How the potentials in different gauges yield the same retarded electric and magnetic fields
|
|
- Marilynn Rodgers
- 5 years ago
- Views:
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
Lagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationarxiv: v1 [physics.class-ph] 28 Dec 2008
Can Maxwell s equations be obtained from the continuity equation? José A. Heras arxiv:0812.4785v1 [physics.class-ph] 28 Dec 2008 Departamento de Física y Matemáticas, Universidad Iberoamericana, Prolongación
More informationConservation Laws. Chapter Conservation of Energy
20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationLecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Baird, faculty.uml.edu/cbaird University of Massachusetts Lowell
Lecture 10 Notes, Electromagnetic Theory II Dr. Christopher S. Bair, faculty.uml.eu/cbair University of Massachusetts Lowell 1. Pre-Einstein Relativity - Einstein i not invent the concept of relativity,
More informationLagrangian and Hamiltonian Dynamics
Lagrangian an Hamiltonian Dynamics Volker Perlick (Lancaster University) Lecture 1 The Passage from Newtonian to Lagrangian Dynamics (Cockcroft Institute, 22 February 2010) Subjects covere Lecture 2: Discussion
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationConservation laws a simple application to the telegraph equation
J Comput Electron 2008 7: 47 51 DOI 10.1007/s10825-008-0250-2 Conservation laws a simple application to the telegraph equation Uwe Norbrock Reinhol Kienzler Publishe online: 1 May 2008 Springer Scienceusiness
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationLie symmetry and Mei conservation law of continuum system
Chin. Phys. B Vol. 20, No. 2 20 020 Lie symmetry an Mei conservation law of continuum system Shi Shen-Yang an Fu Jing-Li Department of Physics, Zhejiang Sci-Tech University, Hangzhou 3008, China Receive
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationQubit channels that achieve capacity with two states
Qubit channels that achieve capacity with two states Dominic W. Berry Department of Physics, The University of Queenslan, Brisbane, Queenslan 4072, Australia Receive 22 December 2004; publishe 22 March
More informationLecture 2 Lagrangian formulation of classical mechanics Mechanics
Lecture Lagrangian formulation of classical mechanics 70.00 Mechanics Principle of stationary action MATH-GA To specify a motion uniquely in classical mechanics, it suffices to give, at some time t 0,
More informationA Second Time Dimension, Hidden in Plain Sight
A Secon Time Dimension, Hien in Plain Sight Brett A Collins. In this paper I postulate the existence of a secon time imension, making five imensions, three space imensions an two time imensions. I will
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationConstruction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems
Construction of the Electronic Raial Wave Functions an Probability Distributions of Hyrogen-like Systems Thomas S. Kuntzleman, Department of Chemistry Spring Arbor University, Spring Arbor MI 498 tkuntzle@arbor.eu
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationA look at Einstein s clocks synchronization
A look at Einstein s clocks synchronization ilton Penha Departamento e Física, Universiae Feeral e Minas Gerais, Brasil. nilton.penha@gmail.com Bernhar Rothenstein Politehnica University of Timisoara,
More informationA simple model for the small-strain behaviour of soils
A simple moel for the small-strain behaviour of soils José Jorge Naer Department of Structural an Geotechnical ngineering, Polytechnic School, University of São Paulo 05508-900, São Paulo, Brazil, e-mail:
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationAnalytic Scaling Formulas for Crossed Laser Acceleration in Vacuum
October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945
More informationarxiv:physics/ v4 [physics.class-ph] 9 Jul 1999
AIAA-99-2144 PROPULSION THROUGH ELECTROMAGNETIC SELF-SUSTAINED ACCELERATION arxiv:physics/9906059v4 [physics.class-ph] 9 Jul 1999 Abstract As is known the repulsion of the volume elements of an uniformly
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationAssignment 1. g i (x 1,..., x n ) dx i = 0. i=1
Assignment 1 Golstein 1.4 The equations of motion for the rolling isk are special cases of general linear ifferential equations of constraint of the form g i (x 1,..., x n x i = 0. i=1 A constraint conition
More informationarxiv: v1 [physics.class-ph] 20 Dec 2017
arxiv:1712.07328v1 [physics.class-ph] 20 Dec 2017 Demystifying the constancy of the Ermakov-Lewis invariant for a time epenent oscillator T. Pamanabhan IUCAA, Post Bag 4, Ganeshkhin, Pune - 411 007, Inia.
More informationMATHEMATICS BONUS FILES for faculty and students
MATHMATI BONU FIL for faculty an stuents http://www.onu.eu/~mcaragiu1/bonus_files.html RIVD: May 15, 9 PUBLIHD: May 5, 9 toffel 1 Maxwell s quations through the Major Vector Theorems Joshua toffel Department
More informationOn the dynamics of the electron: Introduction, 1, 9
On the ynamics of the electron: Introuction,, 9 Henri oincaré Introuction. It seems at first that the aberration of light an relate optical an electrical phenomena will provie us with a means of etermining
More informationCalculus in the AP Physics C Course The Derivative
Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationIntroduction to Markov Processes
Introuction to Markov Processes Connexions moule m44014 Zzis law Gustav) Meglicki, Jr Office of the VP for Information Technology Iniana University RCS: Section-2.tex,v 1.24 2012/12/21 18:03:08 gustav
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk - Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking three-imensional potentials in the next chapter, we shall in chapter 4 of this
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationThe Kepler Problem. 1 Features of the Ellipse: Geometry and Analysis
The Kepler Problem For the Newtonian 1/r force law, a miracle occurs all of the solutions are perioic instea of just quasiperioic. To put it another way, the two-imensional tori are further ecompose into
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More informationTEST 2 (PHY 250) Figure Figure P26.21
TEST 2 (PHY 250) 1. a) Write the efinition (in a full sentence) of electric potential. b) What is a capacitor? c) Relate the electric torque, exerte on a molecule in a uniform electric fiel, with the ipole
More informationIMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More information5-4 Electrostatic Boundary Value Problems
11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 5-4 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149-157 Q: A: We must solve ifferential equations, an apply bounary conitions
More informationOptimization of Geometries by Energy Minimization
Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More information1 Heisenberg Representation
1 Heisenberg Representation What we have been ealing with so far is calle the Schröinger representation. In this representation, operators are constants an all the time epenence is carrie by the states.
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationSpectral Flow, the Magnus Force, and the. Josephson-Anderson Relation
Spectral Flow, the Magnus Force, an the arxiv:con-mat/9602094v1 16 Feb 1996 Josephson-Anerson Relation P. Ao Department of Theoretical Physics Umeå University, S-901 87, Umeå, SWEDEN October 18, 2018 Abstract
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More information18 EVEN MORE CALCULUS
8 EVEN MORE CALCULUS Chapter 8 Even More Calculus Objectives After stuing this chapter you shoul be able to ifferentiate an integrate basic trigonometric functions; unerstan how to calculate rates of change;
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More information11.7. Implicit Differentiation. Introduction. Prerequisites. Learning Outcomes
Implicit Differentiation 11.7 Introuction This Section introuces implicit ifferentiation which is use to ifferentiate functions expresse in implicit form (where the variables are foun together). Examples
More informationBEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS. Mauro Boccadoro Magnus Egerstedt Paolo Valigi Yorai Wardi
BEYOND THE CONSTRUCTION OF OPTIMAL SWITCHING SURFACES FOR AUTONOMOUS HYBRID SYSTEMS Mauro Boccaoro Magnus Egerstet Paolo Valigi Yorai Wari {boccaoro,valigi}@iei.unipg.it Dipartimento i Ingegneria Elettronica
More informationSemiclassical analysis of long-wavelength multiphoton processes: The Rydberg atom
PHYSICAL REVIEW A 69, 063409 (2004) Semiclassical analysis of long-wavelength multiphoton processes: The Ryberg atom Luz V. Vela-Arevalo* an Ronal F. Fox Center for Nonlinear Sciences an School of Physics,
More informationDifferentiation ( , 9.5)
Chapter 2 Differentiation (8.1 8.3, 9.5) 2.1 Rate of Change (8.2.1 5) Recall that the equation of a straight line can be written as y = mx + c, where m is the slope or graient of the line, an c is the
More informationClassical Quantum Theory
J.P. Wesley Weiherammstrasse 4 78176 Blumberg, Germany Classical Quantum Theory From the extensive observations an the ieas of Newton an from classical physical optics the velocity of a quantum particle
More informationOn Characterizing the Delay-Performance of Wireless Scheduling Algorithms
On Characterizing the Delay-Performance of Wireless Scheuling Algorithms Xiaojun Lin Center for Wireless Systems an Applications School of Electrical an Computer Engineering, Purue University West Lafayette,
More informationA Novel Decoupled Iterative Method for Deep-Submicron MOSFET RF Circuit Simulation
A Novel ecouple Iterative Metho for eep-submicron MOSFET RF Circuit Simulation CHUAN-SHENG WANG an YIMING LI epartment of Mathematics, National Tsing Hua University, National Nano evice Laboratories, an
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationarxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003
Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P-3004-516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.e-ph] 23 Sep
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More informationGeneralized-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal
Commun. Theor. Phys. (Beijing, China) 44 (25) pp. 72 78 c International Acaemic Publishers Vol. 44, No. 1, July 15, 25 Generalize-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal
More informationAPPLICATION OF DIFFERENTIAL FORMS IN THE FINITE ELEMENT FORMULATION OF ELECTROMAGNETIC PROBLEMS
Technical article APPLICATION OF DIFFERENTIAL FORMS IN THE FINITE ELEMENT FORMULATION OF ELECTROMAGNETIC PROBLEMS INTRODUCTION In many physical problems, we have to stuy the integral of a quantity over
More informationTime-of-Arrival Estimation in Non-Line-Of-Sight Environments
2 Conference on Information Sciences an Systems, The Johns Hopkins University, March 2, 2 Time-of-Arrival Estimation in Non-Line-Of-Sight Environments Sinan Gezici, Hisashi Kobayashi an H. Vincent Poor
More informationLINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form
LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations
More informationModeling time-varying storage components in PSpice
Moeling time-varying storage components in PSpice Dalibor Biolek, Zenek Kolka, Viera Biolkova Dept. of EE, FMT, University of Defence Brno, Czech Republic Dept. of Microelectronics/Raioelectronics, FEEC,
More informationAverage value of position for the anharmonic oscillator: Classical versus quantum results
verage value of position for the anharmonic oscillator: Classical versus quantum results R. W. Robinett Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 682 Receive
More informationA Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential
Avances in Applie Mathematics an Mechanics Av. Appl. Math. Mech. Vol. 1 No. 4 pp. 573-580 DOI: 10.4208/aamm.09-m0946 August 2009 A Note on Exact Solutions to Linear Differential Equations by the Matrix
More informationExact solution of the Landau Lifshitz equations for a radiating charged particle in the Coulomb potential
Available online at www.scienceirect.com Annals of Physics 323 (2008) 2654 2661 www.elsevier.com/locate/aop Exact solution of the Lanau Lifshitz equations for a raiating charge particle in the Coulomb
More informationEnergy behaviour of the Boris method for charged-particle dynamics
Version of 25 April 218 Energy behaviour of the Boris metho for charge-particle ynamics Ernst Hairer 1, Christian Lubich 2 Abstract The Boris algorithm is a wiely use numerical integrator for the motion
More informationON THE MEANING OF LORENTZ COVARIANCE
Founations of Physics Letters 17 (2004) pp. 479 496. ON THE MEANING OF LORENTZ COVARIANCE László E. Szabó Theoretical Physics Research Group of the Hungarian Acaemy of Sciences Department of History an
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationUNDERSTANDING INTEGRATION
UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,
More informationMark J. Machina CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS"
Mark J. Machina CARDINAL PROPERTIES OF "LOCAL UTILITY FUNCTIONS" This paper outlines the carinal properties of "local utility functions" of the type use by Allen [1985], Chew [1983], Chew an MacCrimmon
More informationMagnetic helicity evolution in a periodic domain with imposed field
PHYSICAL REVIEW E 69, 056407 (2004) Magnetic helicity evolution in a perioic omain with impose fiel Axel Branenburg* Norita, Blegamsvej 17, DK-2100 Copenhagen Ø, Denmark William H. Matthaeus University
More informationGeneralization of the persistent random walk to dimensions greater than 1
PHYSICAL REVIEW E VOLUME 58, NUMBER 6 DECEMBER 1998 Generalization of the persistent ranom walk to imensions greater than 1 Marián Boguñá, Josep M. Porrà, an Jaume Masoliver Departament e Física Fonamental,
More informationThe total derivative. Chapter Lagrangian and Eulerian approaches
Chapter 5 The total erivative 51 Lagrangian an Eulerian approaches The representation of a flui through scalar or vector fiels means that each physical quantity uner consieration is escribe as a function
More informationThe effect of nonvertical shear on turbulence in a stably stratified medium
The effect of nonvertical shear on turbulence in a stably stratifie meium Frank G. Jacobitz an Sutanu Sarkar Citation: Physics of Fluis (1994-present) 10, 1158 (1998); oi: 10.1063/1.869640 View online:
More informationDerivative of a Constant Multiple of a Function Theorem: If f is a differentiable function and if c is a constant, then
Bob Brown Math 51 Calculus 1 Chapter 3, Section Complete 1 Review of the Limit Definition of the Derivative Write the it efinition of the erivative function: f () Derivative of a Constant Multiple of a
More informationPhysics 115C Homework 4
Physics 115C Homework 4 Problem 1 a In the Heisenberg picture, the ynamical equation is the Heisenberg equation of motion: for any operator Q H, we have Q H = 1 t i [Q H,H]+ Q H t where the partial erivative
More informationLogarithmic spurious regressions
Logarithmic spurious regressions Robert M. e Jong Michigan State University February 5, 22 Abstract Spurious regressions, i.e. regressions in which an integrate process is regresse on another integrate
More informationA Simple Model for the Calculation of Plasma Impedance in Atmospheric Radio Frequency Discharges
Plasma Science an Technology, Vol.16, No.1, Oct. 214 A Simple Moel for the Calculation of Plasma Impeance in Atmospheric Raio Frequency Discharges GE Lei ( ) an ZHANG Yuantao ( ) Shanong Provincial Key
More informationBrazilian Journal of Physics, vol. 28, no. 1, March, Electrodynamics. Rua Pamplona,145, S~ao Paulo, SP, Brazil
Brazilian Journal of Physics, vol. 28, no., March, 998 35 One An The Same oute Two Outstaning Electroynamics Antonio Accioly () an Hatsumi Mukai (2) () Instituto e Fsica Teorica, Universiae Estaual Paulista,
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationSources and Sinks of Available Potential Energy in a Moist Atmosphere. Olivier Pauluis 1. Courant Institute of Mathematical Sciences
Sources an Sinks of Available Potential Energy in a Moist Atmosphere Olivier Pauluis 1 Courant Institute of Mathematical Sciences New York University Submitte to the Journal of the Atmospheric Sciences
More informationLATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION
The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische
More informationFree rotation of a rigid body 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012
Free rotation of a rigi boy 1 D. E. Soper 2 University of Oregon Physics 611, Theoretical Mechanics 5 November 2012 1 Introuction In this section, we escribe the motion of a rigi boy that is free to rotate
More informationEntanglement is not very useful for estimating multiple phases
PHYSICAL REVIEW A 70, 032310 (2004) Entanglement is not very useful for estimating multiple phases Manuel A. Ballester* Department of Mathematics, University of Utrecht, Box 80010, 3508 TA Utrecht, The
More informationarxiv: v1 [physics.flu-dyn] 8 May 2014
Energetics of a flui uner the Boussinesq approximation arxiv:1405.1921v1 [physics.flu-yn] 8 May 2014 Kiyoshi Maruyama Department of Earth an Ocean Sciences, National Defense Acaemy, Yokosuka, Kanagawa
More information