Brazilian Journal of Physics, vol. 28, no. 1, March, Electrodynamics. Rua Pamplona,145, S~ao Paulo, SP, Brazil

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1 Brazilian Journal of Physics, vol. 28, no., March, One An The Same oute Two Outstaning Electroynamics Antonio Accioly () an Hatsumi Mukai (2) () Instituto e Fsica Teorica, Universiae Estaual Paulista, ua Pamplona45, S~ao Paulo, SP, Brazil (2) Departamento e Fsica, Funac~ao Universiae Estaual e Maringa, Av. Colombo 5790, Maringa, Pr, Brazil eceive August 27, 997 We show that the same route that leas to Maxwell's electroynamics leas also to Poolsky's electroynamics, provie we start from Poolsky's electrostatic force law instea of the usual Coulomb's law. I. Introuction On the secon eition of Jackson's seminal book on classical electroynamics [], there is a section name \On the Question of Obtaining the Magnetic Fiel, Magnetic Force, an the Maxwell Equations from Coulomb's Law an Special elativity", where it is shown in etail that any attempt to erive Maxwell equations from Coulomb's law of electrostatics an the laws of special relativity ens in failure unless one makes use of aitional assumptions. What hypotheses are these? In an ingenious paper, Kobe [2] gave the answer all one nees to arrive at Maxwell equations is Soon after Kobe's paper, Neuenschwaner an Turner [4] obtaine Maxwell equations by generalizing the laws of magnetostatics, which follow from the Biot- Savart law an magnetostatics, to be consistent with special relativity. The preceing consierations leas us to the interesting question what woul happen if we followe the same route as Kobe i, using an electrostatic force law other than the usual Coulomb's one? We shall show that if we start from the force law propose by Poolsky [5], i.e., (i) Coulomb's law; (ii) the principle of superposition; F(r) = QQ0 4, e,r=a r 2, e,r=a r ra r ; () (iii) the assumption that electric charge is a conserve scalar (which amounts to assuming the inepenence of the observe charge of a particle on its spee [3] ); (iv) the requirement of form invariance of the electrostatic el equations uner Lorentz transformations, i.e. the electrostatic el equations are thought as covariant space-space components of covariant el equations. where a is a positive parameter with imension of length, Q an Q 0 are the charges at r an r = 0, respectively, an F(r) is the force on the particle with charge Q ue to the particle with charge Q 0 - an if we follow the steps previously outline, we arrive at the outstaning electroynamics erive by Poolsky [5] in the early 40s. In other wors, we shall show that the same route that leas to Maxwell equations leas also to Poolsky equations. A notable feature of Poolsky's accioly@axp.ift.unesp.br

2 36 A. Accioly an H. Mukai generalize electroynamics is that it is free of those in- nities which are usually associate with a point source. For instance, () approaches a nite value QQ0 8a as 2 r approaches zero. Thus, unlike Coulomb's law, Poolsky's electrostatic force law is nite in the whole space. In Sec. II we obtain the equations that make up Poolsky's electrostatics. In Sec. III we arrive at Poolsky's el equations by generalizing the equations of Sec. II, so that they are form invariant uner Lorentz transformations. For consistency, we show in Sec. IV that () is inee the electrostatic force law relate to Poolsky's theory. The conclusions are presente in Sec. V. Natural units (~ = c = ) are use throughout. As far as the electromagnetic theories are concerne, we will use the Heavisie-Lorentz units with c =. II. Poolsky's electrostatics As is well-known, the force on a text charge is proportional to its charge, all other properties of the force being assigne to the electric el E(r), which is ene by F = QE(r) ; where F is force on the charge Q situate at r an E(r) is the electric el at the position of this charge ue to all other charges. (The source charge's coorinates will be istinguishe from those of the el point, by a prime.) Accoringly, the electric el ue to a point charge Q 0 situate at r 0 is given by E(r) = Q0, e,=a 4 2, e,=a a ; (2) where = r, r 0. Note that this el is nite in the whole space. By the principle of superposition the electric el prouce by the charge istribution (r 0 )is, e,=a E(r) = 3 r 0 (r0 ) 4 2, e,=a a ; (3) where the integration is carrie out over all space. ( must of course vanish at sucient large r, making the volume of integration less than innity.) The preceing expression for the electric el arising from a charge istribution may be easily expresse as a graient of a scalar integral as follows E(r) =, 3 r 0 (r0 ) r, e,=a 4 (4) Since r acts only on the unprime coorinates, we may take it outsie the integral (4) to obtain E(r) =,r 3 r 0 (r0 ) 4, e,=a (5) Taking into account that the curl of any graient vanishes, we have immeiately r E =0 ; (6) which shows that Poolsky's electrostatic el, like Maxwell's one, is conservative. If the ivergence of (3) is taken, the result is r E(r) = where use has been mae of the ientity = c 3 r 0 (r0 ) r, e,=a 4 2, e,=a a 3 r 0 (r0 ) e,=a ; (7) 4 a 2 r ( A) =r A + r A Now, taking the Laplacian of (7), an using the ientity (see appenix) r 2 e,r=a = e,r=a a 2, 43 () ;

3 Brazilian Journal of Physics, vol. 28, no., March, yiels r 2 [r E(r)] = = a 2 3 r 0 (r0 ) e,=a 4a 2 r2 3 r 0 (r0 ) 4 e,=a a 2 = [r E(r), (r)] a2, 3 r 0 (r0 ) a 2 3 (r, r 0 ) So,, a 2 r2 re(r) =(r) (8) Equations (6) an (8) are the funamental laws of Poolsky's electrostatics. We will igress slightly at this stage to analyze an interesting feature of Poolsky's electrostatics. In this vein, we compute the ux of the electrostatic el across a spherical surface of raius with a charge Q at its center. Using (2) we arrive at the result I E S = Q, + e,=a a which tells us that I E S = 0; a ; Q; a ; (9) Therefore, a of raius a, unlike what happens in Maxwell's theory, shiels its exterior from the el ue to a charge place at its center. We remark that in Maxwell's electrostatics a close hollow conuctor shiels its interior from els ue to charges outsie, but oes not shiel its interior from the el ue to charges place insie it [6]. Note, however, that in orer not to conict with well establishe results of quantum electroynamics, the parameter a must be small. Incientally, it was shown recently that this parameter is of the orer of magnitue of the Compton wavelength of the neutral vector boson z, (z) cm, which meiates the unie an electromagnetic interactions [7]. After this parenthesis, let us return to our main subject. Equations (6) an (8) are now reay to be generalize using special relativity an the hypotheses that electric charge is a conserve scalar. We shall o that in the next section. III. Poolsky's el equations To begin with let us establish some conventions an notations to be use from now on. We use the metric tensor = = 0 B@ CA ; with Greek inices running over 0; ; 2; 3. oman inices - i; j; etc, - enote only the three spatial components. epeate inices are summe in all cases. The space-time four vectors (contravariant vectors) are x = (t; x), an the covariant vectors, as a consequences are x =(t;,x). The four-velocities are foun, accoring to u = x = (; v) ; u = (;,v) ; where is the proper time ( 2 = t 2, x 2 ), an enotes t= =(, v 2 ) =2. Let us then generalize (6) so that it satises the requirement of form invariance uner Lorentz transformations. To o that, we write the mentione equation in terms of the Levi-Civita ensity " nml, which equals +() if n; m; l is an even(o) permutation of ; 2; 3, an vanishes if two inices are equal. The curl equation becomes " k E l =0 (0)

4 38 A. Accioly an H. Mukai It we ene the quantities (0) can be rewritten as F 0i =,F 0i = E i =,E i ; () " k F 0l =0 We imagine now the curl law to be the space-space components of a manifestly covariant el equation (invariance uner Lorentz transformations). As a result, we get F =0 ; (2) where " is a completely antisymmetric tensor of rank four with " 023 = +. Of course, this generalization introuces the components F 00 ; F l0, an F lk, for which at this point we lack aphysical interpretation. Note that the F 0i are not necessarily static anymore. On the other han, as is well-known, the charge ensity is ene as the charge per unit of volume, which has as a consequence that the charge q in an elementof volume 3 x is q = 3 x. Since q is a Lorentz scalar [3], transforms as the time-component of a four-vector, namely, the time-component of the charge-current fourvector j =(; j). The electric charge, in turn, is conserve locally [3], which implies that it obeys a continuity j =0 (3) (8) can now be rewritten as +a E j = j 0 ; i i. Note i =,@ i. Using (), yiels +a F 0j = j 0 In orer that the left-han sie of the preceing equation transforms as the time-component of a four-vector, we must write it as +a2 F 0 = j 0 ; where @ 2 =@t 2, r 2 The requirement of form invariance of this equation uner Lorentz transformations leas then to the following result +a2 F = j (4) Imagine now a particle of mass m an charge Q at rest in a lab frame where there is an electrostatic el E. Newton's secon law allows us to write p t = QE (5) In terms of the proper time this becomes p = QE = Qu 0 E ; where u 0 is the time part of the velocity four-vector u. For the component along e x i irection, we have p i = Qu 0F 0i In orer that the right-han sie of this equation transforms like a space-component of a four-vector, it must be rewritten as p i = Qu F i ; whose covariant generalization is p = Qu F (6) If (6) is multiplie by p = mu, where m is the rest mass, the result is However, 2 (p p )=Qmu u F p p = m 2 2 (, v 2 ) = m 2 2 = 2 = m 2

5 Brazilian Journal of Physics, vol. 28, no., March, Therefore, we come to the conclusion that Using this result Kobe [2] u u F =0 an Neuenschwaner an Turner [3] showe that F is an antisymmetric tensor (F =,F ). Since F is an antisymmetric tensor of secon rank, it has only six inepenent components, three of which have alreay been specie. We name therefore the remaining components B i = 2 ilm F lm (7) Note that F kl =, klj B j. Writing out the components of (7) explicitly, B = F 23 = F 23 =,B ; B 2 =,F 3 =,F 3 =,B 2 ; B 3 = F 2 = F 2 =,B 3 Hence, a clever physicist who were only familiar with Poolsky's electrostatics an special relativity coul preict the existence of the magnetic el B, which naturally still lacks physical interpretation. The content of (2) an (4) can now be seen. For = 0, (2) gives r B =0 ; (8) showing that there are no magnetic monopoles in Poolsky's electroynamics, while for = i we obtain r ; (9) which says that time-varying magnetic els can be prouce be E els with circulation. The components = 0 an = i of (4) give, respectively, ( + a 2 2)r E = ; (20) ( + a 2 2) = j ; (2) which are nothing but a generalization of Gauss an Ampere- Maxwell laws in this orer. For = i, (6) becomes p t = Q (E + v B) ; (22) containing the Lorentz force. For = 0, (5) assumes the form U t = Qv E ; (23) where U = p 0 is the particle's energy. Accoringly, our smart physicist, who was able to preict the B el only from its knowlege of electrostatics an special relativity, can now-by making juicious use of (22) an (23) - observe, measure an istinguish the B el from the E el of (5). The new el couples to moving electric charge, oes not act on a static charge particle, an, unlike the electrostatic el, is capable only of changing the particle's momentum irection. Equations (8-2) make up Poolsky's higher-orer el equations. Of course, in the limit a = 0, all the preceing arguments apply equally well to Maxwell's theory. Two comments t in here () equation (4) is consistent with the continuity equation (3). In fact, if the ivergence of (4) is taken, we obtain F j Since F is an F is ientically zero. On the other han, accoring to j =0. Thus, the equation in han is ientically zero; (2) as was recently shown [8], it is not necessary to introuce a formula for the force ensity f representing the action of the el on a text particle. We have only to assume that (,f ) is the simplest contravariant vector constructe with the current j an a suitable erivative of the el F. Applying this simplicity criterion to Poolsky's electroynamics, we promptly obtain f =,F j ; where, as we have alreay mentione, j =(; j). Therefore, f 0 =,F 0i j i = E j ;

6 40 A. Accioly an H. Mukai an f k =,F k j = F 0k j 0 + F ik j i = F 0k j 0 + kil j i B l =(E + j B) k Thus, the force ensity for Poolsky's electroynamics is the same as that for Maxwell's electroynamics, namely, the well-known Lorentz force ensity. IV. Fining the force law for Poolsky's electrostatics We show now that () is inee the force law for Poolsky's electrostatics. >From (5) it follows that where V (r) E =,rv ; (24) 3 r 0 (r0 ), e,=a 4,, a 2 r2 r 2 V (r) =,(r) Eq. (8) can then be rewritten as For a charge Q at the origin of the raius vector this equation reuces to,, a 2 r2 r 2 V (r) =,Q 3 (r) (25) We solve this equation using the Fourier transform metho. First we ene V ~ (k) as follows V (r) = 3 ke,ikr V ~ (k) ; (26) (2) 3=2 ~V (k) = 3 re ikr V (r) ; (27) (2) 3=2 where 3 k an 3 r, respectively, stans for volumes in the three-imensional k-space an the coorinate space. If we substitute (26) into (25) an take into account that 3 (r) = 3 ke,ikr ; (2) 3 we promptly obtain ~V (k) = Q (2) 3=2 M 2 k 2 (k 2 + M 2 ) where M 2 =a 2. So, V (r) = QM 2 3 e,ikr (2) 3 k k 2 (k 2 + M 2 ) Since the orientation of our coorinate system is arbitrary, we may choose the z-axis along r an obtain ; V (r) = QM 2 (2) 3 0 c + k 2 k e,ikrcos 2 (k 2 + M 2 )k (cos) 2 0 ; where r = jrj, k = jkj, an (; ) are conventional spherical polar coorinates. As a consequence, where But, V (r = QM 2 (2) 2 + = QM 2 r (2) 2 sinkr kr sinx x k k 2 + M 2 x x 2 + M 2 r 2 x x 2 + M 2 r = 2 M 2 r 2 x, x x 2 + M 2 r 2 Therefore, V (r) = Q (2) 2 r [I, I 2 ] ; I = xsinxx I 2 = x 2 + M 2 r 2 = Im sinx x ; (28) x xe ix x x 2 + M 2 r 2 (29) Integral (28) may be foun in any textbook on the theory of functions of a complex variable [9]. It can also be

7 Brazilian Journal of Physics, vol. 28, no., March, carrie out by means of a trivial trick [0]. Inee, let I() = I() =, 0 0 e,x sinx x so that I =2I(0) x e,x sinxx = + 2 I() =, + 2 = C, tan But I() = 0. Therefore C = =2. I() = 2,tan an I =2I(0) =. Integral (29) can be easily evaluate by the metho of contour integration []. Consier in this vein ze iz z z 2 +M 2 r, where the contour of integration 2 was chosen to be the real axis an a semicircle of innite raius in the upper half plane. Along the real axis the integral is I 2. In the large semicircle in the upper half plane we get zero, since exp(iz)! 0 for z! i. The resiue of ze iz z 2 + M 2 r 2 = zeiz 2Mri z, Mri, z + Mri at z = Mri (which is the only pole that lies insie the contour of integration) is e,mr 2. Hence, As a result, I 2 = Im(2i e,rm 2 V (r) = )=e,rm Q, e,rm (2) 2 r = Q, e,r=a 4 r Accoringly, the electric el ue to a charge Q at the origin is given by E(r) =,rv =,r Q = Q 4, e,r=a 4 r, e,r=a r 2, e,r=a ra r r It follows then the force law for Poolsky's electrostatics is F(r) = QQ0, e,r=a, e,r=a r ra ; 4 r 2 r which is nothing but the force law for which wewere looking (see Eq. ()). ecently an algorithm was evise which allows one to obtain the energy an momentum relate to a given el in a simple way [8]. Using this prescription we can show that in the framework of Poolsky's electrostatics the energy is given by E el = 2 3 x E 2 + a 2 (r E) 2 Making use of the expression for the electrostatic el we have just foun, we promptly obtain " el = Q 2 =2a ; which tells us that the energy for the el of a point charge has a nite value in the whole space. This is inee a nice feature of Poolsky's generalize electroynamics. V. Final remarks Despite the simplicity of its funamental assumptions, Poolsky's moel has been little notice. Currently some of its aspects have been further stuie in the literature [7;8;2;3]. In particular, the classical self-force acting on a point charge in Poolsky's moel was evaluate an it was shown that in this moel, unlike what happens in Maxwell's electroynamics, the electromagnetic mass is nite an enters the particle's equation of motion in a form consistent with special relativity. To conclue we call attention to the fact the same assumptions that lea to Maxwell's equations lea also to Poolsky's equations, provie we start from a generalization of the Coulomb's law instea of the usual Coulomb's law. Yet, in spite of the great similarity between the two theories, Poolsky's generalize electroynamics leas to results that are free of those innities which are usually associate with a point source.

8 42 A. Accioly an H. Mukai Appenix An important ientity involving functions An useful ientity involving functions is r 2 e,=a = e,=a a 2, 43 (r, r 0 ) ; where = jr, r 0 j. To prove this let us consier V,=a 3 r 0 If r is not in the region V over which we are integrating, it never coincies with r 0, an it is easily verie by irect ierentiation that r 2 e,=a = e,=a a 2. Any region V that contains r0 may be subivie into a small of raius centere on r surroune by surface S an the remaining volume where r 2 e,=a V,=a Now, accoring to the mean value theorem,,=a 3 r 0 = + 3 r 0 = f(),=a 3 r 0 remaining f(r 0 ) e,=a volume a 2 3 r 0 r e,=a 2 3 r 0 ; = e,=a a 2. Hence, where is some point in the. Applying the ivergence theorem to the last integral, we obtain r e,=a r 3 r 0 e,=a = + e,=a 2 a S0 Note that r r =,r r 0. On the other han, e,=a S 2 + e,=a a As tens to 0, must approach r, so that S S0 =,4,=a e,=a + e,=a a 3 r 0 =,4f(r) Thus, 3 r 0 f(r 0 )r 2 e,=a =,4f(r) + = 3 r 0 e,=a a 2 f(r0 ) 3 r 0 e,=a a 2, 43 (r, r 0 ) f(r 0 )

9 Brazilian Journal of Physics, vol. 28, no., March, eferences. J. D. Jackson, Classical Electroynamics (Wiley, New York, 975), 2n. e., pp D. H. Kobe, Am. J. Phys 54, 63 (986). 3. See ef., pp D. E. Neuenschwaner an B. N. Turner, Am. J. Phys. 60, 35 (992). 5. B. Poolsky, Phys. ev. 62, 66 (942); B.Poolsky an P. Schwe, ev. Mo. Phys, 20, 40 (948). 6. See ef., p J. Frenkel an. B. Santos, \The self-force on point charge particles in Poolsky's generalize electroynamics," (to be publishe). 8. Antonio Accioly, Am. J. Phys. 65, 882 (997). 9. F. W. Byron, Jr, an. W. Fuller, Mathematics of Classical an Quantum Physics (Aison-Wesley Publishing Company, New York, 970), volume two, pp Jon Mathews an.l. Walker, Mathematical Methos of Physics (W.A. Benjamin, Inc., New York, 965) p See ef. 9, pp C. A. P. Galv~ao an B. M. Pimentel, Can. J. Phys. 66, 460 (988); J. Barcelos-Neto, C. A. P. Galv~ao, C. P. Nativiae,. Phys. C52, 559 (99); L. V. Belveere, C. P. Nativiae, C. A. P. Galv~ao,. Phys. C56, 609 (992). 3. A. J. Accioly an H. Mukai,.Phys. C 75, 87 (997).