L3 Electrodynamics

[Zoraida]

Classicalelectromagnetism 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 textbooks. For the undergraduate level, textbooks like The Feynman Lectures on Physics, Electricity and Magnetism, and Introduction to Electrodynamics are considered as classic references and for the graduate level, textbooks like Classical Electricity and Magnetism,[1] Classical Electrodynamics, and Course of Theoretical Physics are considered as classic references.

The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of optics centuries before light was understood to be an electromagnetic wave. However, the theory of electromagnetism, as it is currently understood, grew out of Michael Faraday's experiments suggesting the existence of an electromagnetic field and James Clerk Maxwell's use of differential equations to describe it in his A Treatise on Electricity and Magnetism (1873). The development of electromagnetism in Europe included the development of methods to measure voltage, current, capacitance, and resistance. Detailed historical accounts are given by Wolfgang Pauli,[2] E. T. Whittaker,[3] Abraham Pais,[4] and Bruce J. Hunt.[5]

The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both the velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force.

Although the equation appears to suggest that the electric and magnetic fields are independent, the equation can be rewritten in term of four-current (instead of charge) and a single electromagnetic tensor that represents the combined field ( F μ ν \displaystyle F^\mu \nu ):

where q0 is what is known as a test charge and F is the force on that charge. The size of the charge does not really matter, as long as it is small enough not to influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C (newtons per coulomb). This unit is equal to V/m (volts per meter); see below.

Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential can help. Electric potential, also called voltage (the units for which are the volt), is defined by the line integral

The scalar φ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down into simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del operator) of the potential. Or:

A changing electromagnetic field propagates away from its origin in the form of a wave. These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths. Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves, microwaves, light (infrared, visible light and ultraviolet), x-rays and gamma rays. In the field of particle physics this electromagnetic radiation is the manifestation of the electromagnetic interaction between charged particles.

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.[6] 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

In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics.[1][2][3] In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved.[2] QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electromagnetism giving a complete account of matter and light interaction.[2][3]

The first formulation of a quantum theory describing radiation and matter interaction is attributed to British scientist Paul Dirac, who (during the 1920s) was able to compute the coefficient of spontaneous emission of an atom.[6] He is also credited with coining the term "quantum electrodynamics".[7]

Dirac described the quantization of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, Werner Heisenberg and an elegant formulation of quantum electrodynamics by Enrico Fermi,[8] physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. However, further studies by Felix Bloch with Arnold Nordsieck,[9] and Victor Weisskopf,[10] in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer.[11] At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics.

Difficulties with the theory increased through the end of the 1940s. Improvements in microwave technology made it possible to take more precise measurements of the shift of the levels of a hydrogen atom,[12] now known as the Lamb shift and magnetic moment of the electron.[13] These experiments exposed discrepancies which the theory was unable to explain.

A first indication of a possible way out was given by Hans Bethe in 1947,[14] after attending the Shelter Island Conference.[15] While he was traveling by train from the conference to Schenectady he made the first non-relativistic computation of the shift of the lines of the hydrogen atom as measured by Lamb and Retherford.[14] Despite the limitations of the computation, agreement was excellent. The idea was simply to attach infinities to corrections of mass and charge that were actually fixed to a finite value by experiments. In this way, the infinities get absorbed in those constants and yield a finite result in good agreement with experiments. This procedure was named renormalization.

QED has served as the model and template for all subsequent quantum field theories. One such subsequent theory is quantum chromodynamics, which began in the early 1960s and attained its present form in the 1970s work by H. David Politzer, Sidney Coleman, David Gross and Frank Wilczek. Building on the pioneering work of Schwinger, Gerald Guralnik, Dick Hagen, and Tom Kibble,[25][26] Peter Higgs, Jeffrey Goldstone, and others, Sheldon Glashow, Steven Weinberg and Abdus Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.

Near the end of his life, Richard Feynman gave a series of lectures on QED intended for the lay public. These lectures were transcribed and published as Feynman (1985), QED: The Strange Theory of Light and Matter,[2] a classic non-mathematical exposition of QED from the point of view articulated below.

These actions are represented in the form of visual shorthand by the three basic elements of diagrams: a wavy line for the photon, a straight line for the electron and a junction of two straight lines and a wavy one for a vertex representing emission or absorption of a photon by an electron. These can all be seen in the adjacent diagram.

QED is based on the assumption that complex interactions of many electrons and photons can be represented by fitting together a suitable collection of the above three building blocks and then using the probability amplitudes to calculate the probability of any such complex interaction. It turns out that the basic idea of QED can be communicated while assuming that the square of the total of the probability amplitudes mentioned above (P(A to B), E(C to D) and j) acts just like our everyday probability (a simplification made in Feynman's book). Later on, this will be corrected to include specifically quantum-style mathematics, following Feynman.

There is an infinite number of other intermediate "virtual" processes in which more and more photons are absorbed and/or emitted. For each of these processes, a Feynman diagram could be drawn describing it. This implies a complex computation for the resulting probability amplitudes, but provided it is the case that the more complicated the diagram, the less it contributes to the result, it is only a matter of time and effort to find as accurate an answer as one wants to the original question. This is the basic approach of QED. To calculate the probability of any interactive process between electrons and photons, it is a matter of first noting, with Feynman diagrams, all the possible ways in which the process can be constructed from the three basic elements. Each diagram involves some calculation involving definite rules to find the associated probability amplitude.

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