) µ π (cid:90) [ J ] R d r (cid:48) (cid:19) , (27) B = ∇ × (cid:18) µ π (cid:90) [ J ] R d r (cid:48) (cid:19) , (28)or more compactly, E = −∇ Φ − ∂ A ∂ t , B = ∇ × A , (29)where we have defined the retarded scalar and vector potentials as Φ = π(cid:15) (cid:90) [ ρ ] R d r (cid:48) , A = µ π (cid:90) [ J ] R d r (cid:48) . (30)Here c is the speed of light in vacuum and is defined by c = / √ (cid:15) µ . ¶ ¶ It is interesting to note that (23) and (24) can be used to find the retarded fields of the gravitational theorysuggested by Heaviside [8], which is similar in form to that of Maxwell. If we write F = g and F = k , where thetime-dependent vector fields g and k are respectively the gravitoelectric and gravitomagnetic fields, use equations(23) and (24), and consider the gravitational equations in SI units: ∇ · g = − ρ g / ˜ (cid:15) , ∇ · k = , ∇ × g + ∂ k /∂ t = , ∇ × k − (1 / c ) ∂ g /∂ t = − ˜ µ J g , where ρ g and J g are the mass density and the mass current density, which satisfythe continuity equation ∇ · J g + ∂ρ g /∂ t = g = −∇ V g − ∂ A g /∂ t and k = ∇ × A g , where the retarded scalar and vector potentials are V g = − π ˜ (cid:15) (cid:90) [ ρ g ] R d r (cid:48) , A g = − ˜ µ π (cid:90) [ J g ] R d r (cid:48) . Here the gravitational permittivity constant of vacuum is defined as ˜ (cid:15) = / (4 π G), where G is the universalgravitational constant. The gravitational permeability constant of vacuum is defined as ˜ µ = π S / c . It followsthat ˜ (cid:15) ˜ µ = / c , where c is the speed of light in vacuum. Jefimenko [9], McDonald [10] and recently, Vieira andBrentan [11] have discussed this gravitational theory, whose equations have also been obtained using an alternativeapproach [12]. A Heras and R Heras 7In a second application of (23) and (24) , we make the identifications: F = E and F = B in (23) and (24) and subsequently use the Maxwell equations with material sources (SI units): ∇ · E = (cid:15) ( ρ − ∇ · P ) , ∇ · B = , (31) ∇ × E + ∂ B ∂ t = , ∇ × B − c ∂ E ∂ t = µ (cid:18) J + ∇ × M + ∂ P ∂ t (cid:19) , (32)where P and M are the polarisation and magnetisation vectors, to get the retarded fields [12]: E = − ∇ (cid:18) π(cid:15) (cid:90) [ ρ − ∇ (cid:48) · P ] R d r (cid:48) (cid:19) − ∂∂ t (cid:18) µ π (cid:90) [ J + ∇ (cid:48) × M + ∂ P /∂ t ] R d r (cid:48) (cid:19) , (33) B = ∇ × (cid:18) µ π (cid:90) [ J + ∇ (cid:48) × M + ∂ P /∂ t ] R d r (cid:48) (cid:19) , (34)or more compactly, E = −∇ Φ − ∂ A ∂ t , B = ∇ × A , (35)where we have defined the retarded scalar and vector potentials as Φ = π(cid:15) (cid:90) [ ρ − ∇ (cid:48) · P ] R d r (cid:48) , A = µ π (cid:90) [ J + ∇ (cid:48) × M + ∂ P /∂ t ] R d r (cid:48) . (36)Notice that if the vectors P and M are vanished in (33) and (34) then we recover (27) and (28).In the last application we write F = c E and F = B in (23) and (24), and subsequentlyuse Maxwell’s equations with magnetic monopoles expressed in Gaussian units as ∇ · c E = π c ρ e , ∇ · B = πρ m , (37) −∇ × c E − ∂ B ∂ t = π J m , ∇ × B − c ∂ E ∂ t = π c J e , (38)where ρ e and J e are the electric charge and current densities and ρ m and J m are the magneticcharge and current densities, to directly obtain the retarded fields with magnetic monopoles E = −∇ (cid:18) (cid:90) [ ρ e ] R d r (cid:48) (cid:19) − ∇ × (cid:18) (cid:90) [ J m ] Rc d r (cid:48) (cid:19) − c ∂∂ t (cid:18) (cid:90) [ J e ] Rc d r (cid:48) (cid:19) , (39) B = −∇ (cid:18) (cid:90) [ ρ m ] R d r (cid:48) (cid:19) + ∇ × (cid:18) (cid:90) [ J e ] Rc d r (cid:48) (cid:19) − c ∂∂ t (cid:18) (cid:90) [ J m ] Rc d r (cid:48) (cid:19) , (40)or more compactly: E = −∇ Φ e − ∇ × A m − c ∂ A e ∂ t , B = −∇ Φ m + ∇ × A e − c ∂ A m ∂ t , (41)where we have defined the retarded electric and magnetic scalar potentials as Φ e = (cid:90) [ ρ e ] R d r (cid:48) , Φ m = (cid:90) [ ρ m R d r (cid:48) , (42)and the retarded electric and magnetic vector potentials as A e = (cid:90) [ J e ] Rc d r (cid:48) , A m = (cid:90) [ J m ] Rc d r (cid:48) . (43)Notice that if the magnetic densities ρ m and J m are vanished in (39) and (40) then we recover(27) and (28). A Heras and R Heras 8
4. Pedagogical comment
Standard textbook presentations of electromagnetism follow a di ff erent route to the electricand magnetic fields in the time-independent regime of Maxwell’s equations than in the time-dependent regime of these equations. While in the time-independent regime, the Helmholtztheorem of the vector analysis is commonly applied to find the electrostatic and magnetostaticfields in terms of their respective scalar and vector potentials, in the time-dependent regimeMaxwell’s sourceless equations are commonly used to introduce the scalar and vector potentialswhich are then inserted into Maxwell’s source equations, obtaining two coupled second-order equations involving potentials, which are shown to be gauge invariant. The Lorenz-gauge condition is usually adopted to decouple these second-order equations and as a result,we arrive at two wave equations, which are solved to obtain the retarded scalar and vectorpotentials and by a subsequent di ff erentiation of them we finally obtain the retarded electricand magnetic fields. It is evident that the usual method followed in the time-dependent regimeis somewhat more complicated than the usual method used in the time-independent regimewhich is based in the Helmholtz theorem. It is clear that for pedagogical reasons it is worthformulating an extension of the Helmholtz theorem, which may be useful in the time-dependentregime of Maxwell’s equations. This has been made in the past and di ff erent extensions ofthe Helmholtz theorem to include the time-dependence of the fields have been formulated[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. But, as far as we are aware, noneof these extensions of the Helmholtz theorem has yet been included in standard textbooks.In our opinion, equations (23) and (24) for the retarded fields F and F could be includedin a undergraduate presentation of Maxwell’s equations. Similarly, the analogous equation butfor a single retarded field [18]: F = −∇ (cid:90) [ ∇ (cid:48) · F ]4 π R d r (cid:48) + ∇× (cid:90) [ ∇ (cid:48) × F ]4 π R d r (cid:48) + c ∂∂ t (cid:90) [ ∂ F /∂ t ]4 π R d r (cid:48) , (44)could alternatively be included in such an undergraduate presentation. We notice that ifany of the fields F and F defined by (23) and (24) is vanished then we obtain (44). Thedisadvantage of (23) and (24) is that they are somewhat complicated but their advantage is thatthey directly yield the retarded fields. In contrast, (44) has the advantage of being simpler butthe disadvantage is that it yields the retarded fields in an indirect way.
5. Conclusion
How to solve the time-independent Maxwell’s equations? Answer: using the Helmhotz theoremof the vector analysis. How to solve the time-dependent Maxwell’s equations? Answer: usingthe Helmholtz theorem for two retarded fields, which was formulated in this paper. The key to formulate this generalised theorem was the observation that the curls in the first regimeare decoupled quantities: ∇ × E = ∇ × B = µ J , while the curls in the second regimeare coupled quantities: ∇ × E + ∂ B /∂ t = ∇ × B − (1 / c ) ∂ E /∂ t = µ J . Accordingly,we formulated the Helmholtz theorem for two retarded vectors F and F in terms of their A Heras and R Heras 9divergences: ∇ · F and ∇ · F and coupled curls: −∇ × F − ∂ F /∂ t and ∇ × F − (1 / c ) ∂ F /∂ t .The proof of the theorem required of the uniqueness of the solutions of the homogeneous waveequation, which was explicitly demonstrated in Appendix A. As applications, we applied thetheorem to Maxwell’s equations when they have electric charge and current densities, whenthey additionally have polarisation and magnetisation densities and when they additionally havemagnetic charge and current densities. For each case we obtained the retarded fields in terms oftheir retarded potentials. Standard pedagogy generally considers the Helmholtz theorem for astatic field to justify the mathematical form of Maxwell’s equations in the time-independentregime. Standard pedagogy might also consider the Helmholtz theorem for two retarded vectorfields to justify the mathematical form of Maxwell’s equations in the time-dependent regime. Appendix A. Uniqueness of the solutions of the wave equation
We have shown that the set of functions F and F given (11) satisfies (8) and (9). We have alsopointed out that this set of functions is not generally unique and that the set formed by F + H and F + H also satisfies Eqs. (8) and (9) provided the functions H and H satisfy ∇ · H = , ∇ · H = , (.1) −∇ × H − ∂ H ∂ t = , ∇ × H − c ∂ H ∂ t = . (.2)The uniqueness of F and F will be guaranteed if we are able to show that H and H vanisheverywhere. From (A.1) and (A.2) we obtain the homogeneous wave equations (cid:3) H = (cid:3) H = . Our strategy is now to construct a relation that allows us to discover those initialand boundary conditions that guarantee that the only solutions of these wave equations are thetrivial ones: H = H = . We will see that these conditions are precisely those given inthe formulation of the Helmholtz theorem for two retarded vector fields.In order to proceed in a more rigorous way, we adopt the following notation: the Cartesiancomponents of ∇ F are given by ( ∇ F ) i j = ∂ i F j . Latin indices i , j , k ... run from 1 to 3 and thesummation convention a i a i on repeated indices (one covariant and the other one contra-variant)is adopted. The normal derivative of F at the surface S (directed outwards from inside thevolume V ) is denoted by the vector n ·∇ F and defined by its components as ( n ·∇ F ) i = n j ∂ j F i where n is a unit vector outward to the surface with ( n ) j = n j . Now we can state the followinguniqueness theorem for the solutions of the homogeneous vector wave equation:
Theorem.
If the vector function F ( r , t ) satisfies the homogeneous wave equation (cid:3) F = F and ∂ F /∂ t vanish at t =
0, and the boundary condition that n ·∇ F is zero at the surface S of the volume V , then F ( r , t ) is identically zero.Proof. We write the wave equation (cid:3) F = ∂ j ∂ j F i − c ∂ F i ∂ t = . (.3)We multiply this equation by ∂ F i /∂ t and obtain ∂ F i ∂ t (cid:18) ∂ j ∂ j F i − c ∂ F i ∂ t (cid:19) = . (.4) A Heras and R Heras 10The left-hand side can be re-written as ∂ j (cid:18) ∂ F i ∂ t ∂ j F i (cid:19) − ∂∂ t (cid:26) ∂ j F i ∂ j F i + c ∂ F i ∂ t ∂ F i ∂ t (cid:27) = . (.5)The volume integration of this equation implies ∂∂ t (cid:90) V (cid:18) ∂ j F i ∂ j F i + c ∂ F i ∂ t ∂ F i ∂ t (cid:19) d r = (cid:90) V ∂ j (cid:18) ∂ F i ∂ t ∂ j F i (cid:19) d r . (.6)The volume integral on the right-side can be transformed into a surface integral, ∂∂ t (cid:90) V (cid:18) ∂ j F i ∂ j F i + c ∂ F i ∂ t ∂ F i ∂ t (cid:19) d r = (cid:73) S (cid:18) ∂ F i ∂ t (cid:19)(cid:16) n j ∂ j F i (cid:17) dS . (.7)From the boundary condition that n j ∂ j F i is zero at S it follows that ∂∂ t (cid:90) V (cid:18) ∂ j F i ∂ j F i + c ∂ F i ∂ t ∂ F i ∂ t (cid:19) d r = , (.8)and therefore the integral is at most a function of space g ( x i ). The initial condition that F i iszero at t = ∂ j F i is zero at t =
0, which is used together with thecondition that ∂ F i /∂ t is zero at t = g ( x i ) =
0. Accordingly, (cid:90) V (cid:18) ∂ j F i ∂ j F i + c ∂ i F i ∂ t ∂ F i ∂ t (cid:19) d r = , (.9)or in the more familiar vector notation (cid:90) V (cid:18) ( ∇ F ) + c ( ∂ F /∂ t ) (cid:19) d r = . (.10)Since the volume V is arbitrary the integrand must vanish and considering that F is a realfunction it follows that ∇ F = ∂ F /∂ t =
0, which imply that F = constant. Using thecondition that F is zero at t =
0, it follows that this constant vanishes and then F =
0, whichproves the theorem.The above theorem clearly describes a uniqueness theorem for the solutions of thehomogeneous wave equation. Suppose we have two vector fields F and F that satisfy the same wave equation (cid:3) F = (cid:3) F =
0, the same initial conditions that F , F , ∂ F /∂ t and ∂ F /∂ t vanish at t =
0, and the same boundary conditions that n ·∇ F and n ·∇ F are zeroat the surface S of the volume V . Now, let us write F = F − F . It follows that F satisfies (cid:3) F =
0, the initial conditions that F and ∂ F /∂ t vanish at t =
0, and the boundary conditionthat n · ∇ F zero at the surface S . By the uniqueness theorem we have F = F = F , i.e., the solution is unique.In particular, if the surface goes to infinity then we can assume the boundary conditionsthat ∇ F and ∂ F /∂ t go to zero faster than 1 / r as r → ∞ . Under these conditions, the surfaceintegral in (A.7) vanishes. In this case we also arrive at (A.8) but with the volume V extendedover all space. Using the initial conditions that F and ∂ F /∂ t vanish at t = V extended over all space and therefore F = ∇ F and ∂ F /∂ t are zero in the limit r → ∞ . But F = F = F , i.e., the solution is unique.We now return to the solutions F + H and F + H of (8) and (9). As previously noted: (cid:3) H = (cid:3) H =
0. Therefore we can apply the uniqueness theorem to both H and H A Heras and R Heras 11by assuming the initial conditions that H , H , ∂ H /∂ t , ∂ H /∂ t vanish at t =
0, the boundaryconditions that ∇ H , ∇ H , ∂ H /∂ t and ∂ H /∂ t go to zero faster than 1 / r as r → ∞ , andthe boundary conditions that H and H go to zero as r → ∞ . Under these conditions, theuniqueness theorem states that H = H = . Therefore, the uniqueness of the vector functions F and F of the Helmholtz theoremfor two retarded fields is guaranteed by assuming the initial conditions that F , F , ∂ F /∂ t and ∂ F /∂ t vanish at t =
0, the boundary conditions that ∇ F , ∇ F , ∂ F /∂ t and ∂ F /∂ t go tozero faster than 1 / r as r → ∞ , and the boundary conditions that F and F go to zero as r → ∞ . These boundary conditions imply that the quantities D , D , C and C , considered inthe Helmholtz theorem for two retarded fields, go to zero faster than 1 / r as r → ∞ . References [1] Gri ffi ths D 1999 Introduction to Electrodynamics
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