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Electromagnetic induction in magnetic rod moving with high velocity Prof. E.G. Cullwick, O.B.E., M.A., D.Sc, C.Eng., F.I.E.E., F.R.S.E. Indexing terms: Electromagnetism, Relativity Abstract An account is given of the relativistic component-field electromagnetic theory of moving bodies, which extends orthodox relativity theory by recognising sources of the field, and it is shown to have advantages over orthodox theory in aiding physical comprehension. It is then applied to the determination of the electromagnetic field in a cylindrical rod of nonretentive ferromagnetic material, which can be either a conductor or a dielectric, in motion with high velocity through crossed fields. Particular cases investigated include the electromotive force induced in the rod by its motion through a magnetic field, it being confirmed that this is independent of the relative permeability, and the magnetisation of the rod by its motion through an electrostatic field. Expressions for the polarisation and magnetisation of the moving rod are also obtained. All the results are considerably simplified if the velocity of the rod is small compared with the velocity of light. For low velocities the theory can be presented in simple terms without reference to the theory of relativity, and for this reason, and also because of its physical basis, it is considered to have considerable advantages. List of symbols (SI units) B D E e F Ft H M P v, v c Mo 1 magnetic flux density electric flux density electric field intensity electromotive force (e.m.f.) Lorentz force electromagnetic force magnetic field intensity magnetisation (dipole moment per unit volume) polarisation (dipole moment per unit volume) velocity = velocity of light in vacuo (i-t)V)-" 2 permittivity of free space, or primary electric constant permeability of free space, or primary magnetic constant relative permittivity relative permeability Introduction The phenomenon of electromagnetic induction, discovered by Michael Faraday in 1831, is the rock on which electrical engineering has been built, while the experimental and theoretical investigation of the electromagnetic effects of moving bodies has been fundamental to the development of modern physical theory in the evolution of aether theories and the emergence of the theory of relativity. It was indeed the latter which, in providing a purely mathematical specification of the electromagnetic effects of moving bodies, swept out of respectable science the current physical theories and postulated explanations of such important practical phenomena as the electric field produced by a moving magnet. Engineers and physicists who seek descriptions of natural phenomena in terms of physical concepts, however, have available in the relativistic component-field theory of electromagnetism a theory of the fields of moving bodies which is not only equivalent to the mathematical relativistic theory but also provides much physical insight into the phenomena, and, as will be shown, physically consistent accounts of the action. The development of relativistic electromagnetism largely as a province of mathematics has had the unfortunate result that electromagnetic theory in physical terms, as taught in our universities and colleges, has too often been confined to the prerelativistic form developed by H. A. Lorentz on the foundation of Maxwell's theory, which is commonly known as Maxwell-Lorentz. This is entirely satisfactory and sufficient for the majority of practical applications, but in unusual problems involving moving magnets its application can sometimes be complex and ambiguous. An example is the case of a magnetic cylinder moving concentrically along a currentcarrying conductor,1 and an old problem, considered by Faraday in 1851,2 which has been the source of much discussion is that of a rotating cylindrical magnet.3 In the case of electromagnetic induction in a body which is both magnetic and a dielectric moving through a magnetic field, prerelativistic theory actually gives an incorrect result.4 All these problems are readily analysed and solved by means of the component-field theory.s In this paper the theory will first be explained and compared with orthodox theory, and then used to find the field measures in a magnetic cylindrical rod, which may be either a conductor or a dielectric, moving transversely with high velocity through crossed electric and magnetic fields. This analysis also provides a specific proof, which so far appears to have been lacking, that when a wire moves through a magnetic field the induced e.m.f. is the same whether the conductor is nonmagnetic or magnetic. This fact, first discovered and very simply demonstrated by Faraday in 1831,6 may be inferred from the general laws of electromagnetic induction and the conservation of energy, but it will be proved here directly by finding the magnitude of the electromagnetic field within the moving wire. From the same analysis it will appear that the induced e.m.f. is given by the simple flux-cutting law, even up to velocities comparable with the velocity of light. Another particular case which will be analysed is that of the magnetisation of an iron rod by its motion through an electrostatic field, a phenomenon sought but not detected by Faraday in 1832.7 2 Electromagnetic component-field theory of moving media 2.1 Maxwell's equations and Lorentz transformation Maxwell's equations for the field in free space: curl* = - J - (1) at and curlff=^ (2) at define relationships between the resultant or total measures of the electromagnetic field at a point fixed in the inertial reference system in which measurements are made and the equations used. They do not relate the field to its sources. The principle of relativity of inertial reference systems, first enunciated by Henri Poincare in 1904,8 requires that these equations should be valid, and therefore the velocity of light in vacuo equal to c, in all such systems, and this requirement is met if the measures of space and time defining a given physical point event, as observed in two relatively moving inertial systems, 5 and S', are related by the Lorentz transformation: x = Q(x — vt), = y, z = z, t' = fiit-^ (3) and, reciprocally, , t = t+7x where v is the velocity of the system 5 ' relative to S along the x-axis andj3 = ( l - ^ / c 2 ) - 1 / 2 . It then follows that the electromagnetic field measures at corresponding points of space and time in the two systems must be related by transformations such as Paper 7957S, first received 10th January and in revised form 24th June 1977 En = P(E'n-vxB'), Prof. Cullwick is Emeritus Professor, University of Dundee, and Honorary Professor of Electrical Engineering, University of Kent at Canterbury E - E' E'n =P(En+vxB) PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977 Authorized licensed use limited to: Princeton University. Downloaded on June 04,2010 at 13:02:35 UTC from IEEE Xplore. Restrictions apply. (4) 1105 the suffixes n and v denoting components normal and parallel to the velocity v, respectively. Eqn. 4, and similar transformations for H, B, and D, provide a specification of the electromagnetic field of moving bodies as required by the special or restricted theory of relativity. The specification is obtained by the purely mathematical application of the Lorentz transformation to Maxwell's equations, and the relations such as eqn. 4 are functions of the resultant field measures at a point, and are not related to the physical sources of the field. 2.2 2.2.1 and where M' is the magnetisation. The resultant fields in the rest system S' are then: 01) E' = E'o + E's (12) D' = D'0+D's (13) Component-field theory H' = H^+Hp (14) Transformation of field measures B' = B'0+B'p (15) In the component-field theory we extend relativistic theory to take account of field sources. We suppose a body, which may consist of conductors and dielectrics, nonmagnetic or magnetic, to be stationary in an inertial reference system S' which is moving with constant rectilinear velocity v, in the direction of the x-axis, in a second inertial reference system S. For our purpose, we may identify S as the reference system of the laboratory in which the body is moving and in which observations of the electromagnetic field, at points fixed in the reference system, are made. Let Eo and Do = €o'Eo be the components of the electric field measures, and Ho and Bo = n0H0 be the components of the magnetic field measures, assumed constant and uniform,* due to sources external to the moving body and as observed in the system S. The corresponding fields as observed in the system S1, at points fixed in that system, are given by eqn. 4 and similar transformations. To simplify the expressions we shall suppose that the fields Eo etc. are perpendicular to v. The unchanged parallel components of field, if any, can then be added in any particular case. These external fields then exert polarising and magnetising forces on the moving body which in its rest system S' appear as a polarising field +vxB0) E'o = (5) In the system S the selfcomponents of field E's etc. are observed to be increased by the relativistic factor |3 and also to cause 'motional' fields by the motion of their sources in the moving body. These are: (a) electric motional field Em = -0(vxB'n) (16) (17) (b) magnetic motional field Hm = p(vxD's) (18) (19) It follows that the relativistic field transformations can be expressed in two forms, the first of which in the following equations is the standard orthodox transformation of relativity theory, and the second is obtained by collecting the three components of field, i.e. external, self and motional, as given previously: E = 0(£" -vxB') = EQ + Q(E't -vxB'p) (6) (20) (21) Dand a magnetising field H'o = P(Ho-vxDo) vxE0 D'S=D'P+D'q (9) where E'p, Dp are due to the polarisation of the dielectrics and E'q, D'q are due to the displaced free charges on conductors, and Dp = e0Ep +P', D'q = e0Eq (10) where P' is the polarisation, and u' a' "p. tt p due to the magnetisation, •If the external field varies with time or in space the selfcomponents of field in the moving body will vary with time in S' and will induce field components. These will, in many cases, be negligible, but can readily be included if necessary. See Reference 5, chap. 7, for a more extensive account of the theory. 1106 p +vx D's) (22) (23) (8) We may note that E'o is merely the Lorentz force, or the force on a unit charge moving with velocity v through the fields Eo and Bo, but modified by the relativistic factor j3, since it now appears in S' as a force on a stationary unit charge, i.e. an electric field intensity. H'o is its magnetic 'dual' or force on a unit magnetic pole stationary in S'. D'o and B'o are related to E'o and H'o as in free space. So if we are concerned merely with a theory of moving bodies with low velocities, in which v^.c and 0 can be taken as unity, we can deduce eqns. 5—8 independently of the theory of relativity from the physical laws for the force on a charge and a magnetic pole when moving through an electromagnetic field, and by transferring our viewpoint from one reference system to the other. The field measures at a point inside the body, as observed in its rest system S', consist of the above 'external' field components, given by eqns. 5—8, together with 'self field components arising from the state of electrification, polarisation, and magnetisation of the body, brought into existence by the action of the polarising and magnetising fields. The selfcomponents of field, which will depend on the shape of the body and its orientation to the external fields, may be expressed as D'8 = eQE's +P' , H = 0(H' + v x D') = Ho (7) The second form of these equations can also be obtained by substitution of the values of E' etc. given by eqns. 12—15 in the orthodox transformations and the use of eqns. 5—8. 2.2.2 Magnetisation and polarisation The transformations for the polarisation P and magnetisation M are, in symmetrical form in which M = B — and P = D- M = ${M'-Ho(vxP')} (24) P = 0{P' +eo(vxM')} (25) or, if the unsymmetrical form Mh = B/n0 — H is used, then Mh = (S(M'h-vxP') (26) (27) 2.2.3 Time In these relations all unprimed quantities are those observed in the system S at an instant f, and all primed quantities are observed in the system S' at an instant t', where t and t' are related by the Lorentz transformation, eqn. 3. This applies to all expressions relating S and S' measures, but in the use of the theory in problems of electromagnetic induction involving e.m.fs of the first order only, and tThese symmetrical definitions of M and P are preferred to the alternative unsymmetrical form in which B = n0 (H + M). The dependence of H on the concept of magnetic poles, which is symmetrical with the dependence of E on the concept of electric charge, is inherent in Poisson's classical analysis of a polarised magnetic medium, which remains valid and basic to the theory of magnetism even if we now accept that it is spinning electrons that cause M. The symmetry, moreover, leads to symmetrical transformations as seen in eqns. 24 and 25. SI units provide for both forms: electromagnetic moment, magnetic moment ( A m 3 ) ; magnetisation (A/m); magnetic polarisation (T); magnetic dipole moment (N m 2 /A or Wb m). The latter two conform to the symmetrical form of the definitions of M and P. For a physical explanation of the apparent magnetisation of a moving polarised dielectric, and the apparent polarisation of a moving magnet, see Reference S, chap. 3 and 4. PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977 Authorized licensed use limited to: Princeton University. Downloaded on June 04,2010 at 13:02:35 UTC from IEEE Xplore. Restrictions apply. velocities small compared with c, the distinction between t and t' can be ignored. With constant and uniform fields, of course, the time transformation is of no significance. 2.3 Comparison of orthodox and component-field theories The logic of the component-field theory is very simple: the electric and magnetic fields caused directly by the motion of magnetised and electrified matter arise from the motion of the constituent electric charges and magnetic dipoles (if the pole concept of magnetism is used) in the moving bodies and of nothing else. These motional fields can be expressed, for rectilinear motion, by the vector products of the parent fields and velocities of the bodies, but if the total resultant field in the interior of a moving body is due partly to the body itself and partly to external sources, the true motional field of the body must clearly be a function of the selfcomponent of field alone, and not of the resultant field in the body. This is consistent with Lorentz's extension of Maxwell's theory, in which he regarded electrons, moving freely in a fixed and immobile aether, as the sole sources of the electromagnetic field,9 and in contrast to the earlier theory of Hertz, in which it was supposed that matter in motion carried the enclosed aether along with it,10 This hypothesis of Hertz was found by A. Eichenwald in 1903-04 11 and H.A. Wilson in 190512 to be inconsistent with experiments which were in accord with Lorentz's theory. To compare the use of the two forms of the relativistic field transformations given in eqns. 20—23, consider the simple case of a solid iron sphere, of very high relative permeability, moving with constant rectilinear velocity transversely through an external field Bo with Eo = 0. The magnetising flux density acting on the sphere in its rest system S' is, from eqn. 8, B'o = $Bo that E = -$13(vxBQ) = B x v (38) as in eqn. 36. This, however, is all that we do find. There is no suggestion in eqn. 38 of any displaced charges and the two parts of E are obscured. We cannot derive the induced e.m.f. from the result, and if we attempt to do so we may conclude that there is none. Tofindthe e.m.f. we must go back to the reason for E' being zero and accept the idea of two field components cancelling each other, and so are led to the component-field theory. The orthodox transformations, being the result of a purely mathematical operation on Maxwell's equations, are a poor and indeed misleading guide to physical interpretations. It follows that the component-field theory is not just an alternative method of obtaining relativistic results: each term of the equations has a recognisable physical meaning and the theory brings physics into the transformations where previously it was absent. A further advantage of the component-field theory is that it provides a low-velocity (§ = 1) theory of moving media which can be taught and used without reference to the theory of relativity at all, and which, contrary to prerelativistic Maxwell-Lorentz theory which is still prevalent, is consistent with the principle of relativity and experiment in all known cases, including that of induction in a moving magnetic dielectric. E o°rBo (28) and the resultant field in the sphere, in its rest system, is given by E' = 0 (29) since the iron is a conductor, and B' = 3B'O (30) very closely. To find the electric field E inside the moving sphere as observed in the system S, we may use either form of the field transformation given in eqn. 20. First, consider the component-field form: E = E0+P(E's-vxB'p) (31) The polarising electric field acting on the sphere in its rest system S' is, from eqn. 5, E'o = P(vxB0) (32) and this displaces charges in the sphere, which set up a selfcomponent of field E's cancelling the external component E'o, i.e. E'8 = -E'o = -P(VXB0) (33) Then, since B' = B'o+B'p = 3B'O Fig. 1 Cross-sectional view of cylindrical rod at rest in a transverse electric field EQ or a transverse magnetic field BQ (34) 3 we have Bp = 2B'O E = -i' The use of eqn. 31 shows clearly that the resultant electric field E is made up of two parts with different physical origins: the first, of magnitude one-third of the whole, being the field of the charges displaced in the conducting sphere, while the second, of magnitude two-thirds of the whole, is the motional electric field produced by the motion of the magnetised iron. The extent to which charges are displaced in the sphere is a measure of the induced e.m.f.; so it follows that this, induced by the motion of the sphere through a field Bo, is proportional to Bo and not to the resultant internal field B. In comparison, consider the orthodox form of the field transformation: E = $(E'-vxB') electromagnetic field (35) so that, since Eo — 0, eqn. 31 gives (37) We have E' = 0 and B' = 3B'O and so, using eqn. 28, we find at once Magnetic rod moving with high velocity through an 3.1 Field components in cylindrical rod 3.1.1 Transverse fields Fig. 1 depicts the circular cross-section of a solid long straight rod of magnetic material, of relative permeability /i r , which may be either a dielectric of relative permittivity e r or a conductor. It is stationary in a transverse electric or magnetic field due to external sources which contribute to the resultant field, at an internal point, the 'external' field components: E0,D0 = eoE'o or H'o,B'o = (39) Primed symbols are used since the rod is considered to be stationary in- an inertial reference system S', and unprimed symbols will then denote the field measures as observed in the system S in which the rod is moving. These would be the field measures if the rod were removed, assuming that its removal did not disturb the external field sources: PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977 Authorized licensed use limited to: Princeton University. Downloaded on June 04,2010 at 13:02:35 UTC from IEEE Xplore. Restrictions apply. 1107 Dielectric rod By the usual methods of potential theory it may be shown that the field measures at an internal point are given by: E' = 2E'O er+l The polarisation or magnetisation will be P' = (er-\)D'o = D's or M' = (ptr-l)BQ = B'p (51) or H' = (40) Conducting rod If the rod is a conductor, the electric field will be given by eqns. 45 and 46, and the magnetic field will be the same as in a dielectric rod. or (41) 3.2 Magnetic conducting rod moving across a transverse magnetic field 3.2.1 Fields for internal equilibrium and D' = and the selfcomponents (eqns. 9—11) of field are: E'q =0,E's = Ep = - ^ D'q = 0, D's = Dp = - ^ — D'o er + 1 or or I'o (43) The rod, Fig. 2, is assumed to be long and to move with constant velocity v transversely through a constant and uniform magnetic field Bo due to external sources. The rod is on 'open circuit', carrying no current, and so for internal equilibrium, from eqns. 45, 46 and 5: E' The polarisation or magnetisation will be = 0, E's = -E'o = -P(VXB0) (52) and from eqns. 43 and 8: (44) or Conducting rod If the rod is a conductor the magnetic field will be the same as before, but the electric field measures will be £" = 0, D' = 0, />' = 0 Hr + 1 = 0, D'p = 0, 3.1.2 E't = E'Q--=-E'o D's = D'Q= -D'o (45) E= -p = E'o or (54) and r Tangential fields E' (53) Mr+1 (46) (55) Mr + 1 The Lorentz force is Dielectric rod For a dielectric rod parallel to the external field, the resultant field measures at an internal point are D' = ejDo or Mr + since f0 = 0. Then, from eqns. 20 and 23, we find that and the selfcomponents will be 4 o — p H' = H'o (47) B' = (irB'o (48) F = E + vxB E's = 0 or H'p = 0 (49) Al = (e r -l)^>o or B' = (u (50) (56) confirming the state of internal electrical equilibrium. 3.2.2 and the selfcomponents are = 0 E.M.F. induced in rod The electromagnetically induced e.m.f. along an open path in a conductor is the line integral along the path of the electromagnetic force, Fh which is defined as the force on a unit charge situated at a point on the path and stationary relative to the body, induced by electromagnetic action alone and excluding the electrostatic component of force due to true charges displaced from their neutral positions in the body.13 To find the induced e.m.f. in the present case, therefore, we remove the component E's from the expressions for E and B given by eqns. 54 and 55 and find for the fields to be used in the expression for Ft: (57) and (58) Bt = so that /•j — £/ i = vxB0 (59) If the length of the rod is L, the induced e.m.f. is e = B0Lv Fig. 2 Cross-sectional view of cylindrical rod moving with velocity v perpendicularly to transverse electricfieldEo or to transverse magnetic field Bo 1108 (60) The e.m.f. is thus the same as in a nonmagnetic conductor. The result may be obtained in an elementary way by assuming that the magnetised rod moves through the external component of flux density Bo alone, carrying its selfcomponent Bp with it. It is indeed obvious that the electromagnetic force Ft must be independent of the magnetisation of the rod, since, in the rest system of the latter, the magnetisation causes no force on a stationary unit charge, so there can be no such force in any other inertial reference system in which charge and rod are co-moving. We may briefly sum up these conclusions in the statement that a magnet moving with constant rectilinear velocity cannot induce an e.m.f. in itself. PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977 Authorized licensed use limited to: Princeton University. Downloaded on June 04,2010 at 13:02:35 UTC from IEEE Xplore. Restrictions apply. 3.3 vxH, Magnetic conducting rod moving across transverse electrostatic field The long rod, Fig. 2, moves transversely with constant velocity v through a constant and uniform electrostatic field EQ due to external sources. For internal equilibrium, from eqns. 5 and 46: Eg = ~E0 = —&EQ so that, from eqns. 20—23 (62) H = 02 Ean. 62 also gives the magnetisation Af' in the rest system 5 ' sihce Hp = 0. Since Bo = 0, we then find from eqns. 20 and 23: E= vxB0 (69) 2 \vxE0 (70) "' j B = 02 vxEQ -PQlr- 2 * - ( P " (61) while from eqns. 8 and 50: B'p = (68) D= 2er\vxH0 2 T\ c r° I (63) -f-^ vxD0 -?£il*-'- (71) (72) From eqns. 69 and 70 the Lorentz force is and vxE0 (64) 3.4.2 Thus, the magnetic flux density in the rod consists of two parts: (a) that of the magnetisation, (iBp (b) the motional magnetic field of the charges induced by Eo on the conducting rod. The Lorentz force in the rod, F = E + v x B, is zero. In any practical case, the magnetisation will be extremely small. For example, a rod moving at one-tenth the velocity of light through an electrostatic field of 104 V/m, with relative permeability Mr = 100> would be magnetised to the extent of only about 10"s T. 2 F=E+vxB= (73) -(Eo+vxBo) er+T Comparison with conducting magnetic rod To compare the case of a magnetic dielectric rod moving through an electrostatic field with that of a conducting rod, Section 3.3, we put Bo = 0 in eqn. 70, so that (74) This value of B is smaller than that given by eqn. 64, since the selfcomponent E's is smaller when the rod is a dielectric. The magnetisation is the same in both cases. This difference is shown more clearly by expressing the results in the form: (a) conducting rod (75) (b) dielectric rod er — l)vxE0 (76) In eqns. 75 and 76 the term in (fir — 1) is the magnetisation of the rod, while the second terms give the magnetic field produced by the motion of the charges induced on the rod. 3.4.3 Magnetisation and polarisation The magnetisation and polarisation of the rod may be obtained either by using eqns. 69 to 72 in the relations \ M = P = B - (77) D-e0E or from the S' measures given by eqn. 44 and the transformations in eqns. 24 and 25. The resulting expressions are lengthy, but if v«c they reduce to the simpler forms: x. (78) M = (urand P — Fig. 3 Cross-sectional view of cylindrical rod moving with velocity v perpendicularly to transverse electricfieldEo and longitudinal magnetic 3.5 field Bo 3.4 3.4.1 Magnetic dielectric rod in a transverse electric field and a tangential magnetic field, moving perpendicularly to both fields Electromagnetic field in rod 3.5.1 2(e r -l)Ux/y 0 2(er~l) Magnetic dielectric rod in tangential electric field and a transverse magnetic field, moving perpendicularly to these fields Electromagnetic field in rod The situation is that of Fig. 3 with Eo and Bo interchanged. Following the same procedure, we obtain: The situation is shown in Fig. 3. The rod is long, moving with constant velocity v, and the fields are constant and uniform. From eqns. 42,50,49,43 and 5-8, we find: E[ = E'p = - (65) )Ur - 1 \ V X E o B = 01 H= PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977 (80) Eo- )vxD0 (66) (67) (79) D = 02 t? 2 2 \vxHp Mr+1 Authorized licensed use limited to: Princeton University. Downloaded on June 04,2010 at 13:02:35 UTC from IEEE Xplore. Restrictions apply. (81) (82) (83) 1109 Eqns. 80-83 can also be derived from eqns. 69-72 by interchanging electric and magnetic field measures, er and jur, and changing the sign of the vector product terms. 5 Acknowledgment The author wishes to express his indebtedness to the referees of the paper for helpful comments and suggestions. The Lorentz force is F = E + vxB 3.5.2. - Eo +vxB0 (84) Magnetisation and polarisation As in Section 3.4.3, we find, if v«c, 6 that (85) and (86) P = (eP These expressions can also be derived from eqns. 78 and 79 by appropriate interchange of symbols and signs. 4 Conclusion The foregoing analysis provides proofs of the motional e.m.f. in a magnetic rod and its magnetisation by motion through an electrostatic field. In addition, rigorous expressions are obtained for the electromagnetic field in a magnetic dielectric rod moving with high velocity through constant and uniform crossed fields. These results satisfy the orthodox relativistic field transformations given in eqns. 20—23, as may be confirmed by substitution. Expressions for the magnetisation and polarisation are limited to the case in which v « c, but the rigorous expressions for high velocities can readily be obtained by the use of eqn. 77 and the rigorous expressions for the field measures given by eqns. 69—72 and 80—83. References 1 CULLWICK, E.G.: 'An experiment on electromagnetic induction by linear motion',/. IEE, 1939, 85, pp. 315-318 2 FARADAY, M.: 'Diary' (Bell, 1934), Vol. v, p. 397 3 CRAMP, W., and NORGROVE, E.H.: 'Some investigations on the axial spin of a magnet and on the laws of electromagnetic induction', / IEE, 1936, 78, pp. 481-491, and Discussion ibid, 1936, 79, p. 344 4 LORENTZ, H.A.: 'Lectures on theoretical physics' English translation by L. Silberstein and A.P.H. Trivellis. (Macmillan, 1927-31). Vol. 3, pp. 303-304 5 CULLWICK, E.G.: 'Electromagnetism and relativity' (Longmans Green, 1959, 2nd edn.), chap. 9-11 6 FARADAY, M., 'Experimental researches in electricity', Second series, 6, Dec. 21st, 1831, at the Royal Institution, paragraphs 194-195, reproduced in TRICKER, R.A.R.: 'The contribution of Faraday and Maxwell to electrical science' (Pergamon, 1966), pp. 154-155 7 FARADAY, M.: 'Diary' (Bell, 1934), Vol. i, pp. 425-426 8 POINCARE, H.: 'L'e"tat actuelle et l'avenir de la physique mathe"matique', Bull Sci. Math., 1904, 28, pp. 302-324 and 306 (English translation, 'Monist', 1905, 15, p.l) 9 LORENTZ, H.A.: 'Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten KOrpern' (Leiden, E.J. Brill, 1895) 10 HERTZ, H.: 'The fundamental equations of electro-dynamics of moving bodies', Wiedermann's Annalen, 1890,41, p.369 11 EICHENWALD, A.: 'Uber die magnetischen Wirkungen bewegter Korper im electrostatischen Felde', Ann. Phys., 1903, 11, pp. 1-30, 421-441, and 1904, 13, pp. 919-943 12 WILSON, H.A.: 'On the electric effect of rotating a dielectric in a magnetic field', Philos. Trans. A, 1905, 204, pp. 121-137 13 CULLWICK, E.G.: 'Electromagnetism and relativity' (Longmans Green, 1959, 2nd edn.), p. 126 Correspondence NUMERICAL METHODS IN CONFORMAL TRANSFORMATION Indexing terms: Conformal transformations, Electrostatic fields, Integration, Magnetostatic fields, Numerical methods 00 f = («i + i " w = dw = -(I/A yields the result: fcTJi •'o Abstract Numerical methods can usefully extend the application of conformal transformation. Various approximations have been proposed to improve the calculation of integrals containing singularities, but approximations can be avoided in the manner described. [u- 4. f(l//3,) < fe it"i + (2) In many practical cases of electromagnetic-potential field analysis, conformal transformation can facilitate accurate solution. Methods for dealing with polygonal contours have been given,1'2 but the difficulty of integration between limits that are isolated singular points has persisted. Some approximations have been proposed and their inaccuracies noted. However, the singularities of integrals can be completely avoided without approximation, as follows: The integrals concerned have the form at = c\ n (w- where j3fe is always positive. This integral, is the sum of two integrals of the same form, but, respectively, with the limits: (i) between ut and $(u,- + M1+, ) (ii) between £(M,- + ui+1) and MI+1 . Using the following changes of variable: (i) f = (w w = u. No singularities appear, because in the first integral the (i)th singular point is avoided, and in the second the (i + l)th singular point is avoided. When the variable is at the integration limits, the modulus of the whole expression in eqn. 2 is obtained by replacing the parentheses in the integrals by the corresponding moduli, because eqn. 2 preserves the same arguments of the complex value. 24th March 1977 PROF. A. NICOLAIDE Department of electrical Engineering University of Brasov 2200 Brasov Romania References 1 BINNS, K.J.: 'Numerical methods of conformal transformation', Proc. IEE, 1971,118, (7), pp. 909-910 2 BINNS, K.J., and LAWRENSON, P.J.: 'Analysis and computation of electric and magnetic field problems' (Pergamon, 1973) SPTC99S 1110 PROC. IEE, Vol. 124, No. 11, NOVEMBER 1977 Authorized licensed use limited to: Princeton University. Downloaded on June 04,2010 at 13:02:35 UTC from IEEE Xplore. Restrictions apply.