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Electromagnetic waves with E parallel to B

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1992 J. Phys. A: Math. Gen. 25 5373

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1. Phys. A Math. Gen. 25 1992) 5373.5316. R i m e d in the UK

Electromagnetic waves with E parallel to BJohn E GrayCo de 3-71, Naval Surface Warfare Center, Dahlgren, VA 22448, USAReceived 12 November 1991, in final form 2 lune 1992Abstract. In this paper, two explanations of how to obtain electromagnetic waves withEllS are presented. The nature of these solutions is discussed, with a caveat on whatconstitutes a general solution. Finally, an example of an electromagnetic wave with theelectric field parallel to magnetic field is presented.

1. IntroductionStudents in electromagnetic theory have been taught throughout all levels of theireducation that the only plane wave s olutio ns to M axwells equations are those withthe electric field perpendicular to the magnetic field. This is a universally acceptedmyth that is taught in the s op ho m ore level texts [l], junior-senior level texts [2-51and graduate level texts [6-81. Recently, an article by C hu [9] has purportedly arrivedat a general plane wave solution to Maxwells equations with EIIB. Several au thors[lo 1 1 1 have subsequently published refutations of this claim that miss the point.Rebuttals of the refutations have been published [13 ,14] and several articles haveappeared in the physics education literature [13 ,15] attempting to explain this result.In this paper, an interpretation of electrom agnetic waves with EllB is presented, witha caveat on what constitutes a general solution.2. DiscussionA point that has not been discussed in the literature is the nature and interpretationof the method for obtaining Chus solutions, rather the emphasis has been on howthey can be obtained. We briefly show how the solutions were obtained in order toclarify discussion of the interpretation of th e solution. If on e assumes a h armonic timevariation, with k = o / c , then instead of solving the Helmho ltz equations for the electrican d m agnetic fields, the single equation

V2+ 2 ) F k r ) 0 1 )

F k r )= c l A h r ) + c 2 Vx A k r ) 2)is a linear combina tion of the electric an d magnetic fields. Since c I and c2 are arbitraryconstants, any choice for them will also satisfy 1 ) . The particular combination ofcoefficients c , = 1 and c2 = l / k makes the superposition vector obey the differentialrelation

can be used to obtain the electric and magnetic fields. Th e superposition vector

V x Fk kFh. 3 )0305-4470/92/205373+04$07.50 1992 IOP Publishing Ltd 5373

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5374 J E GrayThis is the relation ship fo r th e superp osition vector that C h u assumes. By interchangingF for A in Z), he vector potential can be expressed in analogous fashion to berepresented by

1kk r ) x F k r )+F k r ) (4)WL1G1T r 1J dll drvlrlnly *T+LU1 U u G y r r r g ,,. s lPULd l lU l l gauge ullulllun prvcb k . A = 0,.. ^-^ c: - --L:.- - ..^^.-..- , - - A:...:-.. ---A:.:.- :.. .which also implies k is perpendicular to the superposition vector, so the fields areTEM. The fields are

6)I

B ,. r , t ) = V x A , r , t ) = V x F k r ) + - V x F k r ) ) = k A k r , I).i iFrom (5) and 6) it is clear that E r , t ) . B r , ) is not equal to zero, the fields are notperpendicular to each other.The vector differential eq ua tio nv x a = h 7)

that produces Chu s EllB solu tion s is a mathematically consistent eq uation which hasa general solution [ 1 2 ](8)1ka = V x C Y )+ V x V Y

where E is a constant vector an d s a solution toV2 4 k 2 Y = 0. 9)

Regardless of the mathematical correctness of (7) or equivalently 3), the physicalvalidity of the solutions is determined by what assumptions what into the derivationof 3) an d how general the so lution is.An important point that is not always emphasized is what a general solution to aphysical problem is. The term general means the equations used to obtain a solutionand the solution itself are the widest possible solutions that can be used to fit arbitrarybut physically realizable bou nd ary conditions. The plane wave solutions to Maxwell sequations in free space, except fo r the h arm on ic time dep endence, are general solutions.Free space mean s the absence of boun dary conditions. Th e solutions obtained by C huare not free space solutions, so are not in the same general class of solutions as theclassical E I B solutions. F was obtained by combinging the wave equations for thevector potential and magnetic field into one equation, equation (1). using linearsuperposition. Specific values for th e superposition coefficients E , = 1, c 2 = I l k ) werechosen so that F obeyed the diff ere ntia l relation in 3). The specific choice of thesuperposition coefficients is a boundary condition (specifically a Dirichlet boundarycondition).The classical E I B solution to M axwell s equations have no uch boundary condtionsused in their derivation, hence are general solutions. This is a point that has beenlacking in the interpretation of th e E parallel to B solutions. Standing wave solutionsto Maxwell s equations are not surprising when bo un dar y conditions are imposed.What is surprising is the claim by some authors that these are free space solutions.

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Elecfromagnetic waves wirh E parallel fo B 5375This claim is what makes these solutions unexpected. Unless very general boundaryconditions are considered, the class of solutions is pathological when com pared to theclass of admissible solutions. n example of a pathological solution is to compare theDAlembertian solution to the wave equation w x, f x - ct) to the harmonic sol-ution w x, ) = A sin(kx - wt).The harm onic solution is a pathological solution, thoughan important solution, of the wave equ ation. Generalizations of the harm onic solutionare obtained by making A a function of k, i.e. A = A k ) , and integrating over allpossible k

To clarify the point that th e C hu type solutions ar e the result of boundary conditions,a specific example is shown. Plane wave solutions can be obtained for F by notingspace is isotropic along the direction of p ropagation, a vector z is chosen along thedirection of k so that k = or propagation in the z direction, the operator V2becomes d2/dz 2, so the solution takes the formF z ) = C cos(kz)+C2sin(kz) (10)Cl = (c11, C12,O) C2=(c2,, C22,O). (11)

c12 = C 2I CII = -c22 (12)

F , z , r ) = [ (c I I , c12,0 ) c o s ( k z ) +( c I 2 , c l , , 0) in( kz)] eiu (13)

where

Requiring (10) to obey (3) imposes the b oun dary conditions

so the solution fo r the sup erposition vector is

an d the fields areE -j ~ F i z , ) B = k F , z , t ) . (14)

These boundary conditions are identical to those for a wave along a string with fixedendpoints, a standing wave. The analogous type of boundary for electromagneticradiation is to confine it between two perfectly reflecting boundaries (the electromag-netic equivalent of a string hanging between two infinitely heavy walls).Electromagnetic waves with El lB are possible as long as they are confined to anenclosed region. This is similar to setting up a standing wave in a cavity resonator.O ne possible way to produce them is split a light beam in o rde r to confine the beambetween two mirrors, as discussed theoretically in [13,141. Actual experimehts havebeen conducted by Bretenaker a nd Le Flock [I61 using lasers to pro duc e, severaldifferent standing wave solutions with EIIB.Zaghloul [13,1 5] has claimed to have generalized Chu s cond itions for E IB. Thisclaim at a generalization is false. The conditions required by Zaghloul,

A ; T ) *AX47 (15)a n d

IAXT) I= lAX5)I (16)are boundary conditions, hence are not in the same class of solutions to Maxwell sequations as the classical E I B solutions. This condition is more restrictive than theChu solution. Requiring the magnitudes of two vector-valued functions to be equalalong two opposite signed caustic surfaces is an even more restrictive boundarycondition th an C hu s result, so Zaghloul s claim to have g eneralized Chu s result is

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5316 J E Graytrue only in a formal sense. The clas s of functions tha t will obey (15) and (16) is evenmore restrictive tha n those that w ill obey (15)-(16) is even more restrictive tha n thosethat obey (13).

4. ConclusionThis paper provides an explanation of a controversy in the well-explored field ofMaxwell s equations. Contrary to wha t is implicitly assume d, electromagnetic waveswith the electric field parallel t o the magn etic field ar e not general solutions to Maxwell sequations in the same sense of the conventional E I B solutions. The Chu result is thesolution to a bo un da ry value problem , hence is in no sense in the same class of solutionsto M axwell s equations as the classical E I B solution are. Th e first solution is obtainedby application of the principle of superposition tha t fixes th e values of the arb itraryconstants used in a general solution to the Helmholtz equation. The second solutionis even more restrictive than the first, because it fixes functions along two differentparame tric lines of spacetime. T hes e standing wave solutions to M axwell s equationsare not surprising, however, when boundary conditions are imposed, and with thisunderstanding, the El lB solutions are not unexpected.

AcknowledgmentTo Miss Doxie Foster, my first science teacher.References

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