How the potentials in different gauges yield the same retarded electric and magnetic fields

Transcription

1 How the potentials in ifferent gauges yiel the same retare electric an magnetic fiels José A. Heras a Departamento e Física, E. S. F. M., Instituto Politécnico Nacional, México D. F. México an Department of Physics an Astronomy, Louisiana State University, Baton Rouge, Louisiana Receive 19 May 2006; accepte 27 October 2006 This paper presents a simple an systematic metho for showing how the potentials in the Lorentz, Coulomb, Kirchhoff, velocity, an temporal gauges yiel the same retare electric an magnetic fiels. The metho uses the appropriate ynamical equations for the scalar an vector potentials to obtain two wave equations whose retare solutions lea to the electric an magnetic fiels. The avantage of this metho is that it oes not use explicit expressions for the potentials in the various gauges, which are generally simple to obtain for the scalar potential but ifficult to calculate for the vector potential. The spurious character of the term generate by the scalar potential in the Coulomb, Kirchhoff, an velocity gauges is note. The nonspurious character of the term generate by the scalar potential in the Lorenz gauge is emphasize American Association of Physics Teachers. DOI: / I. INTRODUCTION As is well known, the avantage of the Coulomb gauge is that the scalar potential in this gauge is simple to obtain, but the isavantage is that the vector potential in this gauge is ifficult to calculate. This characteristic of the Coulomb gauge is why the explicit emonstration that the potentials in this gauge yiel the retare electric an magnetic fiels is not usually presente in textbooks. Some textbooks 1,2 mention a paper by Brill an Gooman 3 in which an elaborate proof that the potentials in the Coulomb an Lorenz gauges yiel the same retare electric an magnetic fiels is presente; this proof is restricte to sources with harmonic time epenence. It is conceptually important to emphasize the fact that the causal behavior of the retare electric an magnetic fiels is not lost when they are expresse in terms of potentials in the Coulomb gauge, espite the result that the scalar potential in this gauge propagates instantaneously, which suggests a lost of causality in the electric fiel. Proofs that the potentials in other gauges such as the temporal or velocity gauges yiel the retare electric an magnetic fiels are omitte in textbooks. This omission is reasonable because these gauges are not usually mentione in textbooks of electroynamics. The velocity gauge is one in which the scalar potential propagates with an arbitrary velocity an the temporal gauge is one in which the scalar potential is ientically zero. In a recent paper Jackson an Okun 4 reviewe the history that le to the conclusion that potentials in ifferent gauges escribe the same physical fiels. In a subsequent paper, Jackson 5 erive novel expressions for the vector potential in the Coulomb, velocity, an temporal gauges an emonstrate how these expressions for the vector potential together with the associate expressions for the scalar potential lea to the same retare electric an magnetic fiels. Jackson emphasize that whatever propagation or nonpropagation characteristics are exhibite by the potentials in a particular gauge, the electric an magnetic fiels are always the same an isplay the experimentally verifie properties of causality an propagation at the spee of light. 5 Rohrlich recently iscusse causality in the Coulomb gauge. 6 The present author has use two methos 7,8 to show that the Coulomb gauge potentials yiel the retare electric fiel an reiscovere 9 the Kirchhoff gauge 10 in which the scalar potential propagates with the imaginary spee ic, where c is the spee of light. In a recent paper, Yang 11 iscusse the velocity gauge. 12 To show that potentials in ifferent gauges yiel the same retare fiels, we usually first erive explicit expressions for the scalar an vector potentials in a particular gauge. The retare fiels are then obtaine by ifferentiation of the expressions for the potentials. The practical ifficulty of this approach is that the erivation of explicit expressions for potentials in most gauges is not simple, particularly for the vector potential. The question arises: Is it necessary to have explicit expressions for the potentials in a particular gauge to show that they lea to the retare electric an magnetic fiels? The answer is negative, at least for the gauges consiere in this paper. In this paper we present a simple an systematic metho to show how the potentials in the Lorentz, Coulomb, Kirchhoff, velocity, an temporal gauges yiel the same retare electric an magnetic fiels. Instea of using explicit expressions for the scalar an vector potentials in these gauges, we use the appropriate ynamical equations of the potentials to obtain two wave equations, whose retare solutions lea to the retare fiels. In Sec. II we efine the Lorentz, Coulomb, Kirchhoff, velocity, an temporal gauges. In Sec. III we review the usual proof that the Lorentz gauge potentials lea to the retare fiels an use the alternative metho to show how these potentials yiel the retare fiels. In Sec. IV we apply the same metho to the Coulomb gauge potentials. In Sec. V we iscuss the steps of the propose metho. In Sec. VI we apply the metho to the Kirchhoff gauge potentials, an in Secs. VII an VIII we apply the metho to the velocity gauge potentials an the temporal gauge vector potential, respectively. In Sec. IX we emphasize the spurious character of the graient of the scalar potential in the Coulomb, Kirchhoff, an velocity gauges an the nonspurious character of the graient of the scalar potential in the Lorenz gauge. We suggest that the Lorenz gauge potentials may be interprete as physical quantities. In Sec. X we present some concluing remarks. 176 Am. J. Phys. 75 2, February American Association of Physics Teachers 176

2 II. ELECTROMAGNETIC GAUGES It is well known that the electric an magnetic fiels E an B are etermine from the scalar an vector potentials an A accoring to E = A, 1a B = A. 1b We use SI units an consier fiels with localize sources in vacuum. The fiels E an B are invariant uner the gauge transformations =, 2a A = A +, 2b where is an arbitrary time-epenent gauge function. The inhomogeneous Maxwell equations together with Eq. 1 lea to the couple equations 2 = 1 A, 0 3a 2 A = 0 J + A + 1 c 2, 3b where 2 2 1/c 2 2 / 2 is the D Alambertian operator an an J are the charge an current ensities, respectively. The arbitrariness of the gauge function in Eq. 2 allows a gauge conition to be chosen. We will consier five gauge conitions, A + 1 c 2 = 0, Lorenz gauge, 1,2 4 A = 0, Coulomb gauge, 3,5 5 A 1 c 2 A + 1 v 2 = 0, Kirchhoff gauge, 9 6 = 0, Velocity gauge, 5,11, L = 1 0, 2 A L = 0 J. 9a 9b The notation L an A L inicates that these potentials are in the Lorenz gauge. An avantage of the Lorenz gauge is that the retare solutions of Eq. 9 are well known, L x,t 3 x 1 x,t R/c, R 10a A L x,t 4 3 x 1 J x,t R/c, R 10b where R is the magnitue of the vector R=x x with x the fiel point an x the source point. The integrals in Eq. 10 are over all space. An avantage of the Lorenz gauge is that it can be written in the relativistically covariant form A =0, where 1/c /, an A /c,a. Greek inices run from 0 to 3; the signature of the Minkowski metric is 1, 1, 1, 1 an summation on repeate inices is unerstoo. Equations 9 an 10 can also be written in a relativistically covariant form. Another characteristic of the Lorenz gauge is that the scalar potential yiels a retare term that can be written as L x,t 3 x Rˆ R2 x,t R/c Rˆ x,t R/c +, 11 Rc R/c where Rˆ =R/R. This form isplays the properties of causality an propagation at the spee of light. We anticipate that the graient of the scalar potential in the other gauges consiere in this paper oes not satisfy the above properties. Given the potentials L an A L, the electric an magnetic fiels can be erive by the usual prescription = 1 E = L A L x 1 R, 3 x 1 R 12a = 0, Temporal gauge. 5 8 We note that the velocity gauge contains the Lorenz gauge v=c, the Coulomb gauge v=, an the Kirchhoff gauge v=ic. III. LORENZ GAUGE The most popular gauge is the Lorenz gauge, which allows us to uncouple Eq. 3 so that the scalar an vector potentials are escribe by symmetrical uncouple equations, which is a characteristic of this gauge. If we assume the Lorenz gauge 4, then Eq. 3 becomes B = A L = x 1 R, 12b where we have introuce the retaration symbol o inicate that the enclose quantity is to be evaluate at the retare time t =t R/c. After an integration by parts, Eq. 12 becomes the usual electric an magnetic fiels E, B 4 3 x 1 R 13b 177 Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras a

3 Equation 13 can also be obtaine without consiering Eq. 10 by first taking the negative of the graient of Eq. 9a an the negative of the time erivative of Eq. 9b, C x,t 3 x 1 R x,t, 20a 2 L 0, 2 A L. The retare solutions of Eq. 14 are given by L 3 x 1, R 15a A L. Alternatively, the curl of Eq. 9b gives 2 A L = 0 J, A L 17 14a 14b 15b 16 Equations 15 an 17 yiel the usual form of the retare electric an magnetic fiels E = L A L B = A L, 18b 18a This alternative metho of obtaining the electric an magnetic fiels, which works irectly with Eq. 9, oes not have any practical avantage for the Lorenz gauge compare to the traitional metho that uses Eq. 10. We will see in the next sections that the alternative metho is avantageous when applie to potentials in other gauges. IV. COULOMB GAUGE A less popular gauge in textbooks is the Coulomb gauge. In this gauge the scalar potential satisfies an instantaneous Poisson equation, which is a peculiar characteristic of this gauge. If we assume the Coulomb gauge in Eq. 5, then Eqs. 3 become the couple equations: 2 C = 1 0, 19a 2 A C = 0 J + 1 C c 2, 19b where we have use the notation C an A C to specify that these potentials are in the Coulomb gauge. As pointe out in Sec. I, the avantage of the Coulomb gauge is that the solution of Eq. 19a is particularly simple to obtain, but its isavantage is that the solution of Eq. 19b is ifficult to calculate. Let us write the solutions of Eq. 19 in the explicit form A C x,t 4 3 x R x,t 1 R/c crˆ x,t R/c + c2 Rˆ R 0 R/c x,t. 20b Equation 20a is a well known instantaneous expression, an Eq. 20b is a novel expression erive recently by Jackson. 5 By making use of Eq. 20, Jackson 5 obtaine the retare electric fiel in the form given by Jefimenko 13 an the usual retare form of the magnetic fiel in Eq. 13b.A isavantage of the Coulomb gauge conition is that it cannot be written in a relativistically covariant form. The scalar potential yiels the instantaneous term C x,t 3 x Rˆ R 2 x,t. 21 This term oes not isplay the properties of causality an propagation at the spee of light, an therefore its explicit presence in the expression for the retare electric fiel E = C A C / seems to inicate an inconsistency. We recall that an instantaneous fiel is in conflict with special relativity, which states that no physical information can propagate faster than c in vacuum. To unerstan the role playe by the acausal term C in the electric fiel E= C A C /, we apply the metho iscusse in Sec. III an show that the potentials C an A C lea to the fiels E an B. In the first step we symmetrize Eq. 19a with respect to Eq. 19b by aing the term 1/c 2 2 C / 2 on both sies of Eq. 19a to obtain 2 C = C 0 c In the secon step we take the negative of the graient of Eq. 22 an the negative of the time erivative of Eq. 19b to obtain two equations involving thir-orer erivatives of the potentials, 2 C c 2 2 C 2, 23a 2 A C 1 c 2 2 C 2. 23b In the thir step we a Eqs. 23a an 23b to obtain the wave equation 2 C A C + 0 0, 24 + C. A C 3 x R 1 1 c 2 25 Equation 25 shows that the term A C / always contains the instantaneous component C, which cancels exactly 178 Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras 178

4 the instantaneous part C of the electric fiel E= C A C /. This well known result 3 has recently been emphasize. 5,14 Equation 25 has also recently been obtaine by applying another more complicate metho. 15 The explicit presence of the acausal term C in the electric fiel is irrelevant because such a term is always cancele by one of the components C of the remaining term efine by A C /. The fiel C is physically unetectable an can be interprete as a spurious fiel. Hence, when Eq. 25 is use in E= C A C /, we obtain the usual retare form of the electric fiel E = C A C. 26 In the fourth step we take the curl of Eq. 19b to obtain the wave equation 2 A C = 0 J, 27 A C 28 Equation 28 is the usual retare form of the magnetic fiel B = A C 29 Note that neither the simple solution 20a of Eq. 19a nor the complicate solution 20b of Eq. 19b was require to show that the Coulomb gauge potentials yiel the electric an magnetic fiels. V. THE FOUR STEPS OF THE METHOD As note in Sec. IV, the metho can be efine by four steps: 1 Apply one of the Lorentz, Coulomb, Kirchhoff, velocity, an temporal gauges to Eq. 3 an symmetrize, if necessary, the gauge equation for the charge ensity with respect to the gauge equation for the current ensity. 2 Calculate the negative of the graient of the gauge an possibly symmetrize equation for the charge ensity an the negative of the time erivative of the gauge equation for the current ensity. As a result, two equations containing thir-orer erivatives of potentials are obtaine: one involving the graient of the charge ensity an the other involving the time erivative of the current ensity. 3 Except for the Lorenz gauge, a the thir-orer equations obtaine in step 2 to obtain a wave equation whose retare solution gives an equation for the time erivative of the vector potential, which is substitute into the expression for the electric fiel in terms of the potentials to obtain the retare electric fiel. 4 Take the curl of the thir-orer equation for the current ensity obtaine in step 2 to obtain a wave equation whose retare solution gives an equation for the curl of the vector potential, which is substitute into the expression for the magnetic fiel in terms of the vector potential to obtain the retare magnetic fiel. For the special case of the Lorentz gauge, it is necessary first to solve Eq. 14 erive in step 2. The retare solutions of Eq. 15 are use in step 3 to obtain the usual retare form of the electric fiel. 16 VI. KIRCHHOFF GAUGE Accoring to Ref. 1, the first publishe relation between the potentials is ue to Kirchhoff 10 who showe that the Weber form of the vector potential A an its associate scalar potential satisfy the equation in moern notation, A 1/ c 2 /=0, that is, Eq. 6, which was originally obtaine for quasistatic potentials in which retaration is neglecte. Of course, the electromagnetic gauge invariance ha not been establishe at that time. The present author has propose calling Eq. 6 the Kirchhoff gauge. 9 In the Kirchhoff gauge 6, Eq. 3 becomes 2 K + 1 c 2 2 K 2 = 1 0, 30a 2 A K = 0 J + 2 c 2 K. 30b We note that Eq. 30a is an elliptical equation, which oes not escribe a real propagation. After the substitution c 2 = ic 2, Eq. 30a may be written as 2 K 1 ic 2 2 K 2 = Equation 31 formally states that K propagates with the imaginary spee ic, which emphasizes the unphysical character of K. The solutions of Eq. 30 can be expresse as 9 K x,t 3 x 1 x,t R/ ic, R 32a A K x,t 4 3 x R x,t 1 R/c crˆ x,t R/c + c i Rˆ x,t R/ ic + c2 Rˆ R/c R R/ ic x,t. 32b It was note in Ref. 9 that the potential K in Eq. 32a exhibits the same form as the scalar potential in the corresponing Lorenz gauge of an electromagnetic theory formulate in Eucliean four-space. 17,18 This interesting result shows how the same potential can be efine in ifferent gauges an in ifferent spacetimes. It is clear that the Kirchhoff gauge cannot be written in a relativistically covariant form. Also, in Ref. 9 it was shown that Eq. 32 yiels the retare electric an magnetic fiels. However, as shown in Ref. 9, the erivation of Eq. 32 is not so simple. We note that the potential K yiels the imaginary term Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras 179

5 K x,t 3 x Rˆ R2 x,t R/ ic Rˆ x,t R/ ic Ric R/ ic The reaer might fin surprising the presence of the instantaneous term C in the retare expression for the electric fiel in terms of the Coulomb gauge potentials E= C A C /. The reaer probably woul fin even more surprising the presence of the imaginary term K in the observable electric fiel in terms of the Kirchhoff gauge potentials E= K A K /. We suspect that, like the instantaneous term C, the imaginary term K oes not play a physical role in the electric fiel. To unerstan the role of K in Eq. 33, we will apply the four step metho given in Sec. V. We will show that the Kirchhoff potentials K an A K lea to the retare fiels E an B. Step 1. After applying the Kirchhoff conition 6 to Eq. 3, we have alreay obtaine Eq. 30. We then symmetrize Eq. 30a with respect to Eq. 30b by aing the term 2/c 2 2 K / 2 on both sies of Eq. 30a to obtain 2 K = K 0 c Step 2. We take the negative of the graient of Eq. 34 an the negative of the time erivative of Eq. 30b to obtain the equations 2 K c 2 2 K 2, 35a 2 A K 2 c 2 2 K 2. 35b Step 3. We a Eq. 35 to obtain the wave equation 2 K A K + 0 0, 36 A K + K. 37 As can be seen, the term A K / in Eq. 37 contains the imaginary component K, which exactly cancels the imaginary part K of the electric fiel E= K A K /. In other wors, the explicit presence of the imaginary term K in the electric fiel is irrelevant because such a term is cancele by one of the components K of the Kirchhoff-gauge vector potential A K /. This result has recently been emphasize. 14 When Eq. 37 is use in the fiel E= K A K /, we obtain the usual retare form of this fiel, E = K A K. 38 Step 4. We take the curl to Eq. 30b to obtain the wave equation 2 A K = 0 J, A K Equation 40 is ientifie with the usual retare form of the magnetic fiel B = A K 41 Therefore, we o not require the complicate Eq. 32 to verify that the Kirchhoff gauge potentials yiel the retare electric an magnetic fiels. VII. VELOCITY GAUGE The velocity gauge v-gauge is one in which the scalar potential propagates with an arbitrary spee. This gauge is not well known espite the fact that it was propose several years ago. 12 The v-gauge is really a family of gauges that contains the Lorenz an Coulomb gauges as particular cases an also inclues the Kirchhoff gauge. 9 The v-gauge has recently been emphasize by Drury, 19 Jackson, 5 an Yang. 11 If we assume the v-gauge efine by Eq. 7, then Eq. 3 become 2 v 1 2 v 2 2 = 1, 42a A v = 0 J + 1 v c 2 1 c2. 42b v The generality of Eq. 42 becomes evient when we observe that it reuces to Eq. 9 when v=c, toeq. 19 when v=, an to Eq. 30 when v=ic. The solutions of Eq. 42 are given by 5 v x,t 3 x 1 x,t R/v, R 43a A v x,t 4 3 x R x,t 1 R/c crˆ x,t R/c + c2 Rˆ x,t R/v v + c2 R/c Rˆ x,t. 43b R R/v As expecte, Eq. 43 reuces to Eq. 10 when v=c, toeq. 20 when v=, an to Eq. 32 when v=ic. Accoring to Eq. 42a, the potential v propagates with an arbitrary spee v, which may be subliminal v c or luminal v=c or superluminal v c incluing the instantaneous limit v =. In Ref. 5 it was shown that Eq. 43 yiels the retare electric an magnetic fiels. Jackson has emphasize: 5 The v-gauge illustrates ramatically how arbitrary an unphysical the potentials can be, yet still yiel the same physically sensible fiels. The velocity gauge cannot be written in a relativistically covariant form. The v-gauge scalar potential v generates the fiel Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras 180

6 v x,t 3 x Rˆ R2 x,t R/v Rˆ x,t R/v Rv R/v This term oes not isplay the property of propagation at the spee of light c. In particular, when v is superluminal, we have a conflict with special relativity. We suspect that the term v oes not play a physical role in the electric fiel. To unerstan the role playe by the term v in the fiel E= v A v /, we apply the four step metho to show that the potentials v an A v yiel the fiels E an B. Step 1. If we apply the velocity conition 7 to Eq. 3, we obtain Eq. 42. We symmetrize Eq. 42a with respect to Eq. 42b by aing the term 1/c 2 2 v / 2 on both sies of Eq. 42a. The resulting equation can be written as 2 v = 1 1 c2 0 c v v Step 2. We take minus the graient of Eq. 45 an minus the time erivative of Eq. 42b. The resulting equations are 2 v c 2 1 c2 2 A v v 2 2 v 2, 46a v c 2 1 c2 v 2. 46b Step 3. We a Eq. 46 to obtain the wave equation 2 v A v 0 + 0, A v + v We observe that the term A v / in Eq. 48 contains the component v, which cancels the term v. In other wors, the explicit presence of a term possessing an arbitrary propagation v in the electric fiel is irrelevant because such a term is cancele by one of the components v of the v-gauge vector potential A v /. This cancellation means that the fiel v is spurious fiel an the propagation at the spee of light c of the electric fiel is not lost. Therefore, when Eq. 48 is use in the expression E= v A v /, we obtain the usual retare form of the electric fiel E = v A v. 49 Step 4. We take the curl of Eq. 42b to obtain the wave equation 2 A v = 0 J, 50 A v 3 x 1 R 51 Equation 51 gives the usual retare form of the magnetic fiel B = A v 3 x 1 R 52 We see that we o not require the complicate Eq. 43 to show that the potentials in the velocity gauge yiel the retare electric an magnetic fiels. VIII. TEMPORAL GAUGE We have pointe out that the scalar potential can be instantaneous, imaginary, an superluminal epening on the gauge Coulomb, Kirchhoff, an velocity gauges, respectively. Now we will see that the scalar potential can also not exist. The temporal gauge is one in which the scalar potential is ientically zero, 5 which means that the electric an magnetic fiels are efine only by the vector potential: E = A T, 53a B = A T. 53b The temporal gauge cannot be written in a relativistically covariant form. The reaer might woner why the scalar potential in the temporal gauge oes not exist, espite the fact that there is a nonzero charge ensity. The simple answer is that the values of the charge ensity o not necessarily lea to a scalar potential in all gauges. The existence of a scalar potential generally epens on the aopte gauge. In other wors, the retare values of the charge ensity always contribute physically to the electric fiel, but they o not lea to a scalar potential in the temporal gauge. If we assume the temporal gauge efine by Eq. 8, then Eq. 3 becomes A T = 1 0, 2 A T = 0 J + A T. The solution of Eq. 54 is given by 5 A T x,t x,t R/c crˆ x,t R/c c2 Rˆ R R/c t t 0 x,t. 54a 54b 55 In Ref. 5 it was emonstrate that the potential A T in Eq. 55 yiels the retare electric an magnetic fiels. To unerstan this result, we apply our metho to show that A T yiels the retare fiels E an B. Step 1. After applying the temporal conition 8 to Eq. 3, we obtain Eq. 54. It is not necessary to symmetrize Eq. 54a with respect to Eq. 54b. Step 2. We take minus the graient of Eq. 54a an minus the time erivative of Eq. 54b to obtain 181 Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras 181

7 A T 0, 2 A T A T. 56a 56b Step 3. We a Eq. 56 to obtain the wave equation 2 A T + 0 0, 57 A T. 58 In the temporal gauge there is no unphysical term on the right-han sie of Eq. 58. When Eq. 58 is use in the expression E= A T /, we obtain the usual retare form of the electric fiel E = A T. 59 Step 4. We take the curl of Eq. 54b to obtain the wave equation 2 A T = 0 J, 60 A T J, 61 which leas irectly to the usual retare form of the magnetic fiel B = A T 62 The complicate form of Eq. 55 is not require to show that the vector potential in the temporal gauge yiels the retare electric an magnetic fiels. IX. THE NONSPURIOUS CHARACTER OF L The fact that the Coulomb gauge scalar potential C propagates instantaneously is of no concern if we assume that the electromagnetic potentials are not physically measurable quantities. As pointe out by Griffiths: 2 The point is that V he Coulomb-gauge scalar potential by itself is not a physically measurable quantity. It follows that the instantaneous term C Eq. 21 must also be an unphysical quantity. The subtle point is that C is part of the physical electric fiel expresse in terms of the Coulomb gauge potentials: E= C A C /. We have pointe out that the presence of C in the electric fiel is irrelevant because it is cancele by a component of the remaining term A C /. We have note that C is a formal result of the theory an has no physical meaning. We have rawn similar conclusions for the term with imaginary propagation K Eq. 33 an for the term with arbitrary propagation v Eq. 44. We cannot come to the same conclusion for the term L Eq. 11 ue to the Lorenz gauge potential L, because this term isplays the experimentally verifie properties of causality an propagation at the spee of light the term L is not cancele by part of the remaining term A L / of the electric fiel. Therefore, we can conclue that L is not a spurious term like the terms C, K, an v. It follows that L can be interprete as a physical quantity. Similarly, the term A L / also satisfies the properties of causality an propagation at the spee of light, which inicates that this term shoul not be interprete as being spurious. Thus, A L / can also be interprete as a physical quantity. The physical character of each one of the terms L an A L / is strongly supporte by the fact that the combination L A L /, that is, the electric fiel, is physically etectable. 20 The Lorenz gauge potentials L an A L naturally yiel the electric an magnetic fiels with the physical properties of causality an propagation at the spee of light, which suggests that the Lorenz gauge potentials an not the Coulomb, Kirchhoff, an velocity potentials can be interprete as physical quantities. X. CONCLUDING REMARKS Jackson has pointe out that: 5 It seems necessary from time to time to show that the electric an magnetic fiels are inepenent of the choice of gauge for the potentials. We have emonstrate that the fiels are inepenent of the choice of gauge for potentials in a variety of gauges Lorentz, Coulomb, Kirchhoff, velocity, an temporal gauges. Our propose metho can be use to emonstrate that the potentials in these gauges yiel the same retare electric an magnetic fiels. Instea of using explicit expressions for the scalar an vector potentials in the various gauges, the metho uses the ynamical equations of these potentials to obtain two wave equations, whose retare solutions lea to the retare fiels. We ientifie the spurious character of the graient of the scalar potential in the Coulomb, Kirchhoff, an velocity gauges an have emphasize the nonspurious character of the scalar potential in the Lorenz gauge. Finally, we suggeste that the Lorenz gauge potentials can be interprete as physical quantities. ACKNOWLEDGMENTS The author is grateful to an anonymous referee for his/her valuable comments an to Professor R. F. O Connell for the kin hospitality extene to him in the Department of Physics an Astronomy of the Louisiana State University. a Electronic mail: heras@phys.lsu.eu 1 J. D. Jackson, Classical Electroynamics, 3r e. Wiley, New York, 1999, p D. J. Griffiths, Introuction to Electroynamics, 3r e. Prentice Hall, Englewoo Cliffs, NJ, 1999, p O. L. Brill an B. Gooman, Causality in the Coulomb gauge, Am. J. Phys. 35, J. D. Jackson an L. B. Okun, Historical roots of gauge invariance, Rev. Mo. Phys. 73, J. D. Jackson, From Lorenz to Coulomb an other explicit gauge transformations, Am. J. Phys. 70, F. Rohrlich, Causality, the Coulomb fiel, an Newton s law of gravitation, Am. J. Phys. 70, J. A. Heras, Comment on Causality, the Coulomb fiel, an Newton s law of gravitation, by F. Rohrlich Am. J. Phys. 70, , Am. J. Phys. 71, J. A. Heras, Instantaneous fiels in classical electroynamics, Europhys. Lett. 69, Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras 182

8 9 J. A. Heras, The Kirchhoff gauge, Ann. Phys. 321, G. Kirchhoff, II. Ueber ie Bewegung er Elektricität in Leitern, Am. J. Phys. 102, Reprinte in Gesammelte Abhanlungen von G. Kirchhoff. A. Barth, Leipzig, 1882, pp K.-H. Yang, The physics of gauge transformations, Am. J. Phys. 73, K.-H. Yang, Gauge transformations an quantum mechanics. II. Physical interpretation of classical gauge transformations, Ann. Phys. N.Y. 101, O. D. Jefimenko, Electricity an Magnetism, 2n e. Electret Scientific, Star City, WV, 1989, p See also Ref. 1, p J. A. Heras, Comment on A generalize Helmholtz theorem for timevarying vector fiels, by Artice M. Davis Am. J. Phys. 74, , Am. J. Phys. 74, J. A. Heras, Comment on Helmholtz theorem an the v-gauge in the problem of superluminal an instantaneous signals in classical electroynamics, by A. Chubykalo et al. Foun. Phys. Lett. 19, , Foun. Phys. Lett. 19, In the propose metho we coul also choose the avance solutions hose evaluate at the avance time t =t+r/c of the wave equations erive in steps 3 an 4. This choice woul lea to the avance form of the electric an magnetic fiels. 17 E. Zampino, A brief stuy of the transformations of Maxwell s equations in Eucliean four space, J. Math. Phys. 27, J. A. Heras, Eucliean electromagnetism in four space: A iscussion between Go an the Devil, Am. J. Phys. 62, D. M. Drury, The unification of the Lorentz an Coulomb gauges of electromagnetic theory, IEEE Trans. Euc. 43 1, Even though the combination L A L / can be etecte, it can be argue that this possibility oes not necessarily imply that each term can be etecte separately. 183 Am. J. Phys., Vol. 75, No. 2, February 2007 José A. Heras 183