REVIEW

Electromagnetic field theoremsProf. B.D. Popovic, D.Sc.

Indexing term: Electromagnetics

Abstract: The paper reviews most of the general electromagnetic field theorems (i.e. theorems not limited tospecial geometrical structures) in a compact and unified way. Applications of all the theorems are illustratedby one or more examples. Although the review is aimed primarily at nonspecialists in electromagnetic fieldtheory, it is believed that specialists might also find it useful. Basic knowledge of Maxwell's equations andvector analysis is assumed.

1 Introduction

In network theory many theorems are well known and usedextensively. Although they are, basically, resulting from thetwo Kirchhoffs laws, our insight into many aspects andproblems of the network theory would hardly be possiblewithout them. This is because such theorems shed a directlight on certain general properties of networks, while theseproperties are hardly visible, if at all, directly from thefundamental network laws.

General theorems are of comparable importance andusefulness in the theory of electromagnetic fields and theirapplications. In the hands of those who have mastered themthey are extremely powerful tools, both for gaining a broadinsight into certain general properties of the electromag-netic field, and for solving various problems. However, theyare certainly not so well known as the network theorems.Consequently, they are relatively rarely applied to their fulladvantage.

This review is aimed at presenting most of the generalelectromagnetic field theorems in one place and in a unifiedway. It is intended more for a wide circle of electricalengineers and advanced students than for specialists inelectromagnetic field theory (although it is hoped thatspecialists will also find it interesting). For this reason, onthe -one hand, proofs of the theorems are simplified asmuch as possible. This resulted in some cases in a certainlack of generality or rigourousness. On the other hand,only very limited historical notes are supplied. As a con-sequence, appropriate credit has certainly not been givento all those who contributed to our knowledge of thetheorems considered in the paper.

None of the theorems presented in the review is new.However, in order to fit better into the framework of thereview, a number of the theorems are proved in a waywhich might be considered novel. Where it was felt that theproof was only a slight modification of the existing proofs,References following the theorem heading are given to themost similar proofs. Such References are omitted in caseswhere the contents and/or the proof of a theorem seemedto be substantially different from those found in theliterature accessible to the author.

2 Outline of basic concepts of macroscopicelectromagnetic field

As a measurable quantity, the electromagnetic field is

Paper 1O75A, first received 14th July and in revised form 30thSeptember 1980. Commissioned IEE ReviewProf. Popovic is with the Department of Electrical Engineering,University of Belgrade, PO Box 816, 11001 Belgrade, Yugoslavia

IEEPROC, Vol. 128, Pt. A, No. 1, JANUAR Y1981

defined in terms of the electric strength, E, the magneticinduction B, and the force exercised by the field on a small('test') charge dQ, moving with a velocity v with respect tothe observer (the Lorentz force)

dF = dQE + dQv* B (1)

Directly or indirectly, the Lorentz force appears to be theonly means for detecting the existence of the electromag-netic field in a region of space.

The sources of the electromagnetic field are electriccharges at rest and in motion. Macroscopically, they aredescribed in terms of the volume charge density p, andcurrent density / . Influence of substance can be reduced tomacroscopic resultants of charges and currents consideredto exist in a vacuum.

In addition to being primarily produced by charges andcurrents, the basic vectors E and B of the electromagneticfield are, in the general case, also interconnected in spaceand time. This interconnection, as well as the connectionwith field sources, is expressed by Maxwell's equations.These equations read

cm\E = -dB/dt

curl/7 = J+bD/dt

(2)

(3)

The magnetic field intensity H and the electric displace-ment D are defined as

H = D = e0E + (4)

The constants ju0 and e0 are known as the permeability andpermittivity of a vacuum, respectively. M is the magnet-isation vector (volume density of magnetic moments m = ISof elemental current loops of magnetised substance), and Pthe polarisation vector (volume density of dipole momentsp = Qd of elemental dipoles inside polarised substance).

The law of conservation of electric charge can beexpressed mathematically as

div/ = -dp/bt (5)

which is known as the continuity equation. Sincediv (curl G) = 0 for any vector function G with continuousfirst and second derivatives, from eqn. 2 we obtain 3 (div B)/dt = 0, i.e. div 5 = C, a constant in time. Since the fieldcannot be of infinite duration, C = 0, and so

divB = 0

In a similar fashion eqns. 3 and 5 yield

divZ) = p

(6)

(7)

470143-702X/81/010047+ 17 $01-50/0

The divergence equations, eqns. 6 and 7, are often con-sidered as the third and the fourth Maxwell's equations.However, it should be noted that eqn. 6 follows fromeqn. 2, and eqn. 7 from eqns. 3 and 5.

Before proceeding further, note that, according to eqns.4 and 7, in a charge-free region, in a vacuum, div E = 0.This means that the vector E, in a charge-free region, can atno point be from all directions directed towards the point,or away from it. Since vector E in eqn. 1 is not due to dQ,the above conclusion applies. A static charge cannot, there-fore, be in stable equilibrium under the influence of theelectric forces only. This statement is known as the Earnshowtheorem

Magnetisation vector M and polarisation vector P reflect,in essence, effects of the Lorentz force in eqn. 1 onelemental particles of substance. They are, therefore, in thegeneral case functions of both E and B. Since elementalparticles have finite masses and are mutually interacting, Mand P are also functions of time derivatives of E and B, aswell as of their magnitudes. The same is true for currentdensity / . For most practical purposes, however, M can beregarded as a function of B and its time derivatives alone,and P and / a function of E and its time derivatives alone.If this dependence for a substance is linear, the substance issaid to be linear. It is usual to write the first of eqns. 4 inthat case in the form

(8)

where n, nx, n2> • • • are constants. Analogous expressionsare valid for D(E) and J(E). If the time variations of thefield and materials involved are such that only the first termin these expressions can be retained, the materials are saidto be linear in the restricted (or narrow) sense.

Energy sources capable of transmitting energy to electriccharges are usually termed 'external' or 'impressed' sourcesof the field. Theoretically, they can be represented either asanalogous to constant-current generators, or to constant-voltage generators of the network theory. In the first case,current distribution of density /,- is imagined, which isindependent of the field. This can be assumed anywhere —inside or outside the substance — and the substance, inprinciple, need not even be conductive. Of course, /,• entersthe right side of eqn. 3 as an additional current density andis accompanied by charges according to the continuityequation. In the second case, nonelectric forces acting oncharges in generating regions are represented by external orimpressed electric field, Et, which is independent ofmagnetic field and of current density, but produces con-duction currents according to the same rule as the trueelectric field [e.g. / = o(E + Et) for a substance linear inthe restricted sense]. It can be meaningfully defined, there-fore, only inside substance. Of course, it does not enter intothe equation curl E = — bB/bt. Sources in the form of im-pressed currents are usually more convenient, and willfurther be used here.

For sinusoidal time variations of the field sources andlinear media, Maxwell's equations and equation of con-tinuity can be written in the complex form

curl E = —jco\iH

curl / / = (o+jcjt)E + Jt

div / = —/cop

(9)

(10)

(11)

All vector quantities and p represent effective values, asusual. Time dependence of the form exp (/cof) is assumed,and \x = n' ~ / / A a = a + jO and e = e' —je" (imaginarypart of a can always be incorporated in the real part of c).Of course, as obvious from eqn. 8,p ,o and e are functionsof angular frequency w. For simplicity, complex quantitieshereafter will not be underlined.

If A is a vector function having the first two spacederivatives, then div(curly4) = 0 identically. Hence it isalways possible to write

B = curl/4 (12)

From eqn. 2 it then follows that E can be expressed as

E = -grad V-bA/bt (13)

where V is an arbitrary scalar function having the necessaryderivatives. The auxiliary functions A and V, from which itis possible to obtain the field vectors E and B, are known asthe magnetic vector potential and the electric scalarpotential, respectively.

Obviously, the potentials A and V are not unique, and anumber of potential functions, defined in a different wayfrom the above, are used. It is, hence, hardly possible toattach to them any physical meaning. Most often potentialsA and Fare used in media linear in the restricted sense, andare then uniquely defined by means of the additionalrelation

div 4 = -eubVlbt (14)

(the Lorentz condition for potentials). If e and JU areconstants and a = 0 (homogeneous lossless medium), it canbe shown that A and V can be expressed as

(15)R

R

where c = (eju)~1/2, and

R = \r-r'\

(16)

(17)

is the distance between the source point (defined by theposition vector r) and field point (defined by the positionvector r). The integration is to be performed over theregion v containing field sources / and p. It is seen that/ and p are sources of the retarded potentials, i.e. thateffects of / and p on potentials propagate away from asource point with a finite velocity c = (eju)~1/2. Accordingto eqns. 12 and 13, however, it is obvious that / and p arenot sources of the retarded field vectors E and B. This factwill further be discussed in Section 5.

Integral expressions for field vectors E and B (or H) interms of / and p are more complicated than those given byeqns. 15 and 16. It is, therefore, usually more convenient tosolve a problem starting from eqns. 15 and 16, than fromexpressions for the field vectors. It should be noted againthat exprs. 15 and 16 are valid only in homogeneous media,linear in the restricted sense.

3 Superposition theorem

Assume that B(H),.D(E) and J(E) are linear functions inthe sense given in eqn. 8, i.e. that the medium is linear.Referring to eqn. 3, let / = /,• +/(£"), /,• representing the

48 IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

density of impressed currents. Assume that

(18)fe=i

where ak are arbitrary constants, and Jik arbitrary functionsof time and space co-ordinates. Suppose that Ek and Hk aresolutions of Maxwell's equations corresponding to sourcesJik. It is a simple matter then to show that

E = £ akEk and H = £ ahHk (19)fe=i fe=i

are solutions of Maxwell's equations corresponding tosources /,-. This is the superposition theorem.

It is often convenient to express the theorem in aslightly different way: if (Ek, Hk), (k = 1, 2 , . . . , n), is aset of possible fields (i.e. fields satisfying Maxwell'sequations), then (E,H) given in eqn. 19 is also a possiblefield, where ak are arbitrary constants.

Superposition is, in general, valid also for potentials Aand Fin linear media. It should be noted, however, that theLorentz potentials, given in eqns. 15 and 16, are definedonly for homogeneous media linear in the restricted sense.

Examples of applications of the superposition theoremare numerous. All kinds of series field solutions (analyticalor numerical) are based on it. As a specific example,approximate determination of the electrostatic field ofcharges on conducting bodies, by means of equivalentsources inside the (removed) body, is essentially based onthe superposition theorem.

Of course, superposition is a very general concept,valid for all phenomena which can be described by linearmathematical relations.

4 Compensation theorem

According to the compensation theorem, the influence ofsubstance on the field can partly or completely be com-pensated by appropriate distribution of impressed currents.This theorem is a generalisation of the same theorem forelectrical networks, one form of which states that anyelement in an electrical network can be substituted by anideal current generator of the same current intensity as inthe element considered. A theorem of this kind for electro-magnetic fields was first proposed by Monteath [1].

Assume that we know the field vectors E and H in aninhomogeneous nonlinear medium, as well as magnetisationvector M, and polarisation vector P, at all points. Let thesources be represented by impressed currents /,-, and letthe current density due to the field be / . At all field pointsvectors E and H satisfy eqns. 2, 3, 6 and 7, with B and Dgiven in eqns. 4 and / substituted by (/ + / , ) . Let thevectors M, P and / be represented as a sum of two arbitraryparts

M = MX+M2, P = PX+P2, J = JX+J2 (20)

In that case

B = = Bx

D = (e0E + Px) + P2 = Dx +P2

Eqns. 2 ,3 ,6 and 7 can be then written in the form

curliT = —dBx/dt—Jmi (Jmi = iiodA

curl H = dDx /dt +JX+ (/,• + /,')

(/,' = J2 +dP2/dt)

(21)

(22)

(23)

= p m (p m = - / x 0 divM2)

divDx = p + p' (p' = - d ivP 2 ) .

(24)

(25)

Field vectors E and H obviously satisfy these equationsalso. However, they can be interpreted as equations of thefield in an inhomogeneous nonlinear medium, the influenceof which is represented by vectors Mx, Px and Jx, in thepresence of impressed electric currents of density (/,- + /,')and fictitious impressed 'magnetic currents' of density Jmi.

We have thus proved the compensation theorem: theinfluence of substance on the field can be partly, or com-pletely, (depending upon the division of M, P and / i n eqns.20) compensated by appropriate distribution of impressedcurrents. In the general case, both electric and fictitiousmagnetic impressed currents are necessary for that purpose.

Note that fictitious magnetic current and chargedensities, as defined in parentheses in eqns. 22 and 24,satisfy the continuity equation

div/mi- = -bpjdt, (26)

analogous to eqn. 5 for electric currents and charges.Most often, the compensation theorem represents the

theoretical basis for 'homogenisation' of an inhomogeneousmedium (usually linear). As we have just proved, in thegeneral-case fictitious magnetic currents and charges alsoneed to be introduced to achieve that aim. Althoughfictitious, they are of extreme usefulness, which is evidenteven from the simple example mentioned. It is, therefore,very convenient to modify Maxwell's equations to incor-porate magnetic currents as well (Reference 2, pp. 69-73).In the light of the compensation theorem, Maxwell'sequations can in general be written in the form

cmlE = -dB/dt-J

cmlH =

(27)

(28)

The total electric current density (/ + /f) satisfies eqn. 5,and magnetic current density Jmi satisfies the analogouseqn. 26.

As the first example of application of the theorem,assume that we wish to determine low-frequency currentdistribution in a nonferromagnetic conducting body,situated in air, in an external electric field (which can beconsidered as the impressed electric field if currents in thebody have negligible influence on the sources of the field).Since time-varying currents are also sources of the electricfield (see eqns. 13 and 15), we know that a secondaryelectric field exists due to currents induced in the body.The medium being inhomogeneous, we cannot computethat electric field using eqns. 13, 15 and 16 (the last twoare not valid in inhomogeneous media). But if we apply thecompensation theorem and consider the induced currents asimpressed currents, these currents are to be imagined in avacuum, and we can compute the field of these currents bymeans of the Lorentz potentials. We next may require thatthese impressed currents satisfy the condition /,- =o(Ei + El), where a is the conductivity of the body, andEl the electric field due to/,-. By means of eqns. 13, 15 and16 we hence obtain an integral equation for current density

Ji-Assume, next, that the body is ferromagnetic (but

approximately linear). We can use the same chain ofreasoning, except that impressed magnetic currents will alsoappear. (This, of course, complicates the integral equation

IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981 49

for current distribution, since magnetic currents are alsosources of the electric field.)

As the final example, computation of current distributionand of near or distant field of antennas is usually performedby means of the retarded Lorentz potential A. Strictly, thisprocedure cannot be used directly, since the medium (in-cluding conducting or dielectric antenna body) is nothomogeneous. If all the antenna currents are considered asimpressed currents, however, the procedure is correct.Thus, the compensation theorem is tacitly used in all suchcomputations.

5 Theorem on sources of retarded fields

As mentioned in Section 2, in homogeneous media, linearin the restricted sense, / and p are sources of retardedpotentials, but not of retarded field vectors E and B. Moreprecisely, effects on potentials due to sources insideelemental volumes propagate away from the sources witha constant velocity c = (e/i)~1/2, which is universallyaccepted as the velocity of propagation of electromagneticdisturbances. However, on the one hand, Lorentz potentialsare just one set of a theoretically infinite number ofpotentials from which the fields can be computed; not allof them have the property of being retarded in the abovesense. On the other hand, it is conceptually probably moreacceptable that retardation is the basic property of fieldvectors, whatever the distance from the elemental sources.We shall now show that it is possible to deduce sources ofretarded field vectors from Maxwell's equations.

Assume that all the field sources are reduced to sourcesin a vacuum (according to the compensation theorem), sothat Maxwell's equations read

cmlE = -nodH/dt-Jmi

curl// = eodE/dt+Ji

di\H = pm/ju0

divE = p/e0

(29)

(30)

(31)

(32)

Taking the curl of the first two equations and noting thatcurl (curl F) = grad(div F) - AF, we obtain that E and Hsatisfy inhomogeneous wave equations

&E oJ{ 1AE - eoMo —T = Mo -T- + curl Jmi + — grad p (33)

Ot ot €Q

o2H oJmi 1eoMo T^" = - curl /,- + e0 —— + — grad pm01 r +

Ot /Lt0

(34)If all sources are within a finite volume v, solutions ofeqns. 33 and 34 are of the form

F(r,t) = - -4n R

(35)

where F stands for E oiH,G for the right sides of eqns. 33and 34, respectively, and r and R are defined in connectionwith eqns. 15 and 16. This represents the expression of thetheorems on sources of retarded fields.

Mathematically, this result has long been known(Reference 4, pp. 192-193) [3]. It seems however, that noparticular physical interpretation has been given to it. Itwas brought to the author's attention by Dr. M.B. Dragovicthat eqn. 35 essentially states that as the real sources of theelectromagnetic field we may consider not / and p, but

rather space and time variations of these quantities (i.e. theright sides in eqns. 33 and 34). This is a very unusual state-ment. For example, according to this approach, sources ofthe electrostatic field in the case of a uniformly-chargedcloud are on its surface (in the region where charge densitydecreases from that inside the cloud to zero), and notthroughout the cloud. In addition, the electric field of suchsources (in electrostatics) decreases not as 1/R2, but as 1/Rwith the distance R from the sources. This is, of course,very difficult to accept, for we are accustomed to consider1/R2 as the 'natural' dependence of the static field on thedistance R. However, in this manner the field vectors in allcases (static, slowly-varying or rapidly-varying fields) havethe same dependence (1/R) on distance from the sources,which is a.very attractive concept indeed.

As a simple example, consider a spherical charged cloudof constant density p and radius a (electrostatic case). Thesources of the electric field are then given by the third termon the right side of eqn. 33. Inside the cloud grad p = 0, sothat the sources are distributed over the cloud surface only.Consider a small pillbox-shaped volume with two bases ofarea dS on the opposite sides of the cloud surface and aheight Ah which tends to zero. Starting from eqn. 35 andusing the identity

= grad(/g)-ggrad/

it is not difficult to prove that such an element produces anelectric field

pdSdE =

where dS is an element of the cloud surface, directed out-wards. Using this formula it is a relatively simple matter toshow that E = Qr/(4ne0r

3) outside, and E = Qr/(4ne0a3)

inside the cloud, Q denoting the total cloud charge, and rthe distance of the field point from the cloud centre. It iswell known, of course, that this result can easily be ob-tained by using the Gaussian law. However, a lengthyprocedure is necessary for obtaining it from the classicalformula for the electric field of a volume element of charge,i.e. from dE = pRdv/(4ne0R

3).The resultant field of any given distribution of currents

and charges is, of course, the same whether computed fromthe retarded potentials, or using eqn. 35. Thus the conceptof sources of retarded fields can hardly predict any neweffect. However, it is useful in various cases. For example,in some instances an integral equation for current distri-bution on antennas and scatterers can be deduced directlyfrom eqn. 35. Also, it is known that wire antennas, withconcentrated or distributed loadings along their length,radiate more efficiently than unloaded antennas; inaddition, there are indications that wire antennas radiatethe major part of energy from their ends [5]. Qualitatively,both phenomena can be attributed to increased gradient ofcharge at discontinuities or along the antenna (in accordancewith the concept of sources of retarded fields). A qualitativeexplanation based on retarded potentials seems to be farfrom straightforward. The author feels that this conceptmight help in the future to understand better variousaspects of electromagnetism, particularly in connectionwith radiating structures.

50 IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

6 Theorem on distribution of energy in electromagneticfield

In order to establish an electromagnetic field, it is necessaryto expend a certain amount of energy. When there is sub-stance in the field, it is clear that inside every elementalvolume containing substance energy must be expended forpolarisation and/or magnetisation. In the case of the field ina vacuum, however, there is no understandable mechanismof energy storage. And yet we know positively that energyis localised in the field even in a vacuum — otherwise trans-mission of energy by means of electromagnetic wavespropagating in a vacuum would not be possible. Thetheorem we shall now prove states that the energy used forestablishing a field is spent throughout the field with acertain density. It also states that it is possible to define theenergy stored in the field only in the case of media linearin the restricted sense.

Consider an electromagnetic field in a medium ofarbitrary electromagnetic properties, produced by im-pressed currents of density / , confined inside a finitevolume. Power density of sources is given by —/,-£", whichfollows from the definition of current density (Jt ^NQVj,where N is the number of free charge carriers Q per unitvolume, and v,- their average velocity due to externalforces) and from eqn. 1 for the Lorentz force. Using eqn.28 this can be expressed in terms of the field vectors E,H and D and current density / due to the field

-JiE = JE + EdDjdt-Ecm\H

If we now note that

establishing the field

dv

fD B

EdtD+ HdtBJ0 Jr\

(38)

-E curl H = div (E x H) - H curl E (36)

and curl E~ — bB/dt (we assume that Jmi = 0), work per-formed by the sources inside a volume v enclosed by asurface S from an instant 10 = 0 when there was no field toan instant t can be written in the form

- f f JiEdtdv = f f JEdtdv +

+ f f (EdD/bt + HbB/dt)dtdv+ f \ (ExH)dSdt

• (37)

Use was made of the divergence theorem to transform thevolume integral of div (E x H) into a surface integral.

Assume now that S is chosen so large that it encloses allthe field (which is always possible, since the field propa-gates away from sources with a finite velocity). The lastterm in eqn. 37 is then zero, and v designates the volumeoccupied by the whole field. The first term on the right sideof the equation then represents the total work done by theelectric field, from t = 0 to the instant considered, to main-tain current in the conducting parts of the system. (Thiscan be shown in the same way as for power density ofsources.) Energy for performing that work was, of course,supplied by the sources. So, by the law of conservation ofenergy, the second term on the right side of eqn. 37 repre-sents the total work of the sources done in establishing thefield existing at the instant considered. Now, E(bD/dt)dt =EdtD (where dtD designates increment of D at a point intime interval dt), and similarly H(bB/bt)dt = HdtB. There-fore the correct result for that work is obtained if it issupposed that it is distributed throughout the field with adensity

IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

This equation represents the mathematical statement of thetheorem on distribution of energy in the electromagneticfield.

While eqn. 37 is certainly correct, interpretation of theintegrand of the second term on its right side, given ineqn. 38, is in a sense arbitrary. For we can add to it anyscalar function having volume integral over the whole fieldequal to zero without violating eqn. 37. However, if weprove that this function is zero in a single case, it should bezero generally. Now, we know from experiments withblack-body absorption that plane electromagnetic waves,propagating in a vacuum with a velocity c0 = (eo//0)~1/2,do indeed possess energy density predicted by eqn. 38. It is,therefore, generally accepted that eqn. 38 is correct. Forstatic fields this interpretation of eqn. 38 was given byMaxwell (Reference 6, sections 630, 631, 634-638), andfor dynamic fields by Poynting [7].

In the general case, as explained, eqn. 38 represents theenergy density required to establish an electromagneticfield, and not the density of energy stored in the field (i.e.energy temporarily located in the field and completely re-coverable when the field is reduced to zero). It should benoted that it consists of two seemingly independent terms,one containing electric field vectors E and D only, and theother containing magnetic field vectors H and B only. (Infact, these vectors are interdependent through Maxwell'sequations.) It is therefore possible to say that the energydensity necessary for establishing an electromagnetic fieldis equal to the sum of energy densities necessary for estab-lishing the electric and magnetic parts of the field.

If the medium is lossless, by the law of conservation ofenergy eqn. 38 represents density of energy stored in thefield, which is briefly referred to as the field energy density.In particular, if materials are linear in the restricted sense, itis easy to show that the two terms on the right side of eqn.38 become eE212, i.e. yM212. General linear materials arealways lossy, and in that case it is strictly not possible todefine field energy density.

If materials exhibit hysteresis behaviour (ferroelectricand ferromagnetic materials), and if E and dtD, i.e. H anddtB, are collinear at the point considered, from eqn. 38 itfollows that average hysteresis power-loss density inperiodic fields is proportional to the area of the hysteresisloop (i.e. of the closed curve traced in the course of time inE — D. i.e. H — B co-ordinate systems). It is interesting tonote that this well-known conclusion is not always valid.For example, it is not strictly true for rotating fields inferromagnetic materials, since then H and dtB are notcollinear.

An interesting theorem, known as 'Thomson's theorem',is valid for energy stored in electrostatic fields in Linearmedia: charges on conducting bodies (the sources of theelectrostatic field) are distributed in such a manner that thefield energy is minimal. A rigorous proof of the theorem isbased, essentially, on eqn. 38 (Reference 8, section 2.11)[9]. For our purpose it suffices to note that for disturbingelectrostatic equilibrium distribution of charges on con-ductors a certain amount of work would have to be doneagainst the electric forces, which would increase the energystored in the field.

Another theorem relating to electrostatic field energy isthe following: introduction of an uncharged conductingbody in an electrostatic field diminishes field energy. Again

51

a rigorous proof can be found elsewhere (Reference 8,section 2.13), but here we shall give only a simple explan-ation. An uncharged conducting body is always attractedtowards the regions of higher field intensities, due toinduced charges on the body (whose total amount is, ofcourse, zero). Therefore electric forces will perform acertain amount of work while the body is being introducedinto the field; the field energy must, therefore, be decreased.The theorem is true also if the body is an unchargeddielectric (Reference 10, pp. 59-60), the qualitative proofgiven above being valid in that case also.

7 Poynting's theorem

Formulated by Poynting in 1884 [7], the theorem essen-tially states that electromagnetic power flow per unit area(W/m2) at a point of the field is given by the vector P =E x H. The vector P is known as 'Poynting's vector'. Thetheorem follows from eqn. 37, the theorem on distributionof energy in the electromagnetic field and the law of con-servation of energy.

Consider a finite domain v of the field enclosed by asurface S. If we interpret the second term on the right sideof eqn. 37 as energy spent by sources to establish the fieldinside v, according to the law of conservation of energy, thelast term (the surface integral) represents energy that left vthrough S. Thus P = E x H can be interpreted as describingthe magnitude, direction and sense of the electromagnetic-power-flow per unit area, which proves the theorem.

Poynting's vector obviously is not unique. According tothe divergence theorem, any vector function having zerodivergence can be added to it without violating the energybalance in eqn. 37. Many different Poynting's vectors cantherefore be defined [ 11 ] . It does not seem possible to findthe correct Poynting's vector experimentally, because nodevice appears to exist which can measure electromagnetic-power-flow density at a point (although we can, of course,measure E and H, and thus also P = E x H). There are,however, indications that interpretation of Poynting'svector according to Poynting's theorem is correct (e.g. againplane electromagnetic-wave absorption). Also, it hasadvantage over all other possible such vectors in being thesimplest.

A frequent form of Poynting's theorem is obtained ifeqn. 37 is differentiated with respect to time and linearmedia in the restricted sense are considered. One obtains

- f JiEdv = f•'v J v

JEdv+ot

f (eE2/2

i (ExH)dSJ o

'I2)dv

(39)

Poynting's theorem gives correct and meaningful results inall cases satisfying the conditions under which it is proved,i.e. that vectors E and H describe the same electromagneticfield. A frequently used example to demonstrate arbitrari-ness of the above interpretation of the Poynting's vector arecrossed static electric and magnetic fields (due, for example,to a charged capacitor and a permanent magnet). Appliedformally, Poynting's theorem in that case states thatelectromagnetic energy is circulating perpetually. Electricand magnetic fields in this case are, however, independent —they are described by pairs of decoupled Maxwell'sequations. Mathematically we can construct the Poyntingvector, but we are then combining independent physical

phenomena. We could, equally, combine, in a convenientmanner, the gravitational and electric or magnetic field, buta meaningful physical result could not thus be obtained.

7.1 Poynting's theorem in complex form [2, 12, 13]

Consider time-sinusoidal steady-state sources in a linearmedium of complex parameters o = o+j0, e = e'—je"and n = fi —j'n". Maxwell's equation in complex formvalid for that case are given in eqns. 9 and 10. We knowthat the real part of the expression —J*E (asterisk denotesconjugate) represents time-average power density of sources,and that imaginary part of —J*E represents amplitude ofthat part of power density of sources which describesperiodic exchange of energy between sources and thesurrounding medium. Using eqn. 9 and the conjugate ofeqn. 10 we can construct an expression in complex formanalogous to that in eqns. 37 or 39. The result is

- f J\Edv = f oE2dv+)CJ f (pB2 -e*E2)dv•>v Jv Jv

(ExH*)dS

The complex vector

P = ExH*

(40)

(41)

is known as the complex Poynting vector.To interpret eqn. 40, let us divide it into real and

imaginary parts. We obtain

Re J - | JfEdv } = | {oE2 + co(p."H2 + e"E2)} dv

+ Re If (ExH*)ds\ (42)

I m j - f JjEdv) = co f Qi'H2-e'E2)dvI "V I JV

(ExH*)ds\ (43)Im

The left side of eqn. 42 represents time-average, or activepower of generators in domain v. The volume integral onthe right side represents time-average power loss. (Inciden-tally, we see that if e = e' —je" and i± - \i —jn", then e"and ii" must be positive for passive media.) The surfaceintegral, according to the law of conservation of energy,must be interpreted as time-average power flow through Sout of the domain. If this last term is positive, we say thatthe surrounding medium represents an active load forgenerators inside S.

As mentioned above, the left side of eqn. 43 representsintegral over v of amplitudes of those parts of power ofelemental generators inside volumes dv which describeperiodic exchange of energy between elemental generatorsand their surroundings. It is termed the reactive power ofgenerators in domain v. The volume integral on the rightside of the equation represents the difference of maximalmagnetic and electric energies in v (// and E are effectivevalues), multiplied by GO. The surface integral represents theintegral of the amplitude of that part of the Poyntingvector which describes periodic flow of energy out of the

52 IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

surface and into it. If it is nonzero, we say that the sur-rounding medium represents a reactive load for generatorsinside S.

Although perhaps not obvious at the first glance, theabove definitions of active and reactive power are identicalto those in network theory. In particular, if S encloses alinear network with time-harmonic generators, surfaceintegrals in eqns. 40, 42 and 43 are zero (by networktheory assumption, the field outside the network elementscan be neglected). It is not difficult to conclude then thateqn. 40 is the general form of the theorem of conservationof complex power, and eqns. 42 and 43 of active andreactive power, respectively, known to be valid for suchnetworks.

Applications of Poynting's theorem are numerous. In themajority of applications the theorem serves to determinelosses or reactive power on the basis of known field vectorsE and H on a closed surface. Determination of powerradiated by antennas, resistance and internal reactance ofconductors of circular cross-section at arbitrary frequency,and determination of dynamic resistance of a bar conductorin a groove of an electrical machine are probably the best-known examples. (In computing resistance and internalreactance of conductors, Poynting's theorem requiressurface integration only, while otherwise much morecomplicated volume integration would be necessary.)However, Poynting's theorem is also one of the basictheorems, using which other useful theorems of the electro-magnetic field can be proved.

8 Uniqueness theorem for solutions of Maxwell'sequations

The uniqueness theorem states that if the field inside aclosed surface S was zero at all points for t > 0, the fieldinside S is uniquely determined by sources in S and bytangential components of E or H on S for / > 0. If themedium is linear in the restricted sense, the field at t = 0need not be zero, but must be known. In the case ofsteady-state time-harmonic fields in a linear medium,complex field vectors inside S aie unique provided thattangential components of E or H on S are known and thatlosses are present, however small.

The proof of the theorem is based on eqn. 37 and thetheorem on distribution of energy in the field. Consider thefield in an arbitrary medium. Imagine a surface S enclosinga domain v with sources Jt. Assume that the field for t < 0was zero at all points of v and on S. This means that /,-, Pand M were also zero for t < 0. Let us determine theconditions under which the field inside S is unique forr > 0 .

Suppose that inside S two different fields are possible(due, for example, to different sources outside S). Sinceinside S the sources of the two fields are the same, thedifferences AE, AB etc. of their field vectors satisfy in Sthe source-free Maxwell's equations. They also satisfy,therefore, eqn. 37 with left side set to zero. The first termon the right of eqn. 37 represents energy transformed intoheat. It is, therefore, always positive. The second termrepresents the work done in establishing the difference fieldin S existing at the instant t; it is, obviously, also positive.The last term in eqn. 37 (the surface integral) can, inprinciple, be both positive and negative, as well as zero.

Suppose that the surface integral in eqn. 37 is zero.Since both remaining terms are positive, and the left sideof the equation equals zero, each must be equal to zero.

IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

Now AJ is due to AE, so AE is certainly zero, and thereforeAJ is also zero. As d(AB)/dt = —curl(AE), AB is a con-stant in time, which is zero for AB = 0 at t = 0. AM is aconsequence of AB, so it is also zero, and thus AH = 0 aswell. We similarly conclude that AD = 0. Thus, a sufficientcondition that the field inside S be unique for t > 0 if itwas zero for t < 0 is that the surface integral for the dif-ference field be zero for t > 0.

The surface integral for the difference field is zero undera number of circumstances. For example, if E or H arespecified on S for t > 0, then AE = 0 or AH = 0. The leaststringent requirement is that the tangential components ofE, or of H, be specified on S for t > 0. Then the differencevectors AE, or AH, are normal to S, and consequently(AEx AH)dS = 0. This completes the proof of thetheorem.

In the case of media linear in the restricted sense, thefield at t — 6 need not be zero, but must be specifiedthroughout v. By the same argument as above, using eqn.39 instead of eqn. 37, the same conclusions are easilyreached.

In the case of complex representation (which is, ofcourse, valid only for media linear in the general sense) westart from eqn. 40. The difference field again satisfies thisequation with left side set to zero. If a i= 0, /u" ̂ 0 ore" ^ 0, and if the surface integral is zero, then the dif-ference field vectors AE and AH are obviously zero. But ifa = IJ." — e" = 0, the proof fails, the equation yieldingonly that in such a case the time-average magnetic-fieldenergy of the difference field in v equals the time-averageelectric field energy of the difference field, i.e. uniquenessdoes not hold (Reference 14, pp. 100—103). This seemingparadox is easy to explain. In sinusoidal steady state, repre-sented by complex field vectors, information about theinitial excitation of the field is lost, and many field con-figurations may be possible. In addition, it is well knownfrom network theory that sinusoidal generators willproduce sinusoidal steady state in lossless networks only ifswitched on at particular instants of time. This is true alsofor sinusoidal field sources in lossless media.

8.1 Uniqueness of solution of electrostatic fields

It is interesting to note that the above proof is not directlyapplicable to electrostatic fields. Only if we consider thatthe process of establishing the final static field in eqn. 37,from which we proved the theorem, is meaningful. Thus,the electrostatic field established in S from zero-field statethrough a transient process is unique if the tangential com-ponent of E is known throughout the process (irrespectiveof the properties of the medium inside S).

In the case of a linear homogeneous medium, uniquenessof the electrostatic field is guaranteed by various, lessstringent conditions [15]. The starting point is Poisson'sequation for electrostatic potential, div(grad F) = —p/e,which is obtained by combining Maxwell's equationdiv D = p with D= eE and E = — grad V. By usingGreen's first identity, the following relation is obtained fora difference potential AV= Vx — V2 of, presumably, twopossible solutions Vx and V2 in 5":

on

where d/dn denotes derivative in the direction of the out-ward normal to S. (Of course, the difference potential AVsatisfies Laplace's equation, i.e. Poisson's equation with

53

p = 0.) A number of possibilities exist for AKto be zero,i.e. for the two solutions to be identical inside S:

(a) If AV = 0 (i.e. Vx = V2) at all points of S, the leftside of the equation is zero. The right side is zero if AV =Vx — V-i = Vo = constant. But we have that Vx = V2 onS, so that Vx = V2 throughout v.

(b) If d(AV)/dn = 0 (i.e. bVjdn = dV2/dn) on S, againAV = Vx — V2 = Vo throughout v\ but now Vo need notbe zero.

(c) If AV=V1-V2 = Vo on S (Vo need not beknown), and

By analogous procedure, combining the second of eqns. 44and the first of eqns. 45, we have

s dn

again AV= Vx — V2 = Vo throughout v\ if S represents asurface of a charged conducting body with a given chargeQ, then

Thus charge distribution on, and the field of, a chargedconducting body (or a collection of such bodies) is unique.

Any combination of the above three conditionsguarantees the unique potential throughout the domain v.

9 Reciprocity theorem [14, 16, 17]

The reciprocity theorem is one of the most powerfultheorems of the electromagnetic field. It is a generalisationof the same theorem valid for electrical networks. In thecommon form, which will be proved below, it is valid forisotropic linear media and time-harmonic fields describedby complex field equations.* Essentially, it states that twofields due to any two sets of time-harmonic sources of thesame frequency, existing in the same medium (simul-taneously, or one at a time), are connected by a specificdifferential, and a corresponding integral scalar relation. Inother words, all possible time-harmonic fields of the samefrequency, existing in the same medium, are to some extentinterdependent.

To prove the theorem, consider two source distributions,(JiX,JmiX) and (/i2,Jmi2), of the same angular frequencyOJ, acting inside the same linear medium. Let the fields dueto these sources be (Ex, J ^ ) and (E2, H2), respectively. Tosimplify notation, letZ = jcofi and Y = (a + /we). Maxwell'sequations in complex form (eqns. 9 and 10 with added im-pressed magnetic current density, Jmi) for the two fieldshave the forms

= -ZHX -Jmix, c\xr\Hx = YEX + Jix (44)

cm\E2 = -ZH2 -Jmi2, cw\H2 = YE2 +Ji2 (45)

If we multiply the first of eqns. 44 by H2 and the second ofeqns. 45 by Ex (scalar product), and have in mind eqn. 36,by subtracting one of the equations thus obtained from theother we get

-divCtf, xH2) = ZHXH2 + YE2EX +JmixH2

+ Ji2Ex (46)

•The theorem can be extended to anisotropic media (Reference 4,p. 234) [18, 19], to arbitrary time variations of the field (but in-cluding advanced as well as retarded fields of sources) [20], toLaplace transforms of the field vectors and linear media in therestricted sense (which was brought to the author's attention byDr. A.R. Djordjevic) and, of course, to time-constant fields as alimiting case.

54

x Hx) = ZH2HX + YEXE2 +Jmi2Hx

+ JixE2 (47)

By subtracting eqn. 46 from eqn. 47 we thus obtain anequation relating the two fields and their sources fromwhich properties of the medium are lost. (This is obtainedonly because we assumed that the fields are of the sameangular frequency and consider complex representation ofthe field equations.) Thus

E2 —j x E2 x — Jm\2Hx (48)

which represents the most general statement of thereciprocity theorem in differential form. By multiplyingthis equation by a volume element dv, integrating over avolume v enclosed by a surface S, and applying the diver-gence theorem to the left side of the equation, the mostgeneral statement of the reciprocity theorem in integralform is obtained as

{(ExxH2)-(E2xHx)}dS =

-JmiiH2)-(Ji2Ex -Jmi2Hx)}dv (49)

Essentially this form of the theorem was proposed byRumsey [18].

Various special cases can be derived from the generalform of the reciprocity theorem. As the first example, if Sencloses no sources, the right side in eqn. 49 is zero. On theother side, if sources of both fields are confined within afinite volume and S expands to infinity, the left side of eqn.49 also becomes zero. This can be shown by considering thefar fields of the two source distributions, or alternatively byassuming that there are losses, however small, in which caseall field vectors decrease exponentially with distance fromthe sources. If the surface integral is zero when S expandsto infinity, however, it is zero whenever 5 encloses all thesources; the right side of eqn. 49 is then the same in all suchcases. Thus

{(Ex x H2)-(E2 xHx)}dS = 0 (50)

(when S encloses all the sources or no sources at all). Thiswas, essentially, the original form of the theorem as derivedby Lorentz (Reference 16, pp. 100-103).

Alternatively, from eqn. 49 we obtain also that

f (/nE2 -JmixH2)dv = f (Ji2Ex -Jmi2Hx)dv(51)

(all sources in v).Many other special forms of the equation expressing the

reciprocity theorem can be devised. For example, if Jmix =Jmi2 = 0, or Ji2 =Jmix = 0, or Jix =Ji2 = 0, eqn. 51 isfurther simplified; if the impressed electric field is used torepresent sources instead of impressed currents, an alter-native formulation is obtained and so on.

Applications of the reciprocity theorem are numerousand diverse. They can be divided into two main categories.In the first category fall certain proofs of general properties

IEEPROC, Vol. 128, Pt. A,No.l, JANUARY 1981

of fields or devices using fields. The second category relatesto solutions of specific field problems, often in a surprisinglysimple manner when compared with any other method;these applications are based on the possibility of using, ineqn. 51, one distribution of sources the field of which weknow (usually an electric current element). In manyinstances the field of the other distribution of sources canthen be calculated relatively easily from eqn. 51. We shallnow illustrate both types of application of the reciprocitytheorem by several examples.

Let us first consider two arbitrarily-oriented currentelements JudVi and Jl2dv2, situated in a homogeneousmedium. We know that they produce identical fields (withrespect to corresponding co-ordinate systems), exceptpossibly for amplitude. According to eqn. 51, this identityof the fields in terms of the reciprocity theorem can beexpressed as /,-1E2 dvx — Ji2Ex dv2.

Consider now an electric-current element Jjidvi, and amagnetic-current element Jmi2dv2, again situated in ahomogeneous medium. Eqn. 51 now yields, for arbitraryorientations of the elements

JixE2dvx = —Jmi2Hxdv2 (52)

This means that E2 due to the magnetic-current elementis the same as — Hx due to the electric-current element,except for the amplitude factor. We shall see in the nextSection that this is a special case of duality in electro-magnetic fields.

As the next example, consider two filamentary-impressedelectric-current elements JiXdvx = JixdSxAlx and Ji2dv2 =Ji2dS2Al2, connected to two arbitrary structures. Thereciprocity theorem in eqn. 51 after simple rearrangementsnow gives

(JildS1)E2Al1 = (Ji2dS2)ExAI2

U2X = E2 Alx represents the voltage across the first currentelement due to the field of the second, and vice versa. Inaddition, JudSi and Ji2dS2 are current intensities Iix and/,-2 in the two filamentary current elements. So Iix U2i =hi U\2 > which is the reciprocity relation of electrical net-works for two constant-current generators. (Note, however,that the proof given above is far more general: couplingbetween the two elements need not be by means of apassive electrical network. For example, the two currentelements can be generators driving two distant antennassituated in an arbitrary linear medium.) There are twointeresting corrolaries following from this result:

(a) Mutual admittance between any two ports of a linear//-port system (not necessarily a network) are equal.

(b) Receiving and transmitting radiation patterns of anantenna are equal; for, if / f l =/,-2, Un and U2l otherwisecould not be equal for all orientations of the two antennas.

As the first example of the usefulness of the reciprocitytheorem for computing the field of a given, relatively-complicated distribution of sources, assume that sources1 are in the form of an electric current element, and sources2 in the form of a small one-turn coil carrying electriccurrent of intensity Ii2 (Fig. 1). Let the cross-sectionalarea of the coil wire be dS2. Reciprocity relation in eqn. 51then becomes

JnE2dvx = f Ji2E1dS2dl2 = Ii2 6 Exdl2'AC JAC

But according to Stokes' theorem, and Maxwell's eqn. 9,

IEEPROC, Vol. 128, Pt. A, No. 1, JANUAR Y 1981

the line integral of Ex equals —jojnHxn&S (see Fig. 1).Thus

JixE2dvx = — l/cj/u -^- nLh

(53)

By comparing eqns. 52 and 53, we see that the field of asmall current loop is the same as that of a magnetic-currentelement.

As the final example, consider a small electric-currentloop in air lying on the plane surface of a thick dielectricslab of permittivity e. We wish to determine the far field ofthe loop in the air. Note that as a boundary-value problemthis would be very difficult to solve.

As the first step, let us substitute the loop by theequivalent magnetic-current element Jmi\dvx, according toeqn. 53. Assume that source 2 is a distant magnetic-currentelement Jmi2dv2. The reciprocity relation in eqn. 51 yields

JmilH2dvx = Jmi2Hxdv2 (54)

The field H2 can be computed relatively easily. We knowthat the field of a magnetic-current element in a homo-geneous medium is dual to the field of a Hertzian dipole(the electric current element); therefore the incident fielddue to the second element will be, locally, a plane wave.The field H2 is the resultant field of that incident planewave and of the wave reflected from the surface of the slab,which can be computed using known formulas for plane-wave reflection. By varying the position and orientation ofthe second magnetic-current element it is possible to deter-mine the required field with relatively little effort (whichwill not be done here).

9.1 Reciprocity theorem for electrostatic fields

A scalar relation exists also between two electrostatic fieldsand their sources existing in the same medium. Let px andp2 be two arbitrary charge distributions, and let Dx, Ex andVx, and D2, E2 and V2, respectively, be the correspondingdisplacement vector, electric-field strength and potential.We have that

divDx

divD2

= Pu

= P2>

Ex

E2

= -grad Vx

= -grad V2

(55)

(56)

Let us multiply the first of eqns. 55 by V2, the first ofeqns. 56 by Vx, and then subtract the second equation,thus obtained, from the first. After simple rearrangementsand noting that div (fF) = f div F + F grad / we obtain

PiV2-p2Vx = div(V2Dx-VxD2)

-(E2DX-EXD2) (57)

A S = n AS

Fig. 1 Illustration of the reciprocity theorem

55

If the medium is linear and isotropic the term in the lastparentheses vanishes, and so we have

-PiVx = diw(V2D1-V1D2) (58)

This is the statement of the theorem in differential form.By multiplying both sides of the equation by dv and inte-grating over a volume v enclosed by a surface 5 we obtainthe statement of the theorem in the integral form

f (piV2-p2V1)dv = & (V2DX - VxD2)dS (59)Jv JS

If S encloses all sources of both fields (both charge distri-butions), the surface integral is the same for all suchsurfaces. Since it is zero when S expands to infinity, it iszero whenever all sources of the two systems are enclosed(or, of course, when S encloses no charges)

& {V2DX -VxD2)dS = 0J s

(60)

(all charges inside or all outside S; linear medium). (Thisexpression corresponds to the general statement in eqn.50.) In that case we have also

f (PiV2-p2Vx)dv = 0Jv

(61)

(all charges inside v; linear media).If one system consists of n point charges Qx,. . . , Qn,

and the other of m point charges Q\, • • • ,Qm, thisbecomes

Q'kvk (62)

where Vj is the potential due to charges Q\, . . . , Q'm at thepoint where the charge Qj is situated, and conversely forVk. This is the, slightly generalised, usual expression of thereciprocity theorem for electrostatic fields, known as theGreen reciprocation theorem (Reference 10, pp. 34—35,54) [21].

As an example of the application of this theorem, con-sider a point charge Qx situated at a point M in a homo-geneous medium in an arbitrarily electrostatic field. Let Sbe a sphere centered at Qx, and let there be no othercharges inside S (Fig. 2). Eqn. 59 then yields

Qi(Y2)atM = <f V2DxdS-vA D:Js J s

dS

V2DxdS (63)

Fig. 2 Average potential on a sphere S in a homogeneous dielectricis equal to the potential at its centre, provided that is encloses nocharges

since V\ is constant on S and S encloses no sources ofZ>2-We know that Dx on S is given byDx = Qxr/(4nr3). SincerdS = rdS (see Fig. 2), we have

)at M — 4nr2 V2dS (64)

This is the expression of another theorem on electrostaticfields: electrostatic potential at any point inside a homo-geneous medium is equal to the average potential of anysphere centered at that point, provided that S encloses nocharges.

10 Theorem on duality of electromagnetic field due toelectric and magnetic sources [14, 22]

If two physical phenomena are described by equations ofthe same form, solutions to the equations will also be of thesame form. It is then said that the equations are duals ofeach other, and the corresponding quantities are termeddual quantities. The concept of duality is very useful, forif we know the solution of one of the two dual problems, weobtain the solution of the other by simple interchange ofsymbols. A theorem we shall now prove states thatequations describing the fields originating from electric andmagnetic sources under certain conditions are dual, andspecifies the dual quantities. Thus if we know a solution ofthe field equations corresponding to electric sources, wecan automatically write down the solution corresponding todual magnetic sources.

Consider first eqns. 27 and 28 for two separate cases:when Jmi = 0, and when /,• = 0. It is easy to conclude thatthese pairs of equations are not dual, unless either / = 0(perfect insulator), o r / i s considered as part of/,-, accordingto the compensation theorem. Thus, in the time-domainduality of fields due to electric and magnetic sources ispossible only under these conditions.

Consider now complex equivalents of these equations,which are given in eqns. 44 (with subscripts T omitted).If we assume that electric and magnetic-impressed currentsare acting one at a time, we obtain for fields (Ee,He) dueto electric currents

= ~ZHe, cmlHe = YEe+Jt (65)

Similarly, for fields (Em,Hm) due to magnetic currents wehave

curl£m = -ZHm - = YEr (66)

By comparing eqns. 65 with eqns. 66 we conclude that thelatter are obtained by a systematic interchange of symbolsin eqns. 65, as follows:

Ji-+±J

(67)

If we wish to obtain eqns. 65 from eqns. 66, it is necessaryto interchange symbols in eqns. 66 according to the samerules, but read from right to left. This completes the proofof the theorem.

As the first example of application of the above dualityrelations, note that in homogeneous media it is possible toobtain E and H due to magnetic sources from potentials Ae

(dual to 4 ) and Vm (dual to V) as

E = - - c u r M c , H = -grad Vm -jcoAe (68)

56 1EEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

Ae is given by eqn. 15 if ju is substituted by e, and /by Jmi.Vm is given by eqn. 16, if e is replaced by n, and p by pm.

As the second example, note that the field of a magnetic-current element (small electric-current loop) is obtainedfrom the field of an electric-current element (Hertziandipole) if interchange of symbols is performed as indicatedin eqn. 67.

As the third example, recall that the boundary conditionfor magnetic-field vectors Hi and H2 at two close points ontwo sides of an electric-current sheet of surface currentdensity Js reads

= Js (69)

in directed from 2 to 1).The analogous boundary condition for the vector E on

the two sides of a magnetic-current sheet of magneticsurface current density Jms is, by duality

E2) = (70)

(n directed from 2 to 1). These expressions will be useful inthe following Section.

As the final example, if medium 2 is a perfect electricconductor, the boundary condition in eqn. 69 becomes

nx H1 = Js (71)

(a perfect electric conductor).We may now define a perfect magnetic conductor by

analogy, requiring that eqn. 70 in that case becomes

- « x (72)

(the definition of a perfect magnetic conductor).Note, finally, that dual quantities in eqn. 67 are not

unique (apart from the signs indicated). For example, com-binations with Y-+—Z and Z-> — Y are theoreticallypossible, but practically meaningless.

11 General equivalence theorem and some special cases

Consider a domain v limited by a surface S' in an arbitraryelectromagnetic field. Let S be a closed surface pressedtightly over S', and v the domain outside S (Fig. 3). Assumethat primary and secondary sources of the field exist bothin v and in v. We wish to replace sources in v by anothersystem of sources which leaves the field in v unchanged.There is an infinite number of such source distributions inv and/or on S'. All such source distributions are said tobe equivalent with respect to domain v. According to theuniqueness theorem, all source distributions in v and onS' which produce the same tangential components of E orH on S are equivalent with respect to v. This is the generalequivalence theorem.

Note that, in principle, the theorem is valid whenever

E.H

ItE.H

p u (w i t h respect , ' f "to domain v) v-^>

s'

Fig. 3 Illustration of the general equivalence theorem

Sources equivalent with respect to domain v limited internally by Sin the general case can be situated in domain v and/or on surface S'-* electric impressed current=> magnetic impressed current

the uniqueness theorem holds, i.e. for arbitrary media inv and v if uniqueness conditions for such a case are met.In practice, however, it is usually very difficult to findequivalent sources except if the medium is linear (time-harmonic and static fields) or linear in the restricted sense(fields with arbitrary time dependence).

Note, also, that we denote by v a domain in which, andby S' a surface on which, equivalent sources are distributed.The surface S pressed tightly over S', limiting the domainv from inside (Fig. 3), will also play an important role infurther discussions. A clear distinction should therefore bemade between S and S'.

It is useful to compare concepts of compensation andequivalence. Compensation implies replacement of inducedsources (or part of them) by the same imaginary impressedsources at the same points. Equivalence implies replace-ment of any sources (impressed and/or induced) by anothersystem of impressed sources, generally distributed in adifferent manner.

The main application of the equivalence theorem is forfinding equivalent sources which take into account theinfluence of substance (or, alternatively, homogenise themedium), so that the formulas for retarding potentials orretarded fields can be used. Such procedures can be dividedinto two principal categories:

(a) those which are aimed at determining equivalentsources inside S'

(b) those which are aimed at determining equivalentsources on S' itself.Both are of invaluable aid in solving many classes of fieldproblems, exactly or approximately, as illustrative exampleswill show.

11.1 Equivalent volume sources (images) [14, 16, 22]

That an infinite number of equivalent sources inside S'might exist is obvious from many well-known simple cases.For example, the electrostatic field outside a uniformly-charged spherical cloud of radius a is exactly the same forall concentric spherical (volume or surface) charge distri-butions of the same or smaller radii, carrying the same totalcharge; the extreme cases are a uniformly charged sphere ofradius a and a point charge at the centre of the cloud. Thereader will easily recall many other simple examples of thistype of equivalence.

An important class of equivalent sources of this type isencountered in so-called image theory, originally due toLord Kelvin [23], and further promoted by Maxwell(Reference 6, chap. XI), in both cases for electrostatic fieldonly. The general philosophy of the image theory isillustrated in Fig. 4. Shown in the Figure is a homogeneousmedium inside which a body (occupying domain v) ofdifferent electromagnetic properties is situated. Outside thebody, in domain v, arbitrary sources exist. According to theequivalence theorem, the influence of the body on the fieldin v can be replaced by certain equivalent sources in v,assuming that the medium in v is now the same as in v. Theequivalent sources are known as images of primary sourcesin the body, since in some simple cases they are analogousto optical images. There are usually many different images.The term 'images' is frequently used only for the simplestequivalent sources of this type.

As the first example, consider sources in the form ofcurrent elements and point charges situated in a vacuumnear a plane surface of perfect electric conductor (Fig. 5)-Boundary conditions require that the tangential component

IEEPROC, Vol. 128, Pt. A, No. 1, JANUAR Y1981 57

of E be zero on S. It is easy to conclude that the same con-dition for E on 5 is obtained if the conductor is removedand the images shown in the Figure introduced instead. Bythe duality theorem, images in a plane surface of a hypo-thetical perfect magnetic conductor are as shown in Fig. 6.

satisfied at a sufficient number of points — the so-calledpoint-matching method). This application of the equiv-alence theorem has become increasingly popular within thelast few years, due to its conceptual and computationalsimplicity.

Etangor Htang

primarysources

Etang o r Htang

, U ,o

(with respect . v ! ' ,to domain v) pnmary\i v'

sources \>^

S

-S'images'

of primarysources

Fig. 4 Illustration of general principle of image theory

<

2 i\ - P i *

\

•Qi

Fig. 5 Illustration of images in plane surface of perfect electricconductor

ll

si

\

Fig. 6 Illustration of images in plane surface of perfect magneticconductor

Because of the finite velocity of propagation of electro-magnetic disturbances, in the case of time-dependent fieldsimages of the above type are possible only in plane surfacesof perfect conductors. In time-constant fields such imagesare possible also for some other simple structures. Well-known examples are electrostatic images in a conductingsphere, in a conducting circular cylinder and in a planesurface dividing two homogeneous dielectric media.

Another important class of problems which can besolved approximately using equivalent volume sources arecharged conductors in electrostatics or grounding con-ductors of moderately complex forms etc. In all these andsimilar cases we remove the body and imagine, inside thedomain occupied earlier by the body, a convenient distri-bution of sources with unknown magnitudes, which wethen determine so that boundary conditions are approxi-mately satisfied (for example, by requiring that they be

/1.2 Equivalent surface sources [14, 16, 22]

Consider now the system of sources sketched in Fig. 7. Wewish to substitute the volume sources in v by surfacesources on S' equivalent with respect to v. S' must in thiscase be smooth.

//'/ <! i —

V // (with respect todomain v )

Fig. 7 Sources inside v can be substituted by surface sources onS' equivalent with respect to v

According to the uniqueness theorem, for the field inv due to sources in v to be unique, it suffices to specifytangential components of E or H on S. The same is true ifthe sources are surface sources on S'. But, according toeqns. 69 and 70, surface sources are determined by thetangential components of E and H on both sides of S'.Therefore we must specify, in addition to tangential com-ponents of the real-field vectors on 5, also tangential com-ponents of E and H just inside S''. As far as the field in v isconcerned, these are completely arbitrary. According to theuniqueness theorem, these components then determineuniquely the field inside S'. Thus with surface sources, bynecessity, we always have unique fields outside S and insideS'.

It is, obviously, most convenient to specify a null fieldinside S'. Because in that case also, as explained, we haveuniqueness requirements simultaneously for two domains(true field external to S and zero field internal to S'), inthe general case we must specify both tangential E andtangential H on S. Only in special cases, when inside S' is aperfect electric or magnetic conductor, does one vectorsuffice, for the following reason. The equivalent surfacesources produce, of course, zero field inside S', and there-fore such conductors have no effect if electric and magneticsurface sources are considered simultaneously. But if weapply the superposition theorem and consider themseparately, they both produce nonzero field inside S'. FromFigs. 5 and 6 it is seen that the electric-current elementpressed over the perfect electric conductor produces nofield (it is cancelled by the image field, for in this case theconductor locally can be considered as plane); the same isvalid for the magnetic-current element pressed over theperfect magnetic conductor. Therefore only one of theequivalent surface currents can be considered to act. Note,

58 IEEPROC, Vol. 128, Pt. A, No. 1, JANUAR Y1981

however, that this current will induce on the surface of theperfect conductor a distribution of current of the othertype identical to the equivalent surface current we removed.

Assuming zero field inside S', eqns. 69 and 70 yieldequivalent surface sources

/ . = nx H, Jms = - (73)

where unit vector n normal to S and S' is directed into v(see Fig. 7). These equivalent surface sources, corre-sponding to zero field inside S', are frequently termedHuygens' sources, because the procedure represents ageneralisation of the optical Huygens' principle. Thesurface S' is then referred to as the Huygens' surface. Sincezero field exists inside a Huygens' surface, we can assumeany medium inside it. This is extremely important, for wecan thus 'homogenise' the medium. Note, however, that Eand H in eqn. 73 are unknown. In practical applications ofthe theorem they are most often estimated. Equivalentsurface sources were first proposed by Love [24], and inthis form by Schelkunoff [25].

Essentially the same reasoning applies for surfaceelectrostatic equivalent sources. Uniqueness of solution ofPoisson's equation is satisfied if the potential on S isspecified, and it is required that it be zero just inside S'covered with equivalent surface sources. We again need twotypes of sources to ensure uniqueness outside S and insideS'. It can be shown that surface charges and surface dipoles(double layer of equal charges of opposite signs) suffice forthat purpose, but we shall not go into details here.

Huygens' sources are used extensively for approximatelydetermining the field of various radiating structures in theform of apertures in otherwise homogeneous media(electromagnetic horns, reflector antennas, apertures in aconducting wall etc.). If we can estimate the E and H fieldson the aperture, we can determine the equivalent surfacesources given in eqn. 73. Since we know the field of electricand magnetic current elements in a homogeneous medium,the solution is obtained by integration.

A special but important case of equivalent surfacesources is the case shown in Fig. 8. In the field of sourcesexisting in a homogeneous linear medium, a body (withoutsources) of different electromagnetic properties is situated.The result obtained by the equivalence theorem in this caseis known as the induction theorem [25].

e ,u ,cr

Fig. 8 System to which induction theorem is applicable

With respect to domain v of the body, Huygens' sourcesare given by eqn. 73 with n substituted by — n, where E andH are true fields on S (tangential components of E and Hare equal on the two sides of S and S'). These sources pro-duce the true field in v and zero field outside. Now, thebody in Fig. 8 is a source of secondary field, known as thescattered or reflected field. Denote the scattered (reflected)field by (Es, H8). The sources of this field being inside S,we can determine equivalent Huygens' sources on S' with

respect to v. Huygens' sources of the scattered field in v are

Js = nx Hs, J^ = -n x Es (74)

These sources produce zero field in v, and scattered field inv. Therefore the sum of the two Huygens' sources

Js tot = -nx (H-Hs), Jms tot = nx (E~ES)(75)

produce real field in the body and scattered field outsidethe body.

Since (E, H) is the field of the sources in the presence ofthe object, and (ES,HS) the field due to the object, the dif-ference fields in eqn. 75 are fields which would exist if theobject were not present. This is usually termed the incidentfield, (Ei, Hi). Thus, surface sources

's tot = —n x Hu 'ms tot = n x (76)

produce the real field in the object and scattered fieldoutside it. This result is known as the induction theorem.

The total sources in eqn. 76 are usually known, since Et

and Ht are field vectors due to known sources in theabsence of the object. However, determination of the fieldof all the sources is far from simple, because the medium inwhich they act is not homogeneous (they act in thepresence of the object).

11.3 Theorem on equivalence of electric andmagnetic impressed currents

This theorem states that in homogeneous linear mediaequivalence can be established between complex electricand magnetic impressed current densities, with respect tothe region outside the currents. Its original form is due toMayes [26]. Such equivalence also exists in linear media inthe restricted sense and arbitrary time variation of thesources [27].

To prove the theorem, consider eqns. 33 and 34 in com-plex version in the case of arbitrary homogeneous media

AE - YZE = I ZJi + - grad p) + (curl Jmi) (77)

AH-YZH = (-curl / , ) + | YJmi + - grad pm\ (78)

As before, Z =/COJU and Y=(p + jcoe). Since superpositionapplies, the field vectors E and H can be considered to havetwo components each: that due to electric sources, and thatdue to magnetic sources. The two sources are described byexpressions in the first and the second parentheses in eqns.77 and 78, respectively.

Consider eqn. 77, and let/m,- be given. We wish to deter-mine electric sources equivalent to given magnetic sources.Noting that from equations curl H = (YE + / , ) and div E =p/e it follows that p = — (e div/,)/r, the Afield of thesesources is identical at all points provided that

ZJ{ - — grad (div/,) = curl/mi- (79)

This equation is not sufficient for determining /,-. We cansolve it for /,• only if we assume that div /,- = /(r), wheref(r) is a suitably-chosen function possessing the necessaryderivatives. The simplest choice is div /,- = 0 (or a constant),in which case the equivalence condition in eqn. 79 becomes

1EEPR0C, Vol. 128, Pt. A, No. 1, JANUAR Y1981 59

- c u r l / m / (80)

This electric current produces an ^-field identical to thatdue to magnetic currents Jmi at all points. From eqn. 78it is evident that the //-field is not the same inside theregion containing the sources. However, according to theuniqueness theorem, outside the sources, both E and Hvectors due to such electric and magnetic currents areidentical.

In analogous manner, starting from eqn. 78 andassuming that div Jmi = 0, we can find the magnetic currentdistribution Jmi, equivalent to a given electric current distri-bution /,-, with respect to the region outside the sources.The result is

= —y curl / , (81)

This completes the proof of the theorem.As an example, consider a straight cylinder with uniform

distribution of impressed electric currents directed alongthe cylinder (Fig. 9). The curl of Jt is zero everywhereexcept on the cylinder surface. There, from the integraldefinition of the curl, we obtain

1 /.• x ncurl/,- = /ii /.A/ =1 * AlAh ' Ah

According to eqn. 81, equivalent magnetic currents aresurface currents, of density

y

which is a solenoid over the cylinder surface carryingmagnetic surface current.

Fig. 9 Cylinder with axial electric impressed currents

12 Theorem on extended boundary conditions

In its original form, the theorem on extended boundaryconditions relates to perfectly-conducting bodies (electricor magnetic) situated in an arbitrary incident field. (Inelectrostatics the body need only be conducting and, ifcharged, the incident field — field of other charges — neednot exist.) It states that correct surface distribution ofsources induced on such a body can be obtained by re-quiring that the total field be zero on any closed surfaceinside the body, and not only by requiring that tangentialcomponents of E, e.g. H, be zero on the surface of the

body. The theorem in this form is due to Waterman [28].In electrostatics it was first proposed by Smythe [29] inconsidering a special case.

We shall prove the theorem in a more general case. Theproof follows from the theorem on equivalent surfacesources described in Section 11.2 and the uniquenesstheorem. Consider a closed smooth surface S enclosingarbitrary field sources. Let (n x Ex) x n and (nxH^xnbe the tangential components of the field vectors on S,and let us substitute sources inside S by equivalent surfacecurrents (electric and magnetic) on a surface So pressed onS from inside. To do this, we know that we must specifytangential components of the field vectors, (nx E2)x nand (n x H2) x n, on a surface S' just inside So (Fig. 10).The equivalent surface currents are then given by eqns.69 and 70.

Fig. 10 Relating to proof of theorem on extended boundaryconditions

Note that, once equivalent sources are constructed,inside 5 ' there are no sources. Therefore, if the uniquenessconditions are met, the field inside S' is unique, sincetangential components of the field vectors on S' arespecified. However, the tangential components of E and Hon any arbitrary closed smooth surface S" inside S' arethen also unique (see Fig. 10). Obviously, the converse isalso true: if the tangential components of E and H arespecified on S" instead of on S', the equivalent surfacecurrents are still uniquely determined. Since the conditionsfor determining surface currents in this case are not statedfor the boundary surface, these are not true boundary con-ditions, and are referred to as the extended boundary con-ditions. Thus, equivalent surface currents are determineduniquely also by extended boundary conditions. This is thegeneral statement of the theorem.

When the field vectors on S are given or estimated, theextended boundary conditions result in unnecessarily com-plicated determination of equivalent surface currents (forwe cannot then use directly eqns. 69 and 70). But if equiv-alent currents need to be determined numerically by meansof an integral equation expressing boundary conditions, thetheorem is of great help, because the singular integralencountered in the true-boundary-conditions integralequation is then eliminated.

The most important application of the theorem seems tobe in numerical analysis of perfectly-conducting smoothbodies in an incident field. Consider such a body made of aperfect electric conductor, and let (Eh H{) be the incidentfield. Induced currents Js on the surface So of the body aresuch that the tangential electric field (n x Es) x n due tothese currents on S (just enclosing So) is equal and oppositeto the tangential component (n x £",) x n of Eh i.e. n x Es =—nxEt on S. We can set up an integral equation in thisway to determine Js, but it is inconvenient for numerical

60 IEEPROC, Vol. 128, Pt. A, No. 1, JANUARY 1981

integration, for it has a singular integrand (field and sourcepoints on S may coincide). By the theorem on extendedboundary conditions this singularity can be avoided, for wecan require that n x Es = — n x E{ on any surface S" insideSo. This greatly facilitates the numerical solution of suchproblems. Of course, accuracy of the numerical solutioncan deteriorate if the surface S" is not smooth and/orextended boundary conditions are specified too far fromthe surface of the body.

As a simple example, consider a charged, smooth con-ducting body in a homogeneous dielectric (electrostaticcase). We wish to determine numerically the approximatedistribution of charges on the surface of the body using thetheorem on extended boundary conditions. The methodthen consists in representing surface charge distribution inthe form of a series of n known functions with n unknowncoefficients, and then determining these coefficients byrequiring that potential at n points of a surface S" insidethe body has the same value. If the points are chosensufficiently far from the surface of the body, integrationimplicit in computation of the potential at these points iseasy to perform numerically, and a system of linearequations in unknown coefficients is obtained.

As another example, in determining current distributionalong cylindrical metallic-wire antennas and scatterers, it ismost often required that the axial component of the totalfield be zero not on the wire surface, but instead along itsaxis. This is a typical application of extended boundaryconditions.

13 Theorem on electrodynamic similitude [27, 30, 31]

The theorem on electrodynamic similitude specifiesnecessary and sufficient conditions for an electromagneticfield in a system to be similar to the field in another,geometrically-similar system. It represents the theoreticalbasis for construction of various electrodynamic models.The general theorem on electrodynamic similitude will beproved first, and then some special cases considered asexamples.

A model yields complete information about the object itrepresents only if by measurement of all the quantities ofinterest in the model we can predict the results ofanalogous measurements in the object. In the case ofelectrodynamic models these quantities can be very diverse:current intensity in a conductor, power losses, efficiency ofan induction furnace, radiated power, distribution of eddy-current losses etc. All these, as well as the other quantities,can be calculated if the field vectors E and H are known atall points. Thus, the theorem on electrodynamic similitudeshould define the properties of the model in order that therelation between field vectors in the model and the objectbe unique, as well as to specify this relation.

H'

Fig. 11 Schematic representation of model (left) and object(right)

It is assumed that n = 0-5, / = 0-5, kE = 2 and kjj = 1 (see eqns. 82,84 and 85)

For practical reasons, the model obviously should begeometrically similar to the object. Let the linear dimen-sions of the model be n times those in the object. (Thus,for n > 1 the model is larger than the object, and for n < 1smaller than the object.) If we use analogous coordinatesystems in the model and the object, the correspondingpoints of the model and the object are determined by therelation

r = nr (82)

{Prime denotes the model (Fig. 11).}In the general case, electromagnetic similitude is realis-

able only if media in the model are linear in the restrictedsense. Let

e\r) = ee(r), ), °V) = «K') (83)where primes again denote quantities referring to themodel, and e, m and s are constants. Let, in addition,density of impressed current in the model be; times that inthe corresponding points of the object, and let allphenomena in the model be T times slower, i.e. let also

J'i(r') = MOO and (84)

(Note that / here does not stand for the imaginary unit.) Wewish that the field vectors in the model and in the object beconnected through the relations

E'(r) = kEE(r) and H'(r) = kHH(r) (85)

where kE and kH are known desired constants. The con-ditions ensuring that eqns. 85 be satisfied at all points,which we shall derive below, are the conditions of electro-dynamic similitude.

Maxwell's equations for the model have the form

curl if = — n —jot

curl'//' = o'E'+J't+e

(86a)

(86b)

The curl' operation refers to a point r of the model. Let ussubstitute in these equations the primed quantities accordingto eqns. 82-85. Note that curl' operation then becomescurl//?, since x = nx, y = ny and z = nz. We thus obtain,after simple rearrangements,

(81a)

curl H =CH

Eqns. 87a and 87b, become Maxwell's equations for thefield in the object at the point r (i.e. eqns. 86a and 86bwithout primes) only if all the expressions in parentheseson their right sides are equal to unity. After elementarymanipulations we thus obtain

(88a)

(886)

(88c)

m

j -

1n

1

n

kE

kH

kH

kE

kH/n

IEEPROC, Vol. 128, Pt. A, No. 1, JANUAR Y1981 61

n k(88c?)

Eqns. 88a-d represent the conditions of electrodynamicsimilitude. When these conditions are satisfied, equationsdescribing electromagnetic fields in the object and in themodel are of the same form. Since the model and the objectare geometrically similar, boundary conditions are identical,and therefore solutions of these equations are also similar,i.e. the field vectors in the model and in the object will berelated through the relations in eqn. 85. Eqns. SSa-dguarantee complete electrodynamic similitude (i.e. cor-relation of all quantities of interest in the two systems). Weshall term them therefore conditions of the strict electro-dynamic similitude.

The right sides of eqns. 88a-d are given (size scale factorn) or specified (field scale factors kE and kH). Thus theseequations yield directly the necessary relations betweenproperties of the media, impressed currents and time scalesin the two systems for strict electrodynamic similitude tobe satisfied. Obviously, for any given n an infinite numberof similar electrodynamic systems is possible, correspondingto different values of kE and kH; in all such cases completeelectromagnetic correlation between the model and theobject is possible.

Quite often we are interested in relative field distri-bution only, the level of excitation being irrelevant. In thatcase eqn. 88c is not of interest, and from the other three wecan eliminate the ratio kE/kH. We thus obtain the con-ditions of approximate electrodynamic similitude

ms

T

me

T2

1(89)

It is usually practically impossible to have m =£ 1 (i.e. //and difficult to realise e> 1 (or \\e> 1). We then adoptm = e = 1, in which case eqns. 89 become

T = n, - l/n (m = e=l) (90)

These conditions require that the frequency in the model,and conductivity at all points of the model, be 1 /n timesthose in the object. If n < 1 (model smaller than theobject), the condition s = l/n is not possible to fulfil if theobject is made of copper or silver, for under normal circum-stances these are the best available conductors. This con-dition is not always critical, but it may be of extreme im-portance in models of resonant structures (e.g. models ofresonant cavities).

From the general conditions of electrodynamic simili-tude in eqns. 88a-d, it is possible to obtain various specialcases. As the first example, consider the field in a good con-ductor, in which, by definition, we can neglect the termedE/dt with respect to oE. In that case conditions for strictsimilitude reduce to eqns. 88a-c. If, in addition, no im-pressed currents are present in the conductor, the first twoconditions remain only. Let us choose kE/kH = l/n, andwe obtain

T = mnJ s = 1 (91)

(with good conductors, kE/kH — l/n).Suppose we make the model from the same conducting

material as the object. Then m = 1 also, and the only con-dition becomes T= n2. This means that similitude in suchcases can be attained by simply decreasing frequency in themodel n2 times with respect to that in the object. Indeed, itis well known that sinusoidal current distribution in good

62

conductors is always described by functions of the argu-ment of the form ryjcono, precisely as required by thesimilitude condition.

As the final example, consider the similitude in time-constant current fields. Time derivatives in eqns. 86a and bare then zero, and similitude conditions reduce to eqns. 886and 88c. If we choose kH = n, eqn. 88c gives / = 1, i.e.impressed current density in respective points of the twosystems should be equal, and from eqn. 88b in that cases = l/kE. For example, the model of a resistor (of anyform) made of the same material (not necessarily homo-geneous) as the resistor (s = 1), has l/n times largerresistivity. This is because current intensity in the model isn2 that in the resistor, and voltage across its terminals isn times that across the resistor terminals (kE = 1, and thesize of the model is n times that of the resistor).

14 Theorem on polarisation of time-harmonicfields [4, 22]

This last theorem of the review is not a theorem relating toelectromagnetic field only; it is valid for all time-harmonicvector fields. Since such electromagnetic fields are of veryfrequent occurence, the theorem is of significant interest inelectromagnetic field theory. It is also of interest because itgives a physical meaning to complex vector functions.

If the direction of a vector function of space and timeco-ordinates at a point P of space is constant in time, wesay that the vector field at that point is linearly polarised. Ifthe tip of the vector describes, in the course of time, acircle centered at the point P, the vector is said to bepolarised circularly at that point. If the curve described bythe tip of the vector is an ellipse centered at the point P, wesay that the vector is elliptically polarised at that point.The theorem we shall now prove states that the mostgeneral polarisation of time-harmonic vector fields iselliptical, and that mathematically such a vector can also berepresented as a complex vector.

To prove the theorem, note first that the most generaltime-harmonic vector function of angular frequency CJ canbe represented as

A(r, t) = Ax(r) cos (cot + otx)ix

+ Ay(r) cos (cot + ay)iy + Az(r) cos (cot + otz)iz

(92)where ix, iy and iz are unit vectors of the JC, y and z axes,respectively. In complex notation this can be written as

= Ax(r)ix +Ay(r)iy + Az(r)iz (93)

where Ax(r) = Ax(r) exp (j<xx), and similarly for Ay(r) andAz(r). Separating real and imaginary parts we finally get

A(r) =Ar(r)+jAt(r) (94)

where Ar(f) and At(r) are real functions of position.If we now go back to time domain from eqn. 94, instead

of the expression in eqn. 92 we obtain an equivalentexpression

Air, t) = Re {A(r) exp (/cor)}

= Ar(r) cos cot — Ai(r) sin cot (95)

It is obvious from this expression that the tip of the vectorA (r, t) describes in the course of time a plane curve, whichlies in the plane of vectors Ar and/I,-. It is also obvious thatthe curve is a closed curve. To prove that it is an ellipse, it

IEE PROC, Vol. 128, Pt. A, No. 1, JANUAR Y1981

suffices to prove that it is described by a second-orderalgebraic equation in any co-ordinate system in the plane ofthe curve.

Let us represent the vectors Ar and Aj with respect to anew co-ordinate system x-y in the plane of the two vectorsin the form

Ar = aix+aiy, At = bix + 0/y (96)

Then the projections Ax and Ay of the vector A(r, t) aregiven by

Ax = a cos cot — b sin cot

Ay = a cos cot — 0 sin cot(97)

If these equations are solved for cos cot and sin cot, andthen use is made of the elementary relation sin2 cot +cos2 cot = 1, there results

(oAx - aAyy + (fiAx - bAyf = (a/3 - bet) (98)

This is a second-order algebraic equation inAx and^4y. Thetheorem is thereby proved.

Elliptically polarised field vectors are encountered inpractice far more often than we are inclined to think. Forexample, the radiation fields of practically all antennas,aside from cylindrical dipoles, are generally ellipticallypolarised, except in some preferred directions; this is mostreadily understood if we imagine just a simple T-antenna.Rotating magnetic field in electrical machines is at leastslightly elliptically polarised at points off the machine axis.In at least some regions of cores of three-phase transformersthe field is polarised elliptically, rather than linearly, whichis, probably, an unusual result for many power engineers.(Incidentally, this to some extent makes eddy-current andhysteresis losses in these regions different from those com-puted on the linear polarisation assumption.) Many otherexamples could be given.

As the final example of the application of the theorem,assume we wish to obtain a rotating magnetic field in space(i.e. the field in which the tip of vector B describes a closedthree-dimensional curve), e.g. for demagnetisation of ferro-magnetic bodies of complex shapes. We might be temptedto try to obtain such a field by using three mutually-orthogonal coils with sinusoidal currents of equal magni-tudes and frequency, but different phases. According to theabove theorem this would not work, as only an elliptically-polarised magnetic field would be obtained.

15 Conclusions

General electromagnetic field theorems are extremely use-ful for both understanding certain field properties in astraightforward manner, and for theoretical and numericalfield analysis. The purpose of this review was to presentmost of the electromagnetic field theorems in a single placeand in a unified way. Although most of the theorems havebeen well known for a long time, it is believed that thereview does contain certain fresh ideas.

16 Acknowledgments

The author expresses his sincere thanks to Drs. M.B.Dragovic and A.R. Djordjevic for many fruitful discussions

during the preparation of the review, and for useful com-ments on the final manuscript.

17 References

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IEE PROC, Vol. 128, Pt. A, No. 1, JANUARY 1981 63