Hertzian Electromagnetic Potentials and Associated Gauge TransformationsAuthor(s): A. NisbetSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 231, No. 1185 (Aug. 22, 1955), pp. 250-263Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/99751Accessed: 04/03/2010 12:44

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T. P. Das and D. K. Roy T. P. Das and D. K. Roy

The authors are very grateful to Professor M. N. Saha, F.R.S., and Dr A. K. Saha for keen interest in their work.

REFERENCES

Banerjee, M. K. & Saha, A. K. 1954 Proc. Roy. Soc. A, 224, 472. Bloch, F. & Rabi, I. I. 1945 Rev. Mod. Phys. 17, 236. Casimir, H. B. G. I940 On the interaction between atomic electrons and nuclei. (Holland):

Teylers Tweede Genootschop, Haarlem. Das, T. P. & Saha, A. K. I955 Phys. Rev. 98, 516 (referred to as paper I). Das, T. P., Saha, A. K. & Roy, D. K. 1955 Proc. Roy. Soc. A, 227, 407 (referred to as paper II). Hahn, E. L. 1950 Phys. Rev. 80, 580. Hahn, E. L. & Maxwell, D. E. 1952 Phys. Rev. 88, 1070. Pound, R. V. 1950 Phys. Rev. 79, 685. Volkoff, G. M. 1953 Canad. J. Phys. 31, 820.

Hertzian electromagnetic potentials and associated gauge transformations

BY A. NISBET

Department of Mathematical Physics, University of Edinburgh

(Communicated by Sir Edmund Whittaker, F.R.S.-Received 18 March 1955)

In spite of the wide use of Hertzian potentials in special problems there appears to be no account of the general theory which is completely satisfactory-especially with regard to (i) the arbitrariness of the potentials and the relation between different equivalent representa- tions of the same electromagnetic field, (ii) the derivation of the Hertzian potentials for such equivalent representations from the physical sources, and (iii) electromagnetic fields not in vacuo. It is the purpose of this paper to fill this gap. It is shown that the Hertzian potentials may be subjected to a new type of gauge transformation which leaves invariant the electromagnetic field they represent. The particular integrals of the inhomogeneous Maxwell equations are generalized, so that they may be subjected to a related gauge transformation which leaves invariant the physical sources of the field; this leads to a treat- ment of (ii) above, which appears to be new. Examples, including the Whittaker and Debye-Bromwich two-scalar representations, are given. Finally, the theorem is established that, for any electromagnetic field in any stationary material medium, the particular integral of Maxwell equations may be so chosen that in general the complementary function can be expressed in terms of only two scalar functions (components of Hertzian potentials), pre- viously only known to hold for source-free regions in vacuo.

1. INTRODUCTION

Since Hertz (1889) first introduced, for the electromagnetic field of an oscillating electric dipole, a potential of the type which now bears his name, considerable use has been made of such potentials. Righi ( 19gI ) noted the vector character of Hertz's electric potential, and introduced the magnetic Hertzian vector potential, showing that in a source-free region in vacuo any electromagnetic field is expressible in terms of either the electric or the magnetic Hertzian potential. He gives a number of

The authors are very grateful to Professor M. N. Saha, F.R.S., and Dr A. K. Saha for keen interest in their work.

REFERENCES

Banerjee, M. K. & Saha, A. K. 1954 Proc. Roy. Soc. A, 224, 472. Bloch, F. & Rabi, I. I. 1945 Rev. Mod. Phys. 17, 236. Casimir, H. B. G. I940 On the interaction between atomic electrons and nuclei. (Holland):

Teylers Tweede Genootschop, Haarlem. Das, T. P. & Saha, A. K. I955 Phys. Rev. 98, 516 (referred to as paper I). Das, T. P., Saha, A. K. & Roy, D. K. 1955 Proc. Roy. Soc. A, 227, 407 (referred to as paper II). Hahn, E. L. 1950 Phys. Rev. 80, 580. Hahn, E. L. & Maxwell, D. E. 1952 Phys. Rev. 88, 1070. Pound, R. V. 1950 Phys. Rev. 79, 685. Volkoff, G. M. 1953 Canad. J. Phys. 31, 820.

Hertzian electromagnetic potentials and associated gauge transformations

BY A. NISBET

Department of Mathematical Physics, University of Edinburgh

(Communicated by Sir Edmund Whittaker, F.R.S.-Received 18 March 1955)

In spite of the wide use of Hertzian potentials in special problems there appears to be no account of the general theory which is completely satisfactory-especially with regard to (i) the arbitrariness of the potentials and the relation between different equivalent representa- tions of the same electromagnetic field, (ii) the derivation of the Hertzian potentials for such equivalent representations from the physical sources, and (iii) electromagnetic fields not in vacuo. It is the purpose of this paper to fill this gap. It is shown that the Hertzian potentials may be subjected to a new type of gauge transformation which leaves invariant the electromagnetic field they represent. The particular integrals of the inhomogeneous Maxwell equations are generalized, so that they may be subjected to a related gauge transformation which leaves invariant the physical sources of the field; this leads to a treat- ment of (ii) above, which appears to be new. Examples, including the Whittaker and Debye-Bromwich two-scalar representations, are given. Finally, the theorem is established that, for any electromagnetic field in any stationary material medium, the particular integral of Maxwell equations may be so chosen that in general the complementary function can be expressed in terms of only two scalar functions (components of Hertzian potentials), pre- viously only known to hold for source-free regions in vacuo.

1. INTRODUCTION

Since Hertz (1889) first introduced, for the electromagnetic field of an oscillating electric dipole, a potential of the type which now bears his name, considerable use has been made of such potentials. Righi ( 19gI ) noted the vector character of Hertz's electric potential, and introduced the magnetic Hertzian vector potential, showing that in a source-free region in vacuo any electromagnetic field is expressible in terms of either the electric or the magnetic Hertzian potential. He gives a number of

250 250

Hertzian electromagnetic potentials

examples of such alternative representations, but does not indicate how the one

potential is derivable from the other for any particular field. The first published use of a representation involving both Hertz vectors appears to have been that of Whittaker (I903), who shows that the field in vacuo of a moving point charge is

expressible in terms of two scalar potentials, essentially electric and magnetic Hertz vectors which are everywhere parallel to the same fixed direction. Debye (I909), and independently Bromwich (I920, but see footnote, ?4-3 below), also used a

representation in terms of another pair of scalar potentials, which were shown by Sommerfeld (1927) to be electric and magnetic Hertz vectors that are everywhere directed radially from a fixed point. The Whittaker and Debye-Bromwich types of

representation, which split the field into so-called transverse electric and transverse

magnetic parts, have had a considerable and fruitful development, especially in connexion with wave-guide problems.

This development has, however, been mainly concerned with special problems, and has not included a full investigation of the arbitrariness of the Hertzian

potentials, and the relations between the potentials used in different equivalent representations of the same field. These points are covered in the present paper by the introduction of a new type of gauge transformation of the Hertzian potentials, which leaves invariant the electromagnetic field they represent.

Whittaker (I903) expressed his two scalar potentials as simple functions of the co-ordinates of the moving point charges which were the sources of his field.

Recently the Debye potentials have been expressed by Bouwkamp & Casimir (I954) as series'expansions, in spherical polar (multipole) solutions of the homogeneous scalar wave equation, whose coefficients are certain integrals over the current sources of the field. In both cases the expressions given are valid only at points outside the source distribution, and their derivations do not rest on inhomogeneous wave equations for the potentials concerned. In fact, inhomogeneous equations for the two scalar potentials, corresponding to the equations

(l/c2)le -V2IIe 4T7P,

(l/c) Im - V2m = 47rM,j

relating the full Hertzian vector potentials to their polarization sources, do not

appear to be known. In this paper such inhomogeneous equations for the Whittaker and Debye-Bromwich potentials are determined giving, in addition to an alternative method of derivation of Whittaker's and Bouwkamp & Casimir's expressions, a means of obtaining expressions valid within the source distribution. These

inhomogeneous equations, and corresponding ones for other Hertzian representa- tions, arise through a generalization of the relation between the physical sources of the field and the effective polarizations used in equations (1.1); this generalization allows these effective polarizations to be subjected to a gauge transformation, related to that for the Hertzian potentials, which leaves invariant the physical sources of the field.

The theory presented in this paper is in such a form that it is applicable to fields in an arbitrary stationary material medium, the physical sources (currents and

polarizations) being left as unspecified functions of the field variables.

251

A. Nisbet

2. HERTZIAN POTENTIALS AND STREAM POTENTIALS

This preliminary section is concerned with the representation of electromagnetic fields, in any material medium, in terms of certain potential functions. First the standard representation in terms of a scalar and vector potential is summarized, for later reference. Then the representation of the field in terms of Hertzian potentials and associated stream potentials is fully treated and generalized in certain respects -essential for the full development of the transformation properties to be discussed in ?3.

If E and H denote the electric and magnetic field vectors, D and B the electric

displacement and magnetic induction and p and j the electric charge and current densities, and Gaussian units (E, D, p in electrostatic units and H, B, j in electro-

magnetic units) are used, Maxwell's equations are

VAE+(l/c)B 0, V.B =O, (2.1)

VAH-(l/c)D =47j, V.D=47rp. (2-2)

The effect of the material medium (assumed stationary throughout) may be described by the electric and magnetic polarizations (induced and permanent) P and M, satisfying

D = E+47rP, H = B-47rM; (2-3)

and by allowing j to include induced conduction currents, as well as prescribed current sources. (All the prescribed sources may of course, if desired, be included in p and j, the permanent parts PO and Mo of the polarizations by terms - V . P and (1/c) PO + V A Mo respectively. P and M are then purely induced polarizations, and D and H differ from their former values by 47rPo and - 47Mo respectively.)

A frequently used representation of the field is in terms of the scalar and vector potentials 0 and A, equations (2.1) being satisfied by

E = -Vo-(l/c)A, B = VA. (2-4)

These potentials are not unique, and if the Lorentz condition

(l/c)q+V.A= 0 (2-5)

is imposed, equations (2.2), with (2-3), lead to separate wave equations for qS and A:

2 A = 4T7(p- V.P), (2-6)

2 A = 447{j + (l/c) P+ V A M}, (2-7)

1 a2 where the notation D2 =- V2 (2-8)

is used. For some purposes it is more convenient to derive the field from electric and

magnetic Hertzian vector potentials, He and IIm. Usually these are introduced separately, following historical precedent; it is preferable, however, to introduce

252

Hertzian electromagnetic potentials

them together.* A general field may be expressed in terms of 1e and Im, through the potentials 0 and A by

= -V.le, A= (1/c) e+V^IIm, (2-9) which automatically satisfy (2.5); or directly by

E = VV.He-(1/c2)iie-(1/C)V ABn, (2.10)

B = (I/c)V A +V (V AIIm), (2-11)

which are seen to satisfy (2.1), or to follow from (2.4) and (2.9). The further equations satisfied by IIH and nm are simplified by expressing p and j in terms.of two other functions Qe and Qm, the stream potentials (Laporte & Uhlenbeck 1931):

p -V.Qe, j = (/c) e+VAQm. (2.12)

Since p and j are connected by the continuity equation,

(l/c) p +V.j = 0, (2-13)

values of Qe and Qm can always be found to satisfy (2-12); in fact - 47Qe and + 47rQm are any particular integrals, for D and H respectively, of equations (2-2), in general not electromagnetic fields for they need not satisfy (2-1) and (2-3). The equations obtained when (2-9) and (2-12) are substituted in (2.6) and (2-7) (or when (2-10), (2-11) and (2-12) are used in the equations obtained by substituting (2-3) in (2-2)) are readily seen to be satisfied when

D2 Ie = 4T(P + Qe), (2-14)

]2 Im = 47T(M + Qm). (2.15)

The Hertzian potentials are therefore solutions of inhomogeneous wave equations with sources P + Qe and M + Qm. The stream potentials Qe and Qm can, therefore, be considered as electric and magnetic polarizations equivalent in their effect to

p and j. From (2 3), using (2-10), (2.11), (2.14) and (2.15), the following expressions for D and H are obtained:

D +47TQe = -(l/c)V A ,, + V A (V A H), (216)

H-47TQm = VV.IIm-(1/C2)flm+(1/c)VAIe,. (2.17)

The similarity of this pair of equations and the pair, (2 10) and (2 1 1), for E and B is to be noted. This symmetry is, however, accentuated if new functions Re and Rm, corresponding to Qe and Qm, are included in (2*10) and (2.11). The solution of the

inhomogeneous Maxwell equations then takes the form

E = 4rRe+VV.le-(1/c2)1e-(l/c)V AIIm, (2.18)

B =-47TRm + (1/c)V A IIe +V A (V AI m), (2.19)

D = - 47Qe -(lc)V AIm +V A(VA He), (2-20)

H =47Q + VV. lm-( /c2) m + (1/c) V A Ie, (2-21)

* They form a six-vector; see Laporte & Uhlenbeck (I93I), and compare Sommerfeld

(I935).

253

254 A. Nisbet

which, with (2-3), gives the wave equations

D-2 Ie = 47(P + Qe + Re), (2.22)

El2 Im = 4T(M + Qm + Rm). (2-23)

For this more general form to hold it is obvious that 47rRe and - 4r7Rm can be taken as any particular integrals, for E and B respectively, of equations (2.1), in general not electromagnetic fields for they need not satisfy (2.2) and (2.3). If terms 47rim and

47TPm, corresponding to fictitious magnetic current and charge densities, are inserted on the right-hand sides of equations (2.1), they give

Pm = V.R v , = -(l/c)Rn+V ARe, (2.24)

so that Re and Rm may be called magnetic stream potentials. The quantities Pm and

ji have of course zero values, but the auxiliary functions Re and Rm need not be zero.

Reasons, other than symmetry, for the inclusion of Re and Rm will become apparent later.

It should be noted that the theory given here holds for electromagnetic fields in

any type of medium. When the medium is non-conducting and non-polarizable (p, j, P and M prescribed), then the right-hand sides of (2-22) and (2-23) are given functions of position and time, and the forms of solution of these equations are well known. Otherwise, the right-hand sides of (2-22) and (2.23) are functions of the

electromagnetic field variables, and hence of HI and rIm; equations (2-22) and (2-23) then lead to integro-differential equations for the Hertz vectors or for the field variables. (See, for example, Born (I933), where the case of an isotropic non-

magnetic non-conducting medium is thus treated, using the electric Hertz vector

only.)

3. GAUGE TRANSFORMATIONS ON THE HERTZIAN AND

STREAM POTENTIALS

For later reference, the gauge transformation on the potentials 05 and A is first

briefly mentioned. The expressions (2.4) for the field in terms of 0 and A are invariant under the gauge transformation of the second kind (Pauli 1940)

= ;0?+(I/c)%, A = A?-V, (3.1)

where 0? and A? are other possible potentials and X is an arbitrary scalar function.

If, further, the condition2 (3 2)

is imposed on X, then the Lorentz condition (2-5) and the wave equations (2-6) and

(2-7) for 0 and A are also gauge-invariant. The Hertzian and stream potentials introduced in the preceding section are also

not unique for any given electromagnetic field. The corresponding gauge transforma- tions-which are of a new kind-on these potentials will now be investigated. In

this section these transformations will be given in their general form, and some

special applications will be discussed in the subsequent sections.

Hertzian electromagnetic potentials

Considering first the stream potentials, it is seen from (2-12) that p and j are invariant when Qe and Qm are replaced by Q? and Q? in accordance with

Qe = QO + V AG, QC e

AG, 1 (3.3) Qm = Q?-(l/c) G- Vg,

where the functions g and G are arbitrary. The transformation (3 3) will be referred to as a gauge transformation of the third kind.* Similarly, from (2-24), Pm and jm are invariant (i.e. still zero) under the dual transformation

Re = R?-( l/c)L- Vl,} el (3'4) R- R?-VAL,

where I and L are arbitrary functions. The relations (3-3) and (3.4) specify how, from one given set of stream potentials, an infinite number of other sets can be

constructed. For example, since Pm and jm are zero it is always possible to choose

the magnetic stream potentials to vanish. Further, from (3-3), it is seen that it is

also always possible to choose Q? to be zero. Equation (2*12) then gives the value

of QO and leads to the specially simple set of stream potentials

QO =c dt -Q, Ro =R0 = R . (3-5)

Since any set of stream potentials given by (3 3) and (3 4) may be used, the solution

of Maxwell's equations embodied in equations (2-18) to (2-23) can be written in the

general form

E =47rR?-(4Tr/c)L-47rV +VV. He-(l/2) He-(l/c)V A nm, (3 6)

B =-47rR? +47TV AL+ (l/c)V A le +V A(V AIm), (3-7)

D =-47TQe- 47rV AG-(l/c)VA it +V A (V A,e), (3-8)

H = 47TQO-(47T/c)G-47rVg +VV. IIm-(l/2) m +(/c) V A^, (3-9)

with ]211e = 4nr{P + Q+R0+V AG- (/c) -Vl}, (3.10)

[2Im= 47r{M+Q? +R?-(1/c)-Vg-V A L}. (3-11)

It is interesting to contrast the roles played by the functions I and L and the

potentials and A. Formally, the field expressions (3 6) and (3 7), with R? = R = 0, for E and B are simply the sum of the expressions (2.10) and (2.11) in terms of HI

and Im and the expressions (2 4) in terms of 4 (= 4nr) and A ( = 4L), all three pairs

satisfying the first two Maxwell equations (2.1). Substitution into the remaining Maxwell equations led, when 0 and A were used alone, to wave equations (2-6) and

(2.7) for (j and A; now, however, the presence of the expressions in HIe and Hn, means

* Compare the gauge transformation of the second kind, (3-1), which is imposed on a four- vector (s0, A), the gauge being a scalar X. Here the transformation of the third kind is applied to a six-vector (Q,, Qm), the gauge being a four-vector (g, G). There is a certain redundancy in the gauge functions g, G, since, for example, g may be absorbed in G, but the complete form

given here and in later transformations in this section is retained to maintain relativistic covariance.

255

that I and L do not satisfy wave equations like 0 and A, but remain arbitrary functions, and occur in (3 10) and (3.11) as sources of the corresponding Hertzian

potential fields. Similar remarks apply to g and G, which would correspond to scalar and vector potentials for D + 47rQ0 and H - 47Qm.

Considering next the Hertzian potentials, it is seen from (2-9) that 0 and A are invariant when a gauge transformation of the third kind is applied to HIe and IIm

(compare (2-12) and (3.3)). An even greater degree of arbitrariness is, however, to be expected, for 0 and A are themselves not unique. It will be seen that Hie and H1m

may be subjected to the transformation (the sum of direct and dual gauge trans- formations of the third kind)

e-n +V ^ r- )A-A(312) nm= II - (I/c) r- Vy -V ̂ A,

where y, r, A and A are arbitrary functions.

First, however, substitution of (3.12) into (2.9) shows that, if y and r are

arbitrary but A and A satisfy

22A -=-(1/c)g, - 21A = D Vg, (3.13)

where C is an arbitrary function (which may of course be zero), then 05 and A

undergo a gauge transformation of the second kind, (3 1), with gauge

X = + (l/c) A+V.A, (314)

which satisfies (3-2). Hence the field expressions (2-10) and (2-11) for E and B are

certainly invariant under the transformation (3-12) subject to the subsidiary con- ditions (3-13). The full implications of (3-12) are best seen, however, from the

expressions for the field variables, explicitly in terms of He and HIm rather than

through 0 and A. Substitution from (3.12) and comparison with (3-3) and (3.4) shows that the set

of equations (2-18) to (2.23) is invariant under transformation (3.12) with arbitrary gauges y, r, A and A, provided the stream potentials are at the same time subjected to transformations (3 3) and (3-4) with gauge functions related to those of (3.12) by

47rg = E27y+(1/c)g, 47rG = 2r - V, (3.15)

47r1 = -2 A+(1/c)g, 47TL = 2A-V , (3.16)

where ? and g are arbitrary functions (which may be zero). In other words, this means that if a given electromagnetic field can be represented by Hertzian potentials f?l and I? for a given choice of stream potentials, then other stream potentials exist for which the representation of the same field is given by the potentials IIe and IIm of equations (3-12).

It should be noted that the above transformation on the Hertzian potentials also

applies to any of the representations in equations (3-6) to (3-11), the values of g, G, 1 and L given by (3-15) and (3-16) then being additive to the arbitrary values of these same quantities which occur in equations (3-6) to (3-11). Further, if A and A are not arbitrary but satisfy (3-13), then (3.16) shows that I and L are zero, so that

256 A. Nisbet

Hertzian electromagnetic potentials

E and B are invariant under (3.12) without any change in the stream potentials Re and Rm (which can therefore remain zero). This result was obtained earlier by considering the potentials 0 and A. Similarly, if (alternatively or in addition) y and r are such that the right-hand sides of (3-15) vanish, D and H are invariant under (3-12) without any change in the stream potentials Qe and Qm.

4. SPECIAL APPLICATIONS

In this section the general theory of the preceding sections is illustrated in a number of special applications, for which it is assumed that the medium is every- where non-conducting and non-polarizable (p,j,P and M, when used, are inde-

pendent of the field variables). The first example is the field of the Hertzian oscillator, the first problem for which a Hertzian potential was ever used; the second and third examples (the Whittaker and Debye-Bromwich potentials) are the original uses of types of solutions of Maxwell's equations leading to what are now called transverse electric and transverse magnetic waves.

Besides using these examples to illustrate Hertzian potential theory for the field at source-free points, it is shown that the transformation theory of ? 3 leads naturally to generalizations of the types of solution discussed to include the field at points within the source distribution. These generalizations illustrate clearly the relation-

ship between the sources of the Hertzian potential fields, the stream potentials, and the true physical sources of the electromagnetic field. They also illustrate a method, which does not appear to have been known before, for deriving, from the physical sources, the special (e.g. Whittaker or Debye-Bromwich type) potentials concerned.

4-1. The Hertzian oscillator

It was for the problem of the oscillating electric dipole that Hertz (i889) first introduced a potential of the type that now bears his name. An oscillating electric

dipole of moment pe-it, where p is constant, situated at the origin, is a source distribution for which p, j (and hence the stream potentials) and M are all zero, while

P = p e-it 6(r), (4-1)

where 6(r) is the Dirac d-function, r being the vector distance from the origin. The field of such a dipole is given, in terms of the solutions

II? = (p/r)ei(kr-wt), I? = 0, (4-2)

(k = o/c) of equations (2-14) and (2-15), by

E-VV.II-(/c2) i , B =(l/c) V Io. (4-3) It is an interesting application of the gauge transformation (3-12) to show that

this same field is expressible in terms of a magnetic Hertzian potential. Taking all

gauge functions zero in (3-12) except

A = lII cdt =-(p/ikr) ei(kr-, (4-4)

Vol. 231. A.

257

17

258 A. Nisbet

gives the alternative potentials

IIe:0, II =-p A -ikr2ei(kr-t), (4.5) e, r 1}

Ci(kr_~ot)

where r is the unit radial vector. The field is then given by (cf. (2-18) and (2.19))

E = -(/c)VA IIm +47TRe, B= V A(V AIm)-47TRm, (4.6)

where, from (3.16) and (3-4), with R? = R? = 0,

Re -(l/c)L, Rm =-VAL, (4.7)

with L= (1/4rT) 22A

= jPcdt = -(p/ik) e-t (r). (48)

Since L is zero except at the origin, the field at all points except the position of the

dipole is given by (4.6) with Re = Rm =0 .* The correct singularities for the field at the origin are only obtained however by including the terms in Re and R,m. This is also readily verified directly; for example, substitution for IIm and Re in the first of expressions (4-6) leads to

E = VA(V A I?) - 47rP, ( e .., [ TE - r)-~+2: ,)(4-9) VV. II?-O( /c2) 0 + [-2 0-47o 4rPT

which agrees with the first of (4-3) on use of the wave equation for II?. An alternative procedure is to use the gauge transformations on the stream

potentials instead of that on the Hertzian potentials. In equations (3-10) and

(3.11), noting that M and the stream potentials are all zero, the choice of

G=g=l=0, (1/c)L=P, (4.10)

removes the source P from (3 10) and introduces a source

-VAfPcddt (4.11)

into (3-11). New Hertzian potentials, solutions of (3.10) and (3.11), can therefore be taken as (4-5), and the corresponding expressions (3-6) and (3-7) for the field are simply (4-6).

4 2. Whittaker's scalar potentials and certain generalizations Whittaker (i903) appears to have been the first to prove the important result

that, in vacuo at points away from the sources, a general electromagnetic fieldt can be expressed in terms of only two scalar functions, each satisfying the homogeneous wave equation. The expressions given for the field variables, in terms of scalar

* This form of (4.6), and an expression for IIm equivalent to (4.5), were given by Righi (I9oI) in the paper mentioned in the introduction.

t Strictly, Whittaker proved the result for the field due to electric point charges moving in any prescribed manner; but see below.

Hertzian electromagnetic potentials

potentials Y and (F and - G of Whittaker's paper) are identical with (2 10) and

(2.11) provided = -k, nm = Sk, (4.12)

where k is a unit vector in a fixed direction (e.g. the z-axis). The functions Y and f are therefore one-component electric and magnetic Hertzian potentials. A com-

parison of certain equations in Whittaker's paper with (2-9) and (3-1) shows that his proof of the above theorem is equivalent to (i) assuming the field is given by certain known potentials 0?, A?; (ii) applying the gauge transformation (3-1) to

give new potentials 0, A; and (iii) showing that the expressions (2 9) for 0 and A in terms of Ie and Im can then be solved, under assumption (4-12), for F, ( and the

gauge function X (fr of Whittaker's paper). Whittaker used for 0? and A? the well- known Lienard-Wiechert expressions for the field of a moving point charge; it is not difficult to see, however, that his analysis, and hence his theorem, holds when

0 and A? have values appropriate to any other electromagnetic field.

By a suitable choice of stream potentials Whittaker's result may be extended to

points within the source distribution. Substitution of (3-5) and (4-12), with k in the direction of the z-axis, in (3*10) and (3.11) leads to two inhomogeneous scalar wave

equations for Y and C, and to four scalar equations connecting the gauge functions and the given source distribution. The x- and y-components of (3.10) and (3.11) are satisfied when Ox, Gy, Lx, Ly are zero and Gz, 1, Lz, g are particular integrals of

aG, Sl f. - + - Px+ jxcdt,

a+ aly

_ = +fjtycdt,

aL ay (4.13)

aL, ag az +ag = My. ax ay

Hence QG and La may be taken as particular integrals of

Q2 2) (VAP)z+f(V Ai)Cdt (414)

(ax2+ y - (V A^M),

and 1 and g then determined from two of equations (4.13). The gauge functions are thus expressible as functions of space and time, through the sources of the electro-

magnetic field. Some of the gauge functions may of course be zero in certain cases- for example, when M = 0, L, and g may be taken as zero. The z-components of

(3.10) and (3.11) give the wave equations for F and C:

[22 = 47r{Pz+ fjcdt-(1/c)4Lz-/}8z}, ozJz= - (4-15)

o2- = 47r{Mz-(l/c) -ag/z} J

17-2

259

The field is then given by (3-6) and (3-7) as

E = VV. (kr)- (1/c2) kJ-(1/c)V A (k!)-(47T/c) kL -47rVI, (416) B (l/c) V A (kk) + V A {V A(k()}+ 47TV A (kLz),

with D and H obtainedfrom (2.3), or (3.8) and (3.9). Equations (4 16) are Whittaker's

expressions with the addition of the terms in L1 and 1. It is seen therefore that a

particular integral of Maxwell's equations can be obtained which enables the com-

plementary function to be expressed, even within the source distribution, in terms of only two scalar wave functions.

4*3. Radial Hertzian potentials

In discussing the scattering of plane waves by spheres, Debye (1909) introduced a solution of Maxwell's equations in terms of two scalar potentials I1I and I2. In an independent investigation of the same problem Bromwich (1920) used a solution* in terms of two scalar potentials U and V. These solutions are identical provided

11 = U/r, H2 =V/r, (4.17)

where r is the distance from the origin. They are also identical with equations (3-6) to (3-9), and the wave equations satisfied by the two potentials are the same as (3-10) and (3-11), when P, M, Q?, Q,RO R, R, G and L are all zero, but

271 = HI, 27g = H2, (4.18)

and le = rIll, Hm = rrt2, (4.19)

where r is the radius vector from the origin. Debye's (and Bromwich's) two scalar

potentials are therefore essentially radial electric and magnetic Hertzian vector

potentials.t The Debye-Bromwich form of solution of Maxwell's equations may, like

Whittaker's solution, be extended to points within the source distribution. If

r, 0, 0 are the usual spherical polar co-ordinates, it is readily verified that a choice of

Go = = Lo= 0, 27Tr = 1+2171', 27Tg= 2+27Tg

enables the field to be expressed, from (3 6) and (3 7) with (3 5), in the form

E=VV.(rn1)- (1/c2) rnl- (1/c) VA (rliH2)-2VH1- (47r/c) ',r--47Vl',

B = (l/c)VA(r )+VA {V (rn2)} + 47TV A(^Lr),

where, from (3-10) and (3.11), IH1 and I12 satisfy the wave equations

22 H,1 = (47r/r){Pr +frcdt-(l/c) Lr-l'/ar}, (421)

D2 rl2 = (47T/r) {Mr-(l/c) r--ag'/lar},

* 'Originally worked out in 1899, and first published as a question in part II of the "Mathematical Tripos", I9Io.' (Bromwich I920, footnote, p. 179.)

t See also Sommerfeld (I927), where the connexion between the Debye and radial Hertz

potentials is pointed out.

260 A. Nisbet

Hertzian electromagnetic potentials 261

and the gauge functions are particular integrals of

1 3ar 1 l f f , r sin O a+ r aO j c + dt,

aG,r 1 at c fl, a + Po+ ocdt)

rtF ~ aO(4.22) 1 aL 1 ag'

rsinO af r a=M

1aL 1 ag'_ r aO r sin 0 O

From these, Gr and Lr also satisfy

a (sin 0 Gr) +- = (V ̂ A P)r + (V ̂j)r c dt, r2sin20 ao sin8] r2sin20 a 2 (4.23)

a1 /n 8aL I a2L. rsi0 inn 0a0) + -rs2 - (V A M)r. r2sin 0 iO r2 sin2 0 = a V M

5. REDUCED FORMS OF THE HERTZIAN POTENTIALS

The gauge transformations detailed in ? 3 show that there is considerable latitude in the representation of an electromagnetic field in terms of potentials, and this has been illustrated in the applications discussed in ? 4. Mention has been made of the theorem that, at source-free points in vacuo, an electromagnetic field can in general be expressed in terms of only two scalar potentials, each satisfying the homogeneous wave equation. Besides the proofs by Whittaker (1903), Debye (1909) and Bromwich ( 920), already referred to, mention should also be made of others by Schelkunoff

(i943) and Green & Wolf (1953).* In ??4-2 and 4-3 it is shown that this theorem also holds for points within a prescribed source distribution in a non-conducting and non-polarizable medium (p, j, P and M independent of the field), in the form:

The particular integral (i.e. the stream potentials) of the inhomogeneous Maxwell equations may be so chosen that the complementary function can in general be expressed in terms of only two scalars (components of Hertzian potentials).

These two potentials satisfy certain wave equations which are in general in-

homogeneous, their inhomogeneous terms being functions of the stream potentials and the sources of the electromagnetic field, and hence known functions of position and time. This follows from the fact that the stream potentials can be determined, as particular integrals of certain partial differential equations (not wave equations), from the known source distribution.

The remaining restrictions may be removed, so that the above theorem holds in any material medium, at points inside or outside the source distribution; now,

* Green & Wolf's solution of Maxwell's equations can be expressed in Hertzian potential form (of the Whittaker type). It was the discovery by the author of this fact, following on a recent revival of interest in the foundations of approximate scalar theory in optics (Luneburg 1947-8; Wolf 1951; Theimer, Wassermann & Wolf 1952),which led to the investiga- tions reported in this paper.

however, the two potentials satisfy certain inhomogeneous wave equations whose

inhomogeneous terms are in general functions of the potentials themselves. The first part of this result follows immediately from transformation (3-12) which

holds irrespective of the type of medium. Only four of the scalar components of the

arbitrary gauge functions are non-redundant, and these can be chosen to reduce to zero four components of the two Hertzian potentials. Representations such as the following can be obtained:

(i) Two-component II: A choice of y = 0, A = 0 and (l/c) I = IIo gives Im = 0; A then remains to reduce one component of HII to zero.

(ii) Similar remarks apply to a two-component ITm.

(iii) One-component lIe and 11m: This may be realized in a variety of ways, including the Whittaker and Debye-Bromwich types of representation. For example, it is

readily seen from (3-12) that, in general, values of A, y, Az, Fr can be determined to make the x- and y-components of each of IIe and IIm zero; or again, that values of

A, y, Ax, ry, can be determined to give a representation with lie directed parallel to the x-axis and IIm to the y-axis.

The second part of the above result follows from (3.10) and (3.11) which readily give the differential equations satisfied by the two remaining components of IIe and

IIm for any of the above types of representation. For example, the Whittaker type representation (4-12) leads to equations (4-13) to (4-16) even in a material medium.

However, in general, through the constitutive relations of the medium, P, M, p and j are functions of the field variables, and hence, by (3 6) to (3 9), of Y and S. Equations (4.13) and (4.14) then give the gauge functions 1, g, L, and z, as integral functions of Sand f, so that (4 15) is in general a pair of integro-differential equations for the

potentials J and (. When the constitutive relations of the medium are introduced through the

dielectric constant, permeability and conductivity being given as functions of

position, an alternative approach is possible which should lead to differential

,equations only, instead of integro-differential equations, for the two scalar potentials. Work on this is in progress and will be reported separately later.

In conclusion, I wish to express my thanks to Dr E. Wolf for stimulating my interest in his work on scalar potentials in which, as mentioned earlier, the investiga- tions reported in this paper had their origin. I am indebted also to him for his continued encouragement and helpful discussions, and to Professor Kemmer, Dr Pursey, Dr Schlapp and Sir Edmund Whittaker, F.R.S., for their interest in this work and some useful discussions and suggestions.

REFERENCES

Born, M. 1933 Optik, ?74. Berlin: Springer. Bouwkamp, C. J. & Casimir, H. B. G. I954 Physica, 20, 539. Bromwich, T. J. I'A. 1920 Phil. Trans. A, 220, 175 (also Phil. Mag. I919, 38, 143). Debye, P. 1909 Ann. Phys., Lpz., 30, 57. Green, H. S. & Wolf, E. I953 Proc. Phys. Soc. A, 66, 1129. Hertz, H. 1889 Ann. Phys., Lpz., 36, 1. Laporte, O. & Uhlenbeck, G. E. 1931 Phys. Rev. 37, 1380.

262 A. Nisbet

Hertzian electromagnetic potentials Hertzian electromagnetic potentials 263 263

Luneburg, R. K. 1947-8 Propagation of electromagnetic waves (Mimeographed lecture notes). New York University.

Pauli, W. 1940 Phys. Rev. 58, 716.

Righi, A. I9go Nuovo Cim. 2, 104. Schelkunoff, S. A. I943 Electromagnetic waves, p. 382. New York: van Nostrand. Sommerfeld, A. 1927 In Frank & v. Mises, Riemann-Webers Differentialgleichungen der

Physik, 2, 497. Braunschweig: Vieweg. Sommerfeld, A. 1935 In Frank & v. Mises, Riemann-Webers Differentialgleichungen der

Physik, 2nd ed. 2, 790. Braunschweig: Vieweg. Theimer, O., Wassermann, G. D. & Wolf, E. 1952 Proc. Roy. Soc. A, 212, 426. Whittaker, E. T. 1903 Proc. Lond. Math. Soc. 1, 367. Wolf, E. I95I Rep. Progr. Phys. 14, 95.

Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry

BY B. A. BILBY, R. BULLOUGH AND E. SMITH

Metallurgy Department, The University, Sheffield

(Communicated by J. M. Whittaker, F.R.S.-Received 28 March 1955)

When describing a crystal containing an arbitrary distribution of dislocation lines it is often convenient to treat the distribution as continuous, and to specify the state of dislocation as a function of position. Formally, however, there is then no 'good crystal' anywhere, and difficulties arise in defining Burgers circuits and the dislocation tensor. The dislocated state may be defined precisely by relating the local basis at each point to that of a reference lattice. The dislocation density may then be defined; it is important to distinguish this from the local dislocation density. The geometry of the continuously dislocated crystal is most conveniently analyzed by treating the manifold of lattice points in the final state as a non-Riemannian one with a single asymmetric connexion. The coefficients of connexion may be expressed in terms of the generating deformations relating the dislocated crystal to the reference lattice. The tensor defining the local dislocation density is then the torsion tensor associated with the asymmetric connexion. Some properties of the connexion are briefly discussed and it is shown that it possesses that of distant parallelism, in conformity with the requirement that the dislocated lattice be everywhere unique.

1. INTRODUCTION

Consider a crystal containing an arbitrary distribution of dislocation lines. In

general, the dislocation distribution will have stresses which are not zero when

averaged over distances I large compared to the mean spacing d of the dislocations. If they can, the dislocations in a real crystal will adjust their positions to reduce these far-reaching stresses (for example, in the phenomenon of polygonization). Several authors have recently discussed the problem of finding the arrangement of dislocations to be expected when a crystal is deformed under given geometrical and

physical conditions, and have emphasized its importance for the quantitative development of dislocation theory (Nye 1953; Read 1953, I954; Basinski & Christian

I954; Paxton & Cottrell I954; Bilby I955). Nye has given an analysis of this

problem when it is assumed that the above average of the far-reaching stresses is

Luneburg, R. K. 1947-8 Propagation of electromagnetic waves (Mimeographed lecture notes). New York University.

Pauli, W. 1940 Phys. Rev. 58, 716.

Righi, A. I9go Nuovo Cim. 2, 104. Schelkunoff, S. A. I943 Electromagnetic waves, p. 382. New York: van Nostrand. Sommerfeld, A. 1927 In Frank & v. Mises, Riemann-Webers Differentialgleichungen der

Physik, 2, 497. Braunschweig: Vieweg. Sommerfeld, A. 1935 In Frank & v. Mises, Riemann-Webers Differentialgleichungen der

Physik, 2nd ed. 2, 790. Braunschweig: Vieweg. Theimer, O., Wassermann, G. D. & Wolf, E. 1952 Proc. Roy. Soc. A, 212, 426. Whittaker, E. T. 1903 Proc. Lond. Math. Soc. 1, 367. Wolf, E. I95I Rep. Progr. Phys. 14, 95.

Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry

BY B. A. BILBY, R. BULLOUGH AND E. SMITH

Metallurgy Department, The University, Sheffield

(Communicated by J. M. Whittaker, F.R.S.-Received 28 March 1955)

When describing a crystal containing an arbitrary distribution of dislocation lines it is often convenient to treat the distribution as continuous, and to specify the state of dislocation as a function of position. Formally, however, there is then no 'good crystal' anywhere, and difficulties arise in defining Burgers circuits and the dislocation tensor. The dislocated state may be defined precisely by relating the local basis at each point to that of a reference lattice. The dislocation density may then be defined; it is important to distinguish this from the local dislocation density. The geometry of the continuously dislocated crystal is most conveniently analyzed by treating the manifold of lattice points in the final state as a non-Riemannian one with a single asymmetric connexion. The coefficients of connexion may be expressed in terms of the generating deformations relating the dislocated crystal to the reference lattice. The tensor defining the local dislocation density is then the torsion tensor associated with the asymmetric connexion. Some properties of the connexion are briefly discussed and it is shown that it possesses that of distant parallelism, in conformity with the requirement that the dislocated lattice be everywhere unique.

1. INTRODUCTION

Consider a crystal containing an arbitrary distribution of dislocation lines. In

general, the dislocation distribution will have stresses which are not zero when

averaged over distances I large compared to the mean spacing d of the dislocations. If they can, the dislocations in a real crystal will adjust their positions to reduce these far-reaching stresses (for example, in the phenomenon of polygonization). Several authors have recently discussed the problem of finding the arrangement of dislocations to be expected when a crystal is deformed under given geometrical and

physical conditions, and have emphasized its importance for the quantitative development of dislocation theory (Nye 1953; Read 1953, I954; Basinski & Christian

I954; Paxton & Cottrell I954; Bilby I955). Nye has given an analysis of this

problem when it is assumed that the above average of the far-reaching stresses is