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- Mathematical physics
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- M. L. Boas-Mathematical Methods in the Physical Sciences - John Wiley
We show that the same route that leads to Maxwell's electrodynamics leads also to Podolsky's electrodynamics, provided we start from Podolsky's electrostatic force law instead of the usual Coulomb's law. On the second edition of Jackson's seminal book on classical electrodynamics  , there is a section named " On the Question of Obtaining the Magnetic Field, Magnetic Force, and the Maxwell Equations from Coulomb's Law and Special Relativity ", where it is shown in detail that any attempt to derive Maxwell equations from Coulomb's law of electrostatics and the laws of special relativity ends in failure unless one makes use of additional assumptions. What hypotheses are these? In an ingenious paper, Kobe  gave the answer: all one needs to arrive at Maxwell equations is.
We show that the same route that leads to Maxwell's electrodynamics leads also to Podolsky's electrodynamics, provided we start from Podolsky's electrostatic force law instead of the usual Coulomb's law. On the second edition of Jackson's seminal book on classical electrodynamics  , there is a section named " On the Question of Obtaining the Magnetic Field, Magnetic Force, and the Maxwell Equations from Coulomb's Law and Special Relativity ", where it is shown in detail that any attempt to derive Maxwell equations from Coulomb's law of electrostatics and the laws of special relativity ends in failure unless one makes use of additional assumptions.
What hypotheses are these? In an ingenious paper, Kobe  gave the answer: all one needs to arrive at Maxwell equations is. Soon after Kobe's paper, Neuenschwander and Turner  obtained Maxwell equations by generalizing the laws of magnetostatics, which follow from the Biot-Savart law and magnetostatics, to be consistent with special relativity.
The preceding considerations leads us to the interesting question: what would happen if we followed the same route as Kobe did, using an electrostatic force law other than the usual Coulomb's one? We shall show that if we start from the force law proposed by Podolsky  , i.
In other words, we shall show that the same route that leads to Maxwell equations leads also to Podolsky equations. A notable feature of Podolsky's generalized electrodynamics is that it is free of those infinities which are usually associated with a point source. For instance, 1 approaches a finite value as r approaches zero. Thus, unlike Coulomb's law, Podolsky's electrostatic force law is finite in the whole space.
In Sec. II we obtain the equations that make up Podolsky's electrostatics. III we arrive at Podolsky's field equations by generalizing the equations of Sec. II, so that they are form invariant under Lorentz transformations. For consistency, we show in Sec. IV that 1 is indeed the electrostatic force law related to Podolsky's theory.
The conclusions are presented in Sec. Natural units are used throughout. As is well-known, the force on a text charge is proportional to its charge, all other properties of the force being assigned to the electric field E r , which is defined by. The source charge's coordinates will be distinguished from those of the field point, by a prime. Note that this field is finite in the whole space.
The preceding expression for the electric field arising from a charge distribution may be easily expressed as a gradient of a scalar integral as follows. If the divergence of 3 is taken, the result is. Equations 6 and 8 are the fundamental laws of Podolsky's electrostatics. We will digress slightly at this stage to analyze an interesting feature of Podolsky's electrostatics.
In this vein, we compute the flux of the electrostatic field across a spherical surface of radius R with a charge Q at its center.
Using 2 we arrive at the result. We remark that in Maxwell's electrostatics a closed hollow conductor shields its interior from fields due to charges outside, but does not shield its interior from the field due to charges placed inside it .
Note, however, that in order not to conflict with well established results of quantum electrodynamics, the parameter a must be small. After this parenthesis, let us return to our main subject. Equations 6 and 8 are now ready to be generalized using special relativity and the hypotheses that electric charge is a conserved scalar. We shall do that in the next section. To begin with let us establish some conventions and notations to be used from now on.
We use the metric tensor. Roman indices - i , j , etc , - denote only the three spatial components. Repeated indices are summed in all cases. The four-velocities are found, according to. Let us then generalize 6 so that it satisfies the requirement of form invariance under Lorentz transformations.
The curl equation becomes. We imagine now the curl law to be the space-space components of a manifestly covariant field equation invariance under Lorentz transformations. As a result, we get. Of course, this generalization introduces the components F 00 , F l0 , and F lk , for which at this point we lack a physical interpretation. Note that the F 0 i are not necessarily static anymore.
The electric charge, in turn, is conserved locally  , which implies that it obeys a continuity equation. Using 11 , yields. In order that the left-hand side of the preceding equation transforms as the time-component of a four-vector, we must write it as.
The requirement of form invariance of this equation under Lorentz transformations leads then to the following result. Imagine now a particle of mass m and charge Q at rest in a lab frame where there is an electrostatic field E. Newton's second law allows us to write. For the component along de x i direction, we have.
In order that the right-hand side of this equation transforms like a space-component of a four-vector, it must be rewritten as. Since F mn is an antisymmetric tensor of second rank, it has only six independent components, three of which have already been specified. We name therefore the remaining components.
Writing out the components of 17 explicitly,. Hence, a clever physicist who were only familiar with Podolsky's electrostatics and special relativity could predict the existence of the magnetic field B , which naturally still lacks physical interpretation.
Accordingly, our smart physicist, who was able to predict the B field only from its knowledge of electrostatics and special relativity, can now-by making judicious use of 22 and 23 - observe, measure and distinguish the B field from the E field of The new field couples to moving electric charge, does not act on a static charged particle, and, unlike the electrostatic field, is capable only of changing the particle's momentum direction.
Equations make up Podolsky's higher-order field equations. In fact, if the divergence of 14 is taken, we obtain.
Thus, the equation in hand is identically zero;. We have only to assume that is the simplest contravariant vector constructed with the current j m and a suitable derivative of the field F mn. Applying this simplicity criterion to Podolsky's electrodynamics, we promptly obtain.
Thus, the force density for Podolsky's electrodynamics is the same as that for Maxwell's electrodynamics, namely, the well-known Lorentz force density. We show now that 1 is indeed the force law for Podolsky's electrostatics. We solve this equation using the Fourier transform method. First we define as follows:.
If we substitute 26 into 25 and take into account that. Since the orientation of our coordinate system is arbitrary, we may choose the z-axis along r and obtain. As a consequence,. Integral 28 may be found in any textbook on the theory of functions of a complex variable . It can also be carried out by means of a trivial trick .
Indeed, let. Integral 29 can be easily evaluated by the method of contour integration . Consider in this vein , where the contour of integration g was chosen to be the real axis and a semicircle of infinite radius in the upper half plane. Along the real axis the integral is I 2.
Recently an algorithm was devised which allows one to obtain the energy and momentum related to a given field in a simple way . Using this prescription we can show that in the framework of Podolsky's electrostatics the energy is given by. This is indeed a nice feature of Podolsky's generalized electrodynamics. Despite the simplicity of its fundamental assumptions, Podolsky's model has been little noticed. Currently some of its aspects have been further studied in the literature [7,8,12,13].
In particular, the classical self-force acting on a point charge in Podolsky's model was evaluated and it was shown that in this model, unlike what happens in Maxwell's electrodynamics, the electromagnetic mass is finite and enters the particle's equation of motion in a form consistent with special relativity. To conclude we call attention to the fact the same assumptions that lead to Maxwell's equations lead also to Podolsky's equations, provided we start from a generalization of the Coulomb's law instead of the usual Coulomb's law.
Yet, in spite of the great similarity between the two theories, Podolsky's generalized electrodynamics leads to results that are free of those infinities which are usually associated with a point source. Appendix: An important identity involving d functions. Applying the divergence theorem to the last integral, we obtain. On the other hand,.
As r tends to 0, x must approach r , so that. Kobe, Am. Phys 54 , See Ref. Neuenschwander and B. Turner, Am. Podolsky, Phys. Podolsky and P. Schwed, Rev. Phys, 20 , 40 Frenkel and R. Santos, "The self-force on point charged particles in Podolsky's generalized electrodynamics," to be published.
You want to teach yourself about general relativity or particle theory, but you could never really find a suitable book to learn from. Or maybe you just hate the textbook in your intermediate physics class and want a better alternative. Well, here's a list of books I've come across in my wanderings through Crerar Library. I've tried to indicate roughly what the book covers and at what level, and I've tried to point out where certain books are dated or otherwise pathological. I also indicate when a book sucks. I'm afraid you'll find that this list is heavily biased towards theory, and general relativy and particle theory in particular.
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M. L. Boas-Mathematical Methods in the Physical Sciences - John Wiley
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