Classical Electrodynamics

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Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical field theory. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics, which is a quantum field theory.

Fundamental physical aspects of classical electrodynamics are presented in many texts, such as those by Richard Feynman, Robert B. Leighton and Matthew Sands,[1] David J. Griffiths,[2] Wolfgang K. H. Panofsky and Melba Phillips,[3] and John David Jackson.[4]

As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of classical electromagnetism. Problems arise because changes in charge distributions require a non-zero amount of time to be "felt" elsewhere (required by special relativity).

Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevant mathematical models of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena, cf.[9] An electrodynamics phenomenon is determined by the particular fields, specific densities of electric charges and currents, and the particular transmission medium. Since there are infinitely many of them, in modeling there is a need for some typical, representative

While the previous two editions use Gaussian units, the third uses SI units, albeit for the first ten chapters only. Jackson wrote that this is in acknowledgement of the fact virtually all undergraduate textbooks on electrodynamics employ SI units and admitted he had "betrayed" an agreement he had with Edward Purcell that they would support each other in the use of Gaussian units. In the third edition, some materials, such as those on magnetostatics and electromagnetic induction, were rearranged or rewritten, while others, such as discussions of plasma physics, were eliminated altogether. One major addition is the use of numerical techniques. More than 110 new problems were added.[11]

According to a 2015 review of Andrew Zangwill's Modern Electrodynamics in the American Journal of Physics, "[t]he classic electrodynamics text for the past four decades has been the monumental work by J. D. Jackson, the book from which most current-generation physicists took their first course."[4]

L.C. Levitt, who worked at the Boeing Scientific Research Laboratory, commented that the first edition offers a lucid, comprehensive, and self-contained treatment of electromagnetism going from Coulomb's law of electrostatics all the way to self-fields and radiation reaction. However, it does not consider electrodynamics in media with spatial dispersion and radiation scattering in bulk matter. He recommended Electrodynamics of Continuous Media by Lev Landau and Evgeny Lifshitz as a supplement.[8][note 1]

Ronald Fox, a professor of physics at the Georgia Institute of Technology, opined that this book compares well with Classical Electricity and Magnetism by Melba Phillips and Wolfgang Panofsky, and The Classical Theory of Fields by Landau and Lifshitz.[note 2] Classical Electrodynamics is much broader and has many more problems for students to solve. Landau and Lifshitz is simply too dense to be used as a textbook for beginning graduate students. However, the problems in Jackson do not pertain to other branches of physics, such as condensed-matter physics and biophysics. For optimal results, one must fill in the steps between equations and solve a lot of practice problems. Suggested readings and references are valuable. The third edition retains the book's reputation for the difficulty of the exercises it contains, and for its tendency to treat non-obvious conclusions as self-evident. Fox stated that Jackson is the most popular text on classical electromagnetism in the post-war era and that the only other graduate book of comparable fame is Classical Mechanics by Herbert Goldstein. However, while Goldstein's text has been facing competition from Vladimir Arnold's Mathematical Methods of Classical Mechanics, Jackson remained unchallenged (as of 1999). Fox took an advanced course on electrodynamics in 1965 using the first edition of Jackson and taught graduate electrodynamics for the first time in 1978 using the second edition.[13]

Andrew Zangwill, a physicist at the Georgia Institute of Technology, noted the mixed reviews of Jackson after surveying the literature and reviews on Amazon. He pointed out that Jackson often leaves out the details in going from one equation to the next, which is often quite difficult. He stated that four different instructors at his school had worked on an alternative to Jackson using lecture notes developed in roughly a decade with the goal of strengthening the student's understanding of electrodynamics rather than treating it as a topic of applied mathematics.[6]

Thomas Peters from the University of ZÃ¼rich argued that while Jackson has historically been training students to perform difficult mathematical calculations, a task that is undoubtedly important, there is much more to electrodynamics than this. He wrote that Modern Electrodynamics by Andrew Zangwill offers a "stimulating fresh look" on this subject.[14]

Addendum The books on classical electrodynamics such as J. D. Jackson, does not mention about $U(1)$ symmetry in the context of gauge invariance (as far as I know). Gauge invariance is simply understood, in classical electrodynamics books, as the invariance of Maxwell's equations under $A_\mu\to A_\mu+\partial_\mu\chi(x)$. There is no sign of U(1) invariance that I can discover here. On the other hand, when something like Dirac equation or Dirac field is brought into the scene then the implementation U(1) transformation is clear. But that is always discussed in quantum field theory books. It appears that it is essential to have a Dirac field to understand U(1) symmetry. So the question is whether it is possible to understand the existence of U(1) symmetry in classical electrodynamics without bringing in the Dirac field into the picture?

A free "$\mathrm{U}(1)$" gauge theory can never tell whether the gauge group is $\mathrm{U}(1)$ or $\mathbb{R}$ because the only field in the theory, the gauge potential $A$, transforms as$ A\mapsto A + \partial_\mu \chi,$ where $\chi$ is just a real-valued function, and the real numbers are the Lie algebra of both $\mathrm{U}(1)$ and $\mathbb{R}$. This is not a classical or quantum property, you simply cannot tell the difference. So, in a sense, asking whether this theory has $\mathrm{U}(1)$ symmetry or not is meaningless - it has $\mathfrak{u}(1)$ symmetry, and there is no meaningful notion of the symmetry group.

Electromagnetism coupled to other fields can tell what the gauge group is, since part of coupling it to other fields is specifying how these fields transform under gauge transformations. There we have a choice between (infinitesimally) $\psi \mapsto \psi + \chi \psi$ and $\psi\mapsto \psi + \mathrm{i}\chi \psi$, which lead to finite transformations $\psi\mapsto \mathrm{e}^{\chi}\psi$ and $\psi\mapsto \mathrm{e}^{\mathrm{i}\chi}\psi$, respectively. The former corresponds to a gauge group $\mathbb{R}$, the latter to $\mathrm{U}(1)$. Again, none of this is classical or quantum.

The reason you likely think that the $\mathrm{U}(1)$ is a quantum feature is that it much more natural in quantum field theory than in classical field theory to have complex-valued fields, but in fact we can consider e.g. classical electromagnetism coupled to a classical complex scalar field and then we are likewise forced to specify the gauge group.

AbstractWithin the frame of classical electrodynamics, nonlinear Thomson scattering by an electron of a tightly focused circularly polarized laser has been investigated. The electron motion and spatial radiation characteristics are studied numerically when the electron is initially stationary. The numerical analysis shows that the direction of the maximum radiation power is in linear with the initial phase of the laser pulse. Furthermore, we generalize the rule to the case of arbitrary beam waist, peak amplitude and pulse width. Then the radiation distribution is studied when the electron propagates in the opposite sense with respect to the laser pulse and the linear relationship still holds true. Last we pointed out the limitation of the single electron model in this paper.

A high Q-factor of the nanocavity can effectively reduce the threshold of nanolasers. In this paper, a modified nanostructure composed of a silver grating on a low-index dielectric layer (LID) and a high-index dielectric layer (HID) was proposed to realize a nanolaser with a lower lasing threshold. The nanostructure supports a hybrid plasmonic waveguide mode with a very-narrow line-width that can be reduced to about 1.79 nm by adjusting the thickness of the LID/HID layer or the duty ratio of grating, and the Q-factor can reach up to about 348. We theoretically demonstrated the lasing behavior of the modified nanostructures using the model of the combination of the classical electrodynamics and the four-level two-electron model of the gain material. The results demonstrated that the nanolaser based on the hybrid plasmonic waveguide mode can really reduce the lasing threshold to 0.042 mJ/cm2, which is about three times lower than the nanolaser based on the surface plasmon. The lasing action can be modulated by the thickness of the LID layer, the thickness of the HID layer and the duty cycle of grating. Our findings could provide a useful guideline to design low-threshold and highly-efficient miniaturized lasers. 781b155fdc