Electrodynamics

Electrodynamics is the branch of electromagnetism that deals with the temporal evolution of systems where electric and magnetic fields interact with moving charges.

Classical Electrodynamics (CED)

Albert Einstein developed the theory of special relativity through an analysis of electrodynamics. During the late 19th century physicists became aware of a contradiction between the accepted laws of electrodynamics and classical mechanics. In particular, Maxwell's equations predicted non-intuitive results such as that the speed of light is the same for any observer and that it does not obey Galilean relativity. It was therefore believed that Maxwell's equations were not correct and that the true equations of electromagnetism contained a term that would correspond to the influence of the luminiferous ether.

After experiments yielded no evidence for the existence of the ether, Einstein proposed the revolutionary idea that the equations of electrodynamics were correct and that some principles of classical mechanics were inaccurate, leading to the formulation of the theory of special relativity.

Some fifteen years before Einstein's work, Emil Wiechert and, later, Alfred-Marie Liénard searched for expressions of electromagnetic fields of moving charges. These expressions, which included the effect of the retardation of the propagation of light, are now known as Liénard-Wiechert potentials. An important fact that emerges from the retardation is that a set of moving electric charges can no longer be accurately described by equations that only depend on the velocities and positions of the particles. In other words, that implies that the Lagrangian must contain dependencies on the internal "degrees of freedom" of the field.

Classical Lagrangian and energy

The Lagrangian of the classical electromagnetic field is given by a scalar built from the electromagnetic field tensor:

Sc,em[chuckles]Fμ μ .. ,Ω Ω ]=− − 116π π c∫ ∫ Ω Ω Fμ μ .. Fμ μ .. dΩ Ω {displaystyle S_{c,em}[F_{mu nu },Omega ]=-{frac {1}{16pi c}}int _{Omega }F_{mu nu }F^{mu nu }dOmega }

In fact this Lagrangian can be rewritten in terms of the electric and magnetic fields to give (in cgs units):

Sc,em[chuckles]E,B,Ω Ω ]=− − 18π π ∫ ∫ R∫ ∫ V(E2− − B2)d3xdt{displaystyle S_{c,em}[mathbf {E}mathbf {Bomega]=-{frac {1}{8pi }{int _{mathbb} {R}}{x1}{x1}{Big}{bc(E}{2}{bf {B}{2}{c}{2}{bc}{c}{c}}{b

Introducing this Lagrangian into the Euler-Lagrange equations, the result is Maxwell's equations and applying a generalized Legrendre transformation, the expression for electromagnetic energy is obtained:

Eem=18π π ∫ ∫ R3(E2+B2)dV{displaystyle E_{em}={frac {1}{8pi }}}int _{mathbb {R} ^{3}}left(mathbf {E} ^{2}+mathbf {B} ^{2}right)dV}

Field evolution equations

The Euler-Lagrange equations applied to the above Lagrangian provide the following evolution equations:

F,γ γ α α β β +F,α α β β γ γ +F,β β γ γ α α =▪ ▪ Fα α β β ▪ ▪ xγ γ +▪ ▪ Fβ β γ γ ▪ ▪ xα α +▪ ▪ Fγ γ α α ▪ ▪ xβ β =0{displaystyle F_{,gamma }{alpha beta } +F_{,alpha }{beta gamma } +F_{beta }{beta }{gamma alpha }}{frab }{frab }}{fab }}{posalf}}}{posal

Which expressed in terms of electric and magnetic fields are equivalent to the following two equations:

► ► ⋅ ⋅ B=0,► ► × × E=− − ▪ ▪ B▪ ▪ t{displaystyle {boldsymbol {nabla }}cdot mathbf {B} =0,qquad {boldsymbol {nabla }}}times mathbf {E} =-{frac {partial mathbf {B}}{partial t}}}}}}

These are the homogeneous Maxwell equations. To obtain the other two it is necessary to consider in the Lagrangian the interaction between matter with electric charge and the electromagnetic field itself.

Quantum Electrodynamics

Quantum electrodynamics, as its name suggests, is the quantum version of electrodynamics. This quantum theory describes the electromagnetic field in terms of photons exchanged between charged particles, in the style of quantum field theory. Therefore, quantum electrodynamics focuses on the quantum description of the photon and its interaction/exchange of energy and linear momentum with charged particles.

It can be noted that the formulation of the theory of special relativity is made up of two parts, one of them «kinematic», described above, and that establishes the bases of the theory of motion – and, consequently, of the whole of the theory – giving them their mathematical expression, and an "electrodynamic" part which, combining the proposals of the first part with the electromagnetic theory of Maxwell, Hertz and Lorentz deductively establishes a certain number of theorems on the properties of light and, in general of electromagnetic waves as well as the dynamics of the electron.

In the part corresponding to electrodynamics, Albert Einstein formulates his theory applying, for an empty space, the transformation of coordinates – which forms the basis of relativistic kinematics– to the equations of Maxwell-Hertz; this application reveals, once again, that the transformation, far from being a simple artifice of calculations, possesses an essential physical sense: the laws of classical electromagnetism determine the properties of two different fields, X,And,Z{displaystyle scriptstyle X,Y,Z} system K{displaystyle scriptstyle K} and the magnetic field of components Bx,Band,Bz{displaystyle scriptstyle B_{x},B_{y},B_{z}}; now, transforming the equations of K{displaystyle scriptstyle K} a K♫{displaystyle scriptstyle K'} and imposing, according to the principles of relativity, that the new components of the fields X! ! ,And! ! ,Z! ! ;B! ! x,B! ! and,B! ! z{displaystyle scriptstyle {bar {X}},{bar {Y}},{bar {Z}}};{bar {B}}_{x},{bar {B}}_{y},{bar {B}}}}_{z}} in K, there are relationships where the transformed components of the electric field and the magnetic field respectively depend, in turn, on the initial components of both fields, which leads with amazing naturality to the theoretical unification of magnetism and electricity. To this end, the necessary relationships in the conditions that are concerned are:

{X! ! =XB! ! x=BxAnd! ! =b(And− − vVBz)B! ! and=b(Band+vVZ)Z! ! =b(Z+vVBand)B! ! z=b(Band− − vVZ){cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF}{cHFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFF00}{cHFFFFFFFF}{cHFFFFFFFF}{cHFFFFFFFFFFFFFFFF}{cHFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00}{cHFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{c}{cH00}{cHFFFFFFFFFFFFFFFFFFFFFF

On the other hand, the distinction between electric force and magnetic force is nothing but a consequence of the state of movement of the coordinate system; in which, the kinematic analysis eliminates the pre-relativist theoretical anomaly: the different explanation of the same phenomenon (electromagnetic induction) is nothing more than an appearance due to ignorance of the principle of relativity and its consequences.

On the other hand, based on the relativistic formulas it is feasible to extend the preceding results to Maxwell's equations when there are convection currents; The conclusion is that the electrodynamics of bodies in Lorentz motion are in accordance with the principle of relativity.

Now, regarding the dynamics of the slowly accelerated electron, which would require a long discussion, we will only cite the following result: if a mass m is attributed to an electron slowly accelerated by an electric field and based on this mass one can evaluate the kinetic energy of an electron, measured in a rest frame relative to which it has been accelerated by the field to a velocity v.

But where the theoretical formulation of the electrodynamics part of special relativity places its accent is in the propagation of electromagnetic waves, from which it is deduced, always following the same method of algebraic application of the Lorentz formulas, the laws of the two best-known optical phenomena and of great importance for astronomy: the Doppler effect (apparent frequency change for a moving source and which we will analyze in the next section) and aberration, already mentioned above.

QED predictions

• The electromagnetic field is interpretable in terms of particles or a few of radiation called photons.
• The gyroscopic factor or "factor g" predicted by theory is more than double the one predicted by classical theory, i.e. the quotient between the magnetic moment and the electron's thorn is more than double the one expected on the basis of classical theory.
• Atoms are stable because they represent stationary states of the atomic system formed by the atomic nucleus, electrons and electromagnetic radiation.