A SYMMETRIC FORMULATION OF MAXWELL'S ...

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Published in Modern Phys. Letters A, Vol 12, No. 28 (1997) 2089-2101

A SYMMETRIC FORMULATION OF MAXWELL'S EQUATIONS Héctor A. Múnera1 Centro Internacional de Física Bogotá D.C., COLOMBIA (South America) and Octavio Guzmán Departamento de Física, Universidad Nacional Bogotá D.C., COLOMBIA (South America)

ABSTRACT The electromagnetic field formed by the pair (E,B) obeying Maxwell's equations (MEs) is reformulated as another pair (P, N) obeying a symmetric set of four equations tautologically equivalent to (MEs). The symmetric set is formed by a pair of induction and a pair of source equations, each pair with exactly the same structure. In contrast to (E,B), charge and current densities contributes equally to both P and N. The equation of continuity is not an independent condition, but it is automatically fulfilled by any set of solutions (P,N,J,) to the symmetric MEs. The symmetric equations in terms of potentials may be explicitly solved for a variety of constraint conditions, thus leading to different classes of solutions. Each class represents a family of physical problems defined by the constraints. One such family is the conventional class of solutions of MEs. Some unexpected results regarding the conventional solutions of MEs in terms of potentials are: (1) it is a particular case, (2) it may contain magnetic scalar potentials, and (3) there is no Coulomb gauge, thus removing the magnetic transversality constraint. It is not known whether the other classes of solutions correspond to physical problems. KEY WORDS: Maxwell's equations, symmetry, monopole, magnetic scalar potential, wave equations, Coulomb gauge, electromagnetic charge, (non)transversality of vector potential. _________ 1

Corresponding author: [email protected].

A symmetric formulation of Maxwell's equations

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A SYMMETRIC FORMULATION OF MAXWELL'S EQUATIONS

I. INTRODUCTION We have recently obtained new solutions for the homogeneous wave equation. (1,2) In considering the applicability of the new functions for solving the wave equations (WEs) associated with Maxwell equations (MEs), we revisited the conventional solution of MEs in terms of potentials and found some interesting surprises. (3) In the process, it became clear that the high symmetry of MEs could be made explicit by a simple trick. The set of four Maxwell's equations (MEs) in CGS units is E   Bw B  + Ew  4J/c E  4 B  0

(4)

(1a) (1b) (1c) (1d)

where time is the geometric variable w  ct. Other symbols and dimensions are: electric and magnetic field, E, B [= dyne esu-1[= esu cm-2, current density J [= esu sec-1cm-2, charge density [= esu cm-3. The set of MEs may be grouped into two pairs of similar structure: two induction equations (Faraday's law eq. 1a and Ampere's law eq. 1b), and two source equations (Coulomb's law eq. 1c, and the magnetic source eq.1d). Many people in the past wondered why each pair does not have exactly the same structure. For instance, in order to make (1d) exactly alike to (1c), Dirac suggested the existence of magnetic sources (monopoles). However, monopoles have never been convincingly observed. Within the previous context, we exhibit here sets of symmetric Maxwell's equations (SMEs) that are completely equivalent to the conventional set (1). In contrast to Dirac's suggestion, a separate magnetic source is not required. The paper is organized as follows. Section II introduces SMEs in terms of fields. Wave equations for the fields are discussed in Section III. Section IV formulates the SMEs in terms of vector and scalar potentials, and section V presents particular solutions, that include the conventional case. Implicit assumptions in the standard representation of E and B are uncovered. Section VI closes the paper.

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II. SYMMETRIC FORMULATIONS OF MAXWELL EQUATIONS Let us accept that the pair of electromagnetic fields (E,B) fulfilling MEs (1) correspond to some physical reality. The same physical entity may be represented by another pair of electromagnetic fields (P,N), obeying the SMEs (2): Induction equations: P   Nw  4J/c N  Pw  4J/c

(2a) (2b)

Source equations: P  4 N  4

(2c) (2d).

Table 1 exhibits another equivalent set of SMEs (P,M), where both source terms are positive, and M =  N. Their properties are alike to those of the set (P,N); no further mention will be made of this representation. For the time being, no special meaning is attached to vector fields (P,N). The symmetric fields are related to the standard electromagnetic fields E and B by PE+B NBE

(3a) (3b).

Then, if we know a solution (P, N) for the set of SMEs, the standard pair (E, B) is E = (P  N)/2 B = (P  N)/2

(4a) (4b).

To be formal let us state the following two propositions: Proposition 1. The SMEs (2) with fields (P, N) defined by eqs. (3) is equivalent to the set of MEs (1).  Proof. Substitute eqs.(3a) and (3b) in all four equations (2). The four equations in set (1) are obtained as follows: Faraday's law (eq. 1a): (2a)  (2b) Ampere's law (eq. 1b): (2a) + (2b) Coulomb's law (eq. 1c): (2c)  (2d) Magnetic source (eq. 1d): (2c) + (2d). Q.E.D.  Proposition 2. The conventional MEs (1) with fields (E, B) defined by eqs. (4) is equivalent to the set of SMEs (2). 

A symmetric formulation of Maxwell's equations

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Proof. Substitute eqs.(4a) and (4b) in all four equations (1). The four equations in set (2) are obtained as follows: First induction equation (2a): (1a) + (1b) Second induction equation (2b): (1a)  (1b) First source equation (2c): (1c) + (1d) Second source equation (2d): (1d)  (1c). Q.E.D.  Table 1 also shows another symmetric version of MEs in terms of (E, B), obtained by substituting eqs. (4) into MEs (2). This set is included for completeness, no further mention will be made. It immediately follows that the continuity equation simply is a tautological condition fulfilled by any set of values (P,N, J, ) satisfying SMEs. Formally, Proposition 3. Continuity condition (5) is automatically fulfilled by any set (P,N, J, ) satisfying either pair (a) or pair (b): (a) Second induction equation and first source equation, or (b) First induction equation and second source equation.  Proof. (a) Operate with w over (2c), operate with  over (2b) and add to get J + c w = 0

(5).

(b) Operate with w over (2d), operate with  over (2a) and substract to get (5). Q.E.D.  Since only one induction and one source equation are required for continuity, then Corollary 3.1. A bonafide solution of MEs must have the same values of (P,N, J, ) in parts (a) and (b) of Proposition 3.

III. WAVE EQUATIONS FOR ELECTROMAGNETIC FIELDS The symmetric wave equations (WEs) corresponding to SMEs (2) are  P   (4/c) [ c  TJ]  N   (4/c) [ c + TJ]

(6a). (6b)

where the d'Alembertian operator is   2  2w2 [= cm-2, and T   w [= cm-1.

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Proposition 4. Any pair of vectors (P, N) solution of the SMEs (2) automatically is a solution to the WEs (6a) and (6b).  Proof. Operate with w over (2b) and substitute Nw from (2a). Expand the double cross product and substitute P from (2c). Rearrange to get (6a). Likewise, operate with w over (2a), substitute Pw from (2b). Expand the double cross product and substitute N from (2d). Rearrange to get (6b). Q.E.D.  Note that only one source equation enters into the derivation of each WE. As a consequence, it is expected that WEs (6a) and (6b) are slightly decoupled from the viewpoint of the sources. Indeed, the converse of Proposition 4 requires continuity as an implicit coupling condition: Proposition 5. Any pair of vectors (P, N) solution to the WEs (6a) and (6b) are a solution of SMEs (2) provided that  and J fulfill the continuity condition (5).  Proof. Operate with w over (6b) and with  over (6a). Add assuming that the order of the rotational and the d'Alembertian operators may be interchanged. Expand the double cross product. Collect terms to get [P + Nw] = (4 /c) [ J   (J + c w)] Invoke continuity. The remaining expression is a d'Alembertian operating over eq. (2a). Likewise, for (2b) the process is similar. Operate with w over (6a) and with  over (6b). Substract assuming that the order of the rotational and d'Alembertian operators may be interchanged. Expand the double cross product, collect terms and invoke continuity as before. Q.E.D.  From eqs. (4) and (6), the standard WEs in terms of (E,B) ( 5) obtain. They are shown in Table 1 (third column, at the end of the section on electromgantic fields).

IV. REPRESENTATION BY VECTOR AND SCALAR POTENTIALS Firstly, we state without proof one general mathematical property of vectorial fields (electromagnetic or not). Proposition 6. General expression for a field. Any vector field F(r,w), sufficiently differentiable, is the sum of a gradient and a curl (Kellog(6), exercise 7, page 76) F  A + U

















(7). 

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In particular, let F = Fxi + Fyj + Fzk be an electromagnetic field, then A = Axi + Ayj + Azk is a vector potential and U is a scalar potential, both having dimensions of energy per unit charge [= gauss cm = erg esu-1. Specializing to (P, N) and (E, B): P  AP  UP  AP SP N  AN  UN  AN SN

 

 

 

 

 

 

(8a), (8b),

E  AE  UE  AE SE B  AB UB  AB SB

 

 

 

 

 

 

(9a), (9b),

where S  U is the contribution of the scalar potential to the respective electromagnetic field. In general, source equations for (P, N) (eqs. 2c and 2d) are nonsolenoidal so that there is no doubt regarding the presence of SP and SN. On the B contrary, the conventional practice is to ignore S for the solenoidal magnetic field, B B i.e. S  U =0. Some conditions leading to such constraint are clear by calculatiing UB from (10b). Invoking eqs. (4) one immediately obtains XE = (XP XN/2 XB = (XP XN/2

 

 

 

 

 

 

 

 

(10a), (10b),

where X represents any variable in (A, S, V, U).Let us substitute eqs. (8) into SMEs (2) to get: [AP + ANw]   SNw  4J/c   VN(r,w) [AN  APw]  SPw  4J/c   VP(r,w) 2UP  4 2UN  4

(11a) (11b) (11c) (11d),

where VP(r,w)  SPw 4J/c and VN(r,w)  SNw  4J/c. There is a surprise. Source expressions (11c) and (11d) are Poisson equations. Therefrom, invoking eqs. (10), Poisson eqs. immediately follow for UE and UB, without a Coulomb gauge condition  AB = 0 as in standard textbooks (for instance, Jackson (5) , ch.6). Hence, the electric scalar potential UE does not impose transversality on AB. Individual vector potentials AP and AN appearing in (11a) and (11b) may be decoupled to obtain [AN  4J/c]  VNw [AP  4J/c]  VPw

(12a) (12b).

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It is noteworthy that the expressions for vector potentials AN and AP are independent. On the contrary, the conventional solution contains only one equation for the magnetic vector potential AB (eq. 32,section V.D below).

V. PARTICULAR SOLUTIONS In the general case where VNw  0 and VPw  0, it is difficult to obtain explicit expressions for AN and AP from eqs.(12a) and (12b). However, it is possible to obtain solutions for special cases, as discussed next. A. Class U: uncoupled vector potentials Let CU be a class of solutions of SMEs defined by Viw 0

i = (N, P)

(13).

Substituting in (12) and invoking vector analysis,  Ai  4J/c  Fi

i =N,P

(14)

where AN and AP are uncoupled, and Fi are arbitrary scalar functions (with the dimensions of field). Constraints (13) are but a special case of continuity as seen next. Proposition 7. The continuity condition (5) is equivalent to Vi = 0, i = N,P

(15). 

Proof. For i = N: replace  from eq. (11d) into the continuity eq. (5) and rearrange to get (15). Likewise, for i = P, replace eq. (11c) to get (15). Q.E.D.  Corollary 7.1. Any solution of SMEs fulfills (Viw) = 0

(16)



Proof. A generic solution of eqs. (15) is V i = Vi(w). Take the partial time derivative of eqs. (15) and invert the order of the operators. Q.E.D.  Corollary 7.2. A particular solution of eq. (16) is condition (13).



A symmetric formulation of Maxwell's equations

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Corollary 7.2 means that condition (13) is not particularly strong. It simply defines a subclass of functions consistent with continuity. B. Class N: AN independent Sections B through E involve the coupling of AN and AP. Let CN be the class of solutions of SMEs, defined by VN(r,w) = 0, any VP

(17).

Note that CN is not a subclass of CU. Substituting (17) into (11a) one obtains a coupling condition between AN and AP: AP  ANw  U1

(18),

where U1 = U1(r,w) is an arbitrary scalar function (with the same units of potential). Substitute (18) into (8a) to get P   ANw U1 UP   ANw P









(19),

where P  U1  UP. Eqs (18) and (19) mean that, given condition (17),AP is no longer independent. The WE for the independent AN is easily obtained: operate with w on (18), substitute in (11b) and rearrange  AN  4J/c  LN (20), where the symmetric Lorentz function LN   AN  Pw has dimensions of field. Note that (20) is eq. (14) with LN substituted for FN . Hence, condition VN = 0 (eq. 17), which is stronger than condition VNw 0 (eq. 13), brings in structure into FN. The companion WE for vector potential AP is no longer independent, as shown next. Proposition 8. If VN = 0, WE (12b) reduces to [AN  4J/c  LN]w  0

(21). 

Proof. Write (12b) as [AP] (4/c)J/c  VE(r,w)w. Operate with  over constraint (17) to get that J = 0. From coupling condition (18), AP ANw U1. Substitute into (12b). Operate with  over (18) to get 2U1= ( AN)/w. Substitute and rearrange to get (21). Q.E.D.  Corollary 8.1. Eq. (20) automatically solves eq. (21).



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C. Class P: AP independent This is the dual of previous class N. Let CP be the class of solutions of SMEs, defined by VP(r,w) = 0, any VN

(22).

Substitute (22) into (11b) to obtain another coupling condition between AN and AP: AN  APw  U2

(23),

where U2 = U2(r,w) is an arbitrary scalar function (with the same units of potential). Substitute (23) into (8b) to get N   APw U2 UN  APw N









(24),

where N (U2  UN). Eqs (23) and (24) mean that, given condition (22),AN is no longer independent. As before, the WE for the independent AP is  AP  4J/c  LP

(25),

where the symmetric Lorentz function is LP   AP  Nw. As in previous case, the WE (12a) for the dependent potential AN reduces to [AP  4J/c  LP]w  0

(26).

D. Class B: AB independent Let CB be the class of solutions of SMEs, defined by VN VP = 0

(27).

As seen in the following, this is the class of conventional solutions.To solve this class and the next, let us rewrite the induction equations (11), by subtracting member to member (11a  11b), and adding likewise (11a + 11b) to get: (TAPTAN)   (VP  VN) ( TAP TAN)   (VP  VN) Eqs. (28) may be written in terms of AE and AB by invoking eq. (10) to get

(28a) (28b)

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[AE + ABw]   (VP  VN)/2   VB [AB  AEw]  + (VP  VN)/2  VE

(29a) (29b)

It is immediately obvious that eqs. (29) are the same as eqs. (11) with the transformation N  B, and P  E. Hence, the same solutions discussed above in sections A through C are applicable to eqs. (29). In particular, comparing eqs. (17) and (27), it follows that condition (27) is VB(r,w) = 0, any VE. Thus, all equations in section B, transformed as N B, and P E, apply here. Specifically, B

E

Coupling condition between A and A : AE  ABw  U1

(30),

Dependent field: E   ABw U1 UE   ABw E









(31),

where E  U1  UE. And, WE for independent vector potential  AB  4J/c  LB

(32).

where LB   AB  Ew. As announced, eqs. (31) and (32) are the conventional expressions. For completeness, it is noted that the same equations above may be derived by direct substitution, without invoking the transformation N B, and P E. Also, nowhere in the derivation appears a restriction regarding the existence of magnetic or electric scalar potentials. E. Class E: AE independent This is the dual of the conventional class B. Let CE be the class of solutions of SMEs, defined by VP VN = 0

(33).

By analogy with previous section, the solutions to this class are the same as those of class P (section C) with the transformations N  B, and P  E. Table 2 summarizes the results of this section.

VI. CONCLUDING REMARKS We exhibited symmetric sets of equations tautologically equivalent to Maxwell's set of equations (MEs). They arrange into a pair of induction equations with exactly the

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same structure, with current density J contributing to the induction of both electromagnetic (EM) fields; and into a pair of source equations with exactly the same structure, where charge density  also contributes to both EM fields. In this sense, both J and  could be interpeted as electromagnetic current and charge densities, rather than electric current and charge densities. It is shown that the equation of continuity is a tautological condition, automatically fulfilled by the symmetric MEs. In the context of symmetric MEs, the absence of electromagnetic charge (=0) does not imply J = 0. Hence, internal currents are allowed in charge-free space. However, in the conventional interpetation, charge-free space implies =0, J = 0. Tables 1 and 2 summarize the main symmetric equations. Despite the quantitative equivalence between the symmetric and conventional sets of MEs, there are several qualitative differences between sets (2) and (1): 1) The electric charge density  is a source for both fields (P, N). In set (1), B has no source. 2) The electric current density J contributes induction for both fields (P,N). In set (1) J contributes to the magnetic induction only (Ampere's law). 3) In the presence of net charge ( 0), all source equations in (2) are non-solenoidal; whereas in (1) magnetic field B is solenoidal. 4) As a consequence, scalar potentials may be associated with both fields P and N. However, in the received view, there is no magnetic scalar potential associated with B. Contrarywise, it is argued here that the general solution involves nontrivial scalar magnetic potentials (eq. 9b). 5) The general formulation of electromagnetic fields in terms of vector and scalar potentials leads to identifying a hidden restriction for the validity of the conventional representation of E in terms of potentials: constraint condition (27), leading to coupling condition (30). 6) The general formulation leads to Poisson equations for all scalar potentials (eq. 11c and 11d) that are independent of Coulomb's gauge. Hence, the transversality restriction on the vector potentials is removed. The existence of magnetic scalar potentials may shed some light on the current controversy regarding longitudinal magnetic fields. (7-9) All other interpretational matters are deferred for future work.

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REFERENCES 1. H.A. Múnera, D. Buriticá, O. Guzmán, and J.I.Vallejo, "Soluciones no convencionales de la ecuación de ondas viajeras", Revista Colombiana de Física 27, No. 1 (1995) 215-218. 2. H.A. Múnera, O. Guzmán, "New explicit nonperiodic solutions of the homogeneous wave equation", to appear in Found. Phys. Lett. 10, No. 1 (1997, in the press). 3. H. A. Múnera, and O. Guzmán, "A generalized solution of Maxwell's equations in terms of potentials", submitted to Found. Phys. Lett. (November 1996). 4. E.M. Purcell, Electricity and Magnetism, Berkeley Physics Course, vol 2, 2nd ed, ch. 9, McGraw-Hill Book Co (1985) 484 pp. 5. J. D. Jackson, Electrodinámica Clásica, translation into Spanish of Classical Electrodynamics, Editorial Alhambra, Madrid (1966) 623 pp. 6. O.D. Kellogg, Foundations of Potential Theory, J. Springer (1929). Unabridged republication by Dover Publications (1953) 384 pp. 7. M.W. Evans, "Reply to criticisms of the B(3) field", Found. Phys. Lett. 8 (Dec. 1995) 563-573. See additional references therein. 8. S.J. van Enk, "Is there a static magnetic field of the photon?", Found. Phys. Lett. 9 (April 1996) 183-190. 9. M.W.Evans, "One photon and macroscopic B cyclic equations: reply to van Enk", Found. Phys. Lett. 9 (April 1996) 191-204.

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TABLE I. SYMMETRIC MAXWELL EQUATIONS Description E B Induction equation 1 Induction equation 2 Source equation 1 Source equation 2 Continuity equation Induction 1  2 Induction 1  2 Wave equation 1 Wave equation 2 Vector field 1 Vector field 2 Induction equation 1 Induction equation 2 Source equation. 1 Source equation 2 Induction 1 + 2 Induction 1  2

P,N P,M E,B Expressions in terms of electromagnetic fields (P + M)/2 E (P  N)/2 B (P  N)/2 (P  M)/2   P Nw  4J/c P  Mw  4J/c T E  T B  4J/c N Pw  4J/c M  Pw 4J/c T E TB  4J/c P  + 4 P  4 E  4 B N  4 M  4 E  4 + B J + c w = 0 J + c w = 0 J + c w = 0 (P N) (P N)w  8J/c (P M)  (PM)w E  Bw (P N)  (N+P)w (PM)  (PM)w  8J/c B  Ew 4J/c  P   (4/c) [ c  TJ]  P   (4/c) [ c  TJ]  E   (4/c) [ c + Jw]    N   (4/c) [ c + T J]  M   (4/c) [ c + T J]  B   (4/c) [ c + J] Expressions in terms of vector and scalar potentials P = AP  UP P = AP  UP E = AE  UE N = AN  UN M = AM  UM B = AB  UB (AP + ANw)   VN (AP  AMw)  + VM (TAE TAB) (UB/w)(VPVN)/2 (AN  APw)  + VP (AM  APw)   VP (TAE TAB) (UB/w) (VPVN)/2 2UP  + 4 2UP  + 4 2UE  4 2UB 2UN  4 2UM  4 2UE  4 + 2UB (TAP TAN)  VP VN (TAPTAM)  VM VP (AE  ABw) (UB/w) (TAPTAN)  (VP +VN) (TAPTAM)  VM + VP (AB AEw )   (VPVN)/2

Notation: T   w;   2  2w2; VN = (UNw)  4J/c; VP = (UPw)  4J/c; VM = (UMw)  4J/c

A symmetric formulation of Maxwell's equations

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TABLE II. PARTICULAR SOLUTIONS OF SYMMETRIC WAVE EQUATIONS IN TERMS OF POTENTIALS Description

Constraint(s)

Coupling condition

Uncoupled class CU VPw = 0 VNw = 0 Coupled class CN VN = 0 any VP Coupled class CP VP = 0 any VN Coupled class CB VP  VN = 0

NONE

Coupled class CE

AB   AEw U

VP  VN = 0

AP   ANw U1 AN   APw U2 AE   ABw U

Electromagnetic fields P  AP  UP N  AN  UN P   (ANw P) N  AN  UN P  AP  UP N   APw N E   (ABw E) B  AB  UB E  AE  UE B   AEw B

Wave equations

AP  4J/c  FP AN  4J/c  FN AN  4J/c  LN AP  4J/c  LP AB  4J/c  LB AE  4J/c  LE

Notation: P U1  U P, N U2  U N),E U  U E, B U  U B LN   ANPw, LP   APNw, LB   ABEw, LE   AEBw