Energy and momentum for the electromagnetic field described by three outstanding electrodynamics Antonio Accioly
Citation: American Journal of Physics 65, 882 (1997); doi: 10.1119/1.18676 View online: http://dx.doi.org/10.1119/1.18676 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/65/9?ver=pdfcov Published by the American Association of Physics Teachers
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35 Energy and momentum for the electromagnetic field described by three outstanding electrodynamics Antonio Acciolya) Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP, Brazil ͑Received 27 September 1996; accepted 22 March 1997͒ A prescription for computing the symmetric energy–momentum tensor from the field equations is presented. The method is then used to obtain the total energy and momentum for the electromagnetic field described by Maxwell electrodynamics, Born–Infeld nonlinear electrodynamics, and Podolsky generalized electrodynamics, respectively. © 1997 American Association of Physics Teachers.
I. INTRODUCTION In Sec. II we outline the algorithm in hand and then apply it to obtain the total energy and momentum for the sourceless There are two ways of obtaining the expression for the Maxwell field. Section III is devoted to the electrodynamics total energy and momentum related to a given relativistic of Born–Infeld and Podolsky, in this order. We present our field such as the Maxwell field. The first, which incidentally conclusions in Sec. IV. is described in most college or university textbooks on phys- In what follows we will work in natural units where ប ics, is based on mechanics and employs concepts such as ϭcϭ1 and use the Heaviside–Lorentz units with c replaced force, work, and so on. The second way is to make use of the by unit for the electromagnetic theories. Our conventions for so-called symmetric energy–momentum tensor. Unfortu- relativity follow nearly all recent field theory texts. We use nately, the usual prescription for calculating this tensor is the metric tensor rather cumbersome for those who are not facile with field 1000 theory. As a consequence, the study of this powerful and elegant method of easily calculating the total energy and 0Ϫ10 0 ϭϭ , momentum for localized fields is, in general, relegated to 00Ϫ10 graduate courses. Our aim here is to present an elementary ͩ 00 0Ϫ1ͪ algorithm for computing the symmetric energy–momentum tensor from the field equations and apply it to obtain the total with Greek indices running over 0, 1, 2, 3. Roman indices— energy and momentum for the electromagnetic field related i, j, etc.—denote only the three spatial components. Re- to the three most famous known electrodynamics, i.e., the peated indices are summed in all cases. electrodynamics developed by Maxwell,1 Born–Infeld,2 and 3 Podolsky, respectively. By elementary we mean that no ef- II. THE ALGORITHM fort will be expended on field theoretic technicalities and that correspondingly no formal sophistication will be demanded Suppose one wants to find the symmetric energy– of the reader. momentum tensor for some free relativistic field. We take for
882 Am. J. Phys. 65 ͑9͒, September 1997 © 1997 American Association of Physics Teachers 882
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35 granted that this tensor, which from now on will be denoted Lorentz scalar. This field, which corresponds to electrically by T (TϭT), obeys the differential conservation neutral particles, is known to satisfy the Klein–Gordon law:1 equation4 ᮀϩm2 x ϭ0, ͒ T ϭ0. ͑1͒ ͑ ͒͑ץ ’’If we introduce a ‘‘source . ץץϭץץ The continuity equation ͑1͒ yields the conservation of total where ᮀϭ energy and momentum upon integration over all of three- j(x) for the field, the field equation turns out to be space at fixed time. Indeed, ͑ᮀϩm2͒͑x͒ϭ j͑x͒. d 3 3 0 3 i The simplest expression for the ‘‘force density’’ f built . iTץT ϭ d xT ϩ d xץ0ϭ d x ͵ dt ͵ ͵ with a given scalar current j(x) and a derivative of the scalar For localized fields, the second integral—a divergence— field is of the form 1 .͒͑ᮀϩm2͒ץϭ͑ץgives no contribution, and taking into account that f ϭ j Note that for some j there is one and only one derivative of Pϭ d3xT0, ͑2a͒ ͵ the field that couples in the simplest way to j so that the resulting expression is a contravariant vector. where (ii) The Maxwell field. In the absence of sources, Maxwell 5,6 Pϭ͑⑀,P͒, ͑2b͒ equations are F ϭ0, ͑4͒ץ -is the four-momentum of the field, and ⑀ and P are, respec tively, the total energy and three-momentum of the field, we F ϩ F ϩ F ϭ0, 5 ͒ ͑ ␣ ץ ␣ץ ␣ץ find where F denotes the antisymmetric electromagnetic field dP ϭ0, tensor (FϭϪF). Introducing a source j for the elec- dt tromagnetic field, the field equations take on the form or, equivalently, F ϭ j , ͑6͒ץ d⑀ dP F␣ϭ0. ͑7͒ץF␣ϩץFϩ␣ץ .ϭ0, ϭ0 dt dt The force density is given in this case by If we ‘‘switch on’’ a variable external ‘‘current’’ which acts F. ͑8͒ ץ f ϭF jϭF as a source for the field, the field equation will be modified ␣ ␣ ␣ and the energy–momentum tensor will no longer satisfy ͑1͒, (iii) The complex Klein–Gordon field. This field is known as in the case of the free field. Instead, it will now obey the to obey the equations4 equation: ͑ᮀϩm2͒͑x͒ϭ0, ͑ᮀϩm2͒*͑x͒ϭ0, Tϭ f , ͑3͒ ץ where the * field is the complex conjugate of the field. where the contravariant vector f , which has the dimension The complex Klein–Gordon field corresponds to electrically of force density, describes the interaction of the field with the charged particles. Let us then introduce a ‘‘source’’ j*(x) current. From ͑3͒ we promptly obtain for the * field. As a consequence, dP ͑ᮀϩm2͒*ϭ j*, ͑ᮀϩm2͒ϭ j. ϭ d3xf. dt ͵ The ‘‘force density,’’ which of course is a real quantity, has This equation tells us that the rate of change of the four- the form *ץϩ jץ*momentum of the field is equal to the ‘‘total force’’ ͐d3xf , f ϭ j which is the analogue of the Minkowski equation for a single 2 2 .*͑ᮀϩm ͒ ץ͑ᮀϩm ͒*ϩ ץ ϭ 1 particle. Introducing an energy–momentum tensor TM for the current, i.e., for the matter, by the definition (iv) The Dirac field. The corresponding field equations 4 are , TM ϭϪfץ ,͑x͒Ϫm͑x͒ϭ0ץ ␥i we can rewrite ͑3͒ in the form of a continuity equation: ¯ ¯ ,͑x͒i␥ ϩm͑x͒ϭ0ץ .͑T ϩTM ͒ϭ0ץ where ¯ϵ†␥0 is the adjoint spinor to . The indices label- Thus the total energy and momentum of the field and the ing spinor components and matrix elements were suppressed matter are conserved. ¯ † 0 We return now to Eq. ͑3͒. Let us then postulate that f is for the sake of simplicity. We assume that jϭ j ␥ is the ¯ the simplest contravariant vector constructed with a given ‘‘source’’ for the field. As a result, current and a suitable derivative of the field. We qualify this i ¯ ϩm¯ϭ¯j, i Ϫm ϭϪj. ץ ␥ ␥ ץ .statement by means of some examples (i) The real Klein–Gordon field. In a sense the simplest The ‘‘force density,’’ which incidentally is an Hermitian field theory is that of a real field (x) that transforms as a quantity, can be written as
883 Am. J. Phys., Vol. 65, No. 9, September 1997 Antonio Accioly 883
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35 , j Note that, contrary to what happens in usual field theory¯ ץϩ ץf ϭ¯j ␣ ␣ ␣ here we do not have to postulate the equation of motion of ¯ ¯ ¯ ;Ϫm͒. the material bodies under the influence of the interactionץ ␥͑i␣ץϪ␣ץ␥ ϩm͒ץϭ͑i i.e., we do not have to introduce a formula for the force We hope the previous examples have clarified what ‘‘the density representing the action of the field on each body. In simplest contravariant vector constructed with a given cur- fact, the simplicity criterion we have previously introduced rent and a suitable derivative of the field’’ means. Note that determines automatically the force density. Consider, in this there is nothing ambiguous in the prescription for building vein, the Maxwell theory. From ͑9͒, the force density is the force density out of the current and the derivative of the given by field. ␣ ␣ Based on the previous considerations, we now find T for f ϭF j , the free Maxwell field. We do that in two steps. whereupon (i) We ‘‘turn on’’ a current j: From ͑3͒ and ͑8͒, we have jϭ͑,j͒, F. ͑9͒ ץ T ϭf ϭF jϭFץ ␣ ␣ ␣ ␣ where is the charge density. On the other hand, the com- The right-hand side of this equation can be expanded to ponents Ei(Bi) of the electric field ͑magnetic field͒ can be F ͒F, related to the components of the electromagnetic field tensor ץF F͒Ϫ͑͑ ץFϭ ץ F ␣ ␣ ␣ as follows:5,6 and, exchanging the dummy indices and on the second i 0i term, we get E ϭF , ͑14͒ 1 1 i ilm B ϭ 2⑀ Flm , ͑15͒ . F␣͒FץF␣ϩץ͑F␣F ͒Ϫ 2͑ץF ϭץF␣ ilm Using ͑7͒, we can replace the term in parentheses with where ⑀ is the Levi–Civita density which equals ϩ1 (Ϫ1) if i, l, m is an even ͑odd͒ permutation from 1, 2, 3, F , leading to␣ץϪ and vanishes if two indices are equal. Therefore, 1 FF 0 0i␣ץ͑F␣F ͒ϩ 2ץF ϭץF␣ f ϭF jiϭE–j, .F F͒͑ ץF F͒ϩ 1͑ ץϭ ␣ 4 ␣ and But, k k k0 ki k0 kil f ϭF jϭF j0ϩF jiϭF j0ϩ⑀ jiBl 10͒͑ . ץϭ␦ ץϭ ץ ϭ ץ . ␣ ϭ͑E͒kϩ͑j؋B͒k ␣ ␣ ␣ Thus So, 1 ␣ F F ϩ ␦ F F ͔. ͑11͓͒ ץF ϭ ץ F , ␣ 4 ␣ f ϭ͑E–j,Eϩj؋B͒␣ From ͑9͒ and ͑10͒ it follows that which is nothing but the well-known Lorentz force density. F Fϩ 1␦ F F͔. Let us then find the total energy and momentum for the͓ ץT ϭץ ␣ ␣ 4 ␣ Maxwell field with zero source. Using ͑2a͒, ͑2b͒, ͑13͒, ͑14͒, If we now multiply both sides of the above equation by ␣, and ͑15͒, yields we immediately obtain 1 1 3 0i ij   1  ⑀fieldϭ d x F Fi0ϩ F Fij ͬ ͓FF ϩ 4 FF ͔. ͑12͒ ͵ ͫ2 4ץT ϭ ץ Note that 1 ϭ d3x ͓E2ϩB2͔, ͑16͒ ␣  ␣  ͵ F␣ϭF , ␦␣ ϭ . 2
Equation ͑11͒ can be rewritten as i 3 ji 3 i Pfieldϭ d xF0 jF ϭ d x͑E؋B͒ . ͑17͒   1  ͵ ͵ .͔ ͓FF ϩ 4 FFץT ϭץ Equation ͑17͒ can be rewritten as (ii) We ‘‘turn off ’’ the current: As a result, f ϭ Tϭ F Fϩ 1 F F ϭ0. 3 ,͓ 4 ͔ Pfieldϭ ͵ d x͑E؋B͒ץ ץ So, the fully contravariant form of the symmetric energy– from which we trivially obtain the vector S representing en- momentum tensor for the Maxwell field is ergy flow, i.e., the Poynting vector,   1  T ϭFF ϩ 4 FF . ͑13͒ SϭE؋B. ͑18͒ We may now present a simple algorithm for computing Note that SiϭT0i. the symmetric energy–momentum tensor from the field equation. Multiply the equation for the field in hand by a suitable derivative of this field so that the resulting expres- III. THE NONLINEAR ELECTRODYNAMICS OF sion contains only one free spacetime index and then rewrite BORN–INFIELD AND THE HIGHER ORDER it as a four-derivative. Of course, this algorithm is nothing ELECTRODYNAMICS OF PODOLSKY but an ‘‘operational’’ summary of the main points we have just discussed. As such, its main role is to work as a mne- We shall now analyze two interesting attempts to modify monic device for the reader. Maxwell electrodynamics in order to get rid of the infinities
884 Am. J. Phys., Vol. 65, No. 9, September 1997 Antonio Accioly 884
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35 that usually plague that theory. The first was made by Born7 Expanding the square root and keeping only terms of second in the early 30s. He proposed that the field equations con- order, we recover the expression for the total energy of the cerning Maxwell theory, Eqs. ͑4͒ and ͑5͒, should be replaced Maxwell field; i.e., by 1 ⑀ ϭ d3x ͑E2ϩB2͒. F field ͵ 2 ϭ0, ͑19͒ ץ 2 ͫͱ1ϩF/a ͬ The total momentum of the field and the Poynting vector, in turn, are given, respectively, by F␣ϭ0, ͑20͒ץF␣ϩץFϩ␣ץ i ͑EϫB͒ E؋B 1 where Fϵ 2FF ͑F is the antisymmetric electromag- i 3 Pfieldϭ d x , Sϭ . netic field tensor͒ and a is a constant with dimension of ͵ B2ϪE2 B2ϪE2 2 1ϩ 1ϩ (length) . Note that Eq. ͑19͒ is highly nonlinear. Let us then ͱ a2 ͱ a2 find the symmetric energy–momentum tensor for this theory. Since Born’s equations can be derived via a variational prin- For this purpose, multiply Eq. ͑19͒ by F␣ : ciple from a Lagrangian density L, in the limit of small field (0) 1 2 2 F strengths, L has to approach L ϭ 2(E ϪB ), which is .ϭ0 ץ F ␣ 2 exactly the Lagrangian density leading to Maxwell’s equa- ͫͱ1ϩF/a ͬ 7 tions. The Lagrangian density related to Born’s equations is Using ͑20͒ we obtain LϭϪa2ͱ1ϩF/a2. ͑23͒ F Expanding the square root, we obtain␣ץ F F␣F F . ϩ ץϭ ץ F ␣ 2 2 2 ͩͱ1ϩF/a ͪ ͫͱ1ϩF/a ͬ 2ͱ1ϩF/a 2 1 2 2 LϭϪa ϩ 2͑E ϪB ͒ϩ••• . On the other hand, This Lagrangian density, as was expected, differs from the Maxwell one by a constant term. Now, the usual procedure 2F͒͑ ץ F 1 ץF ␣ ␣ 2 2 for calculating the symmetric energy–momentum tensor, and ͱ1ϩF/a␣ץ ϭ ϭa 2ͱ1ϩF/a2 4 ͱ1ϩF/a2 consequently the total energy of the field, is entirely based on the Lagrangian density, which incidentally is not a physically 2 2 ͓␦␣ͱ1ϩF/a ͔. ͑21͒ measurable quantity in the usual sense but a convenientץ ϭa mathematical tool. As a consequence, the symmetric Thus the symmetric energy–momentum tensor is given by energy–momentum tensor computed via ͑23͒ leads to the ␣ same troublesome value for the energy as before. Is it pos- F F T␣ϭ ϩ␣a2ͱ1ϩF/a2. ͑22͒ sible to avoid this problem without further redefining the ͱ1ϩF/a2 symmetric energy–momentum tensor? Yes, we redefine the Lagrangian density as follows: Substitution of ͑22͒ into ͑2a͒ leads to the expression for the total energy of the field: Lϭa2͓1Ϫͱ1ϩF/a2͔, ͑24͒ 2 2 E2 B ϪE leading in first approximation to 3 ϩ 2 ϩ ⑀fieldϭ d x a 1 2 . ͵ 2 2 ͱ a 1 2 2 B ϪE Lϭ 2͑E ϪB ͒ϩ••• . ͫͱ1ϩ 2 ͬ a The T computed using ͑24͒ gives a finite value for the Developing ⑀field in powers of a: energy. Note, however, that the original Lagrangian density proposed by Born ͑23͒ as well as the Lagrangian density ͑24͒ 1 yield the same field equations ͑19͒. In a sense this way of ⑀ ϭ d3x a2ϩ ͑E2ϩB2͒ϩ••• , field ͵ ͫ 2 ͬ sweeping delicate questions under the carpet, despite being mathematically elegant, disguises the underlying physics of we see that, as a first approximation, the new ⑀field differs the process. from the old ͑16͒ by a constant. Yet, this is an infinite con- The fact that Lagrangian density ͑23͒ does not reproduce stant! Fortunately, only energy differences are observable. Maxwell Lagrangian density in the limit of small field Hence, this embarrassing infinite constant is harmless and strengths, among other things, led Born and Infeld to refor- easily removed by redefining the symmetric energy– mulate Born’s original theory. Since the Lagrangian density momentum tensor to be L must be a Lorentz scalar and the electromagnetic field has only two gauge invariant Lorentz scalars, namely, ␣ F F Tϭ ϩa2͓ͱ1ϩF/a2Ϫ1͔. Fϭ 1F FϭB2ϪE2, ͱ1ϩF/a2 2 2 1 2 2 The total energy of the field is now given by G ϭ͑ 4F*F ͒ ϭ͑E–B͒ , 1 E2 B2ϪE2 where *F ϭ 2⑀ F is the dual field strength tensor, L ⑀ ϭ d3x ϩa2 1ϩ Ϫ1 . can be a function of F and G2 only. Born and Infield opted field 2 2 ͱ 2 ͵ B ϪE ͩ a ͪ for the following function of the invariants, as their Lagrang- ͫͱ1ϩ 2 ͬ ian density2 a
885 Am. J. Phys., Vol. 65, No. 9, September 1997 Antonio Accioly 885
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35 F G2 An attractive feature of this theory is that it can account for a 2 Lϭa 1Ϫͱ1ϩ 2Ϫ 4 , stable electron with a finite size and finite self-energy. ͫ a a ͬ The second notable attempt to modify Maxwell electrody- where a has the meaning of a maximum field strength. So, namics was made by Podolsky3 in the early 40s. In general- for fields much weaker than a we recover Maxwell Lagrang- izing the equations of Maxwell electrodynamics, Podolsky ian density by expanding the square root and keeping only made every effort to leave the ordinary assumptions of Max- terms of second order. They chose this Lagrangian density in well theory as nearly unaltered as possible. His higher-order analogy to the relativistic Lagrangian for a free particle, L electrodynamics is described by the equations 2 2 F ϭ0, ͑25͒ץϭm͓1Ϫͱ1Ϫv ͔, which in the nonrelativistic limit (vӶ1) ͑1ϩa ᮀ͒ reduces to the well-known expression Lϭmv2/2. Note that F␣ϭ0, ͑26͒ץF␣ϩץFϩ␣ץ just as the limiting velocity vϭ1 is of no importance in classical mechanics, the maximum field strength a is irrel- where a is a real parameter with dimension of length and evant for classical electrodynamics. The field equations for  It is worth mentioning that the usual prescription . ץץᮀϵ Born–Infeld theory are for obtaining the symmetric energy–momentum tensor for FϪ*FG/a2 higher-order field theories is difficult to handle even for ϭ0, those that are familiar with field theory. Fortunately our ץ recipe allows us to carry out this computation in a simple F G2 way. 1ϩ Ϫ ͫ ͱ a2 a4 ͬ To find T for Podolsky generalized electrodynamics we multiply ͑25͒ by F␣ . Using ͑11͒ we obtain .F␣ϭ0ץF␣ϩץFϩ␣ץ .Fϭ0 ץF Fϩ 1␦F F͔ϩF a2ᮀ͓ ץ Using our algorithm we find that the symmetric energy– ␣ 4 ␣ ␣ momentum tensor for that theory is given by The second term can be transformed as follows:
2 2 2 2 2 . F␣ᮀF ͔aץ͑F␣ᮀF ͒ϪץF ϭ͓ץF F ϩG /a F G a F␣ᮀ 2 T ϭ ϩ a 1ϩ 2Ϫ 4 . F G2 ͱ a a From ͑26͒ and the above equation we obtain 1ϩ 2Ϫ 4 ͱ a a 2 2 1 ͒ 2FᮀF ͑␣ץ͑F␣ᮀF ͒ϩץ͓ F ϭaץa F␣ᮀ In deriving this result, use has been made of the identity 1 .͔ F␣ץϪ 2F ᮀ *FF ϭϪG␦ . On the other hand, Just as we have done before, we redefine the symmetric energy–momentum tensor to avoid troublesome constants: 1 1 ͒␣FץF␣ϩץFϭ 2F ᮀ͑␣ץ2F ᮀ 2 2 2 F ץF FϩG /a F G ϭϪF ᮀ Tϭ ϩa2 1ϩ Ϫ Ϫ1 . ␣ 2 ͱ a2 a4 . FᮀF ץFᮀF ͒ϩ͑ ץF G ͫ ͬ ϭϪ 1ϩ Ϫ ␣ ␣ ͱ a2 a4 Similarly, The energy, the momentum, and the Poynting vector, are now given, respectively, by  F␣ץ ץ FץF ᮀF␣ϭץ ͒ F ץF ϩ ץ͑ץF ץE2ϩ͑E B/a͒2 ϭϪ 3 –  ␣  ⑀fieldϭ d x ͵ 1 2 2 2   .͒F ץ Fץ2 ͑␣ץF␣͒Ϫ ץ Fץ͑ץB ϪE ͑E–B͒ ϭϪ 1ϩ Ϫ ͫͱ a2 a4 Hence,
2 2 2 2 2 ͓F␣ᮀF ϩF ᮀF␣ץ F ϭaץB ϪE ͑E–B͒ a F␣ᮀ ϩa2 1ϩ Ϫ Ϫ1 , ͱ 2 4  1 F ϩ 2␦␣͑FᮀFץ␣F ץa a ͪ Ϫ ͩ  .F͔͒ ץ Fץϩ ͬ The symmetric energy–momentum tensor can then be ex- E؋B pressed as 3 Pfieldϭ ͵ d x , 2 2 2 B ϪE ͑E B͒ 1 a2 – ␣ ␣ ␣ ␣ 1ϩ Ϫ T ϭF Fϩ FF ϩ ͓FᮀF ͱ a2 a4 4 2  2 ␣ᮀ ᮀ ␣ F͔Ϫa ͓F FϩF F ץ FץE؋B ϩ ,F͔ ץ␣F ץSϭ . ϩ 2 2 2  B ϪE ͑E–B͒ 1ϩ Ϫ which agrees with the Podolsky result. ͱ a2 a4 In electrostatics this gives for the energy,
886 Am. J. Phys., Vol. 65, No. 9, September 1997 Antonio Accioly 886
This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35 1 IV. CONCLUSION E ϭ d3x ͕E2Ϫa2͓͑ E͒2ϩ2E 2E͔͖. ͑27͒ field ͵ 2 “– –“ We have devised an algorithm for computing the symmet- Taking into account that “؋Eϭ0, and assuming that E“–E ric energy–momentum tensor from the field equations. The vanishes at infinity faster than 1/r2, one finds that Eq. ͑27͒ prescription can be used for both ordinary and higher-order can easily be put in the form field theory. It does not matter whether the field in hand is a tensor or a spinor field. 1 A last remark: The algorithm can be easily extended to E ϭ d3x͓E2ϩa2͑ E͒2͔, ͑28͒ curved space.8,9 field 2 ͵ “– which is obviously positive. In the static case the scalar po- tential due to a point charge turns out to be3 ACKNOWLEDGMENTS Q ϭ ͑1ϪeϪr/a͒, 4r I am very grateful to the referees for their helpful sugges- tions and comments. which approaches a finite value Q/4a as r approaches zero. Making use of EϭϪ , we easily obtain the expression for a͒Electronic mail: [email protected] “ 1 the electrostatic field due to a point charge: A. O. Barut, Electrodynamics and Classical Theory of Fields and Par- ticles ͑Dover, New York, 1980͒; see also references therein; J. D. Jackson, Classical Electrodynamics ͑Wiley, New York, 1975͒, 2nd ed. Q 1 1 1 r 2M. Born and L. Infeld, ‘‘Foundations of the New Field Theory,’’ Proc. R. E͑r͒ϭ ϪeϪr/a ϩ . ͑29͒ 4 ͫr2 ͩ r2 arͪͬ r Soc. London, Ser. A 144, 425–451 ͑1934͒. 3B. Podolsky, ‘‘A Generalized Electrodynamics Part I—Non-Quantum,’’ Substituting ͑29͒ into ͑28͒ and performing the integration, Phys. Rev. 62, 68–71 ͑1942͒. we find that the energy for the fields of a point charge is 4F. Mandl and G. Shaw, Quantum Field Theory ͑Wiley, New York, 1984͒. 5 given by Q2/2a. Thus, unlike Maxwell electrodynamics, D. H. Kobe, ‘‘Generalization of Coulomb’s law to Maxwell’s equations using special relativity,’’ Am. J. Phys. 54, 631–636 ͑1986͒. Podolsky generalized electrodynamics leads to a finite value 6D. E. Neunschwander and B. N. Turner, ‘‘Generalization of the Biot– for the energy of a point charge in the whole space. Of Savart law to Maxwell’s equations using special relativity,’’ Am. J. Phys. course, the momentum of the field in this case is equal to 60, 35–38 ͑1992͒. zero. The expressions for the total energy and momentum of 7M. Born, ‘‘On the Quantum Theory of the Electromagnetic Field,’’ Proc. the field in the general case are not very illuminating, so they R. Soc. London, Ser. A 143, 410–437 ͑1933͒. 8 will not be displayed here. A. Accioly, A. D. Azeredo, C. M. L. de Araga˜o, and H. Mukai, ‘‘A simple It is worth mentioning that our method tells us that the prescription for computing the stress-energy tensor,’’ Class. Quantum Grav. ͑to be published͒. force law for Podolsky generalized electrodynamics is pre- 9A. Accioly and H. Mukai, ‘‘On the dissimilarity between the algorithms cisely the Lorentz force. If we consult Podolsky’s paper we for computing the symmetric energy–momentum tensor in ordinary field find that this is the very force law he assumed for his theory. theory and general relativity,’’ Nuovo Cimento B ͑to be published͒.
HIGH VOLTAGE ‘‘Oh, and about lab rules,’’ he added. ‘‘The first rule is ‘It’s your job and not mine.’ The second is ‘I don’t take any excuses,’ and the third is ‘Lab hours are 7:30 to 5:30, with half an hour for lunch.’ You should be able to get all your lab work done between those hours. Harvard’s Physics Department lost a student last year when he fell asleep into his experiment’s high-voltage power supply at 2:00 in the morning. He was found dead the next day. We don’t want that kind of thing to happen here in the Sloan Lab.’’ Me neither. ‘‘Uh huh’’. ‘‘You can do computer work at night and read journal articles. And try taking a swim after dinner for half an hour or so. I’ve found that exercising at that time makes me need less sleep and wakes me up so I can work another four or five hours. Come on downstairs—I’ll show you your cell.’’
Pepper White, The Idea Factory—Learning to Think at MIT ͑Penguin Books, New York, 1991͒,p.79.
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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.236.163 On: Wed, 02 Apr 2014 20:35:35