Classical electromagnetism

The classical electrodynamics is the branch of physics that deals with moving electric charges and with time-varying electric and magnetic fields. The electrostatics as a special case of electrodynamics deals with static electric charges and their fields. The underlying fundamental force of physics is called electromagnetic interaction.

The theory of classical electrodynamics was formulated by Maxwell mid-19th century using the Maxwell equations named after him. The investigation of the Maxwell equations for moving reference systems led Albert Einstein in 1905 to formulate the special theory of relativity. In the course of the 1940s to combine quantum mechanics and electrodynamics in quantum electrodynamics, whose predictions match very closely with measurement results achieved.

An important form of electromagnetic fields, the electromagnetic waves, which is one of well-known representative of visible light. Although the physical principles are given for the description of electromagnetic waves through the electrodynamics, provides their research represents a separate branch of physics, optics.

Classical Electrodynamics

Basic equations

Temporal changes of the magnetic flux cause electric vortex field.

Lorentz force on a moving charge with velocity v q.

The interaction of electromagnetic fields and electric charges is fundamentally due to the microscopic Maxwell equations

And the Lorentz force

Determined.

This results in using the material equations of electrodynamics, the macroscopic Maxwell equations. These are equations for the effective fields that occur in matter.

Continue playing ( derivable from ) play an important role:

Potentials and wave equation

The homogeneous Maxwell equations

And

May, in accordance with the introduction of the electromagnetic potentials

And

Be solved in a star- shaped region identical ( Poincaré lemma). This refers to the so-called scalar potential and the vector potential. Since the physical fields are given only by derivatives of the potentials, one has some freedom to modify the potentials and still recover the same physical fields. For example, arise and the same field, if you by

Is related. It also requires that gives the same field in such a transformation, must be as

Transform. Such a transformation is called a gauge transformation. In electrodynamics, two calibrations are often used. First, the so-called Coulomb gauge or Strahlungseichung

And second, the Lorenz gauge

The Lorenz gauge has the advantage to be relativistically invariant and structurally not change when changing between two inertial frames. The Coulomb gauge is indeed invariant non- relativistic, but used more in the canonical quantization of electrodynamics.

Substituting the - and - fields and the vacuum material equations in the inhomogeneous Maxwell equations, and calibrates the potentials according to the Lorenzeichung, decouple the inhomogeneous Maxwell equations and the potentials satisfy inhomogeneous wave equations

Herein, the D' Alembertoperator.

Special cases

Electrostatics is the special case of non-moving electrical charges and static (not changing with time) electric fields. They can also be used within limits, as long as the speed and acceleration of the charges of the fields and the changes are small.

The magnetostatics deals with the special case of constant currents in a total of uncharged conductors and constant magnetic fields. It can be used for sufficiently slowly varying currents and magnetic fields.

The combination of the two, electromagnetism, electrodynamics can be described as the not too strong accelerating charges. Most operations in electrical circuits (such as coil, capacitor, transformer ) can already described at that level. A stationary electric or magnetic field is close to its source, such as the earth's magnetic field. However, a changing electromagnetic field can be removed from its origin. The field is an electromagnetic wave in the interaction between the magnetic and electric field. This emission of electromagnetic waves is neglected in electrostatics. The description of the electromagnetic field here is limited so be prepared for the near field.

Electromagnetic waves, however, are the only form of the electromagnetic field, which can exist independently of a source. Although they are generated by sources, but can continue to exist after their generation regardless of the source. Since light can be described as an electromagnetic wave, and the look is ultimately a special case of electrodynamics.

Electrodynamics and theory of relativity

In contrast to classical mechanics, electrodynamics is not Galilean invariant. This means, if one assumes, as in classical mechanics, an absolute, Euclidean space and absolute time that is independent, then the Maxwell equations are not valid in all inertial systems.

Simple example: A constant speed flying, charged particle is surrounded by an electric and a magnetic field. A flying at the same speed, the same charged particle undergoes by the electric field of the first particle repulsive force, since like charges repel each other; at the same time it learns through its magnetic field an attractive Lorentz force which partially compensates for the rejection. At the speed of light, this compensation would be complete. In the inertial frame in which to rest both particles, there is no magnetic field and thus no Lorentz force. There is only the repulsive Coulomb force so that the particles will be accelerated more than in the original reference frame in the moving both charges. This contradicts the Newtonian physics, where the acceleration does not depend on the reference system.

This finding initially led to the assumption that in electrodynamics a preferred frame of reference would be (ether system ). Attempts to measure the velocity of the Earth relative to the ether, but failed, such as the Michelson -Morley experiment. Hendrik Antoon Lorentz solved this problem with a modified ether theory ( Lorentz ether theory ), but this was superseded by Albert Einstein with his special theory of relativity. Einstein replaced Newton's absolute space and absolute time by a four-dimensional space-time. In the theory of relativity takes the Lorentz invariance, which is filled with the electrodynamics in the place of Galilean invariance.

In fact, it can reduce the acceleration and thus the magnetic force in the above example to explain as a consequence of the length contraction and time dilation when transformed back to the statements made in the moving system observations in a stationary system. In a sense, therefore, suggest the existence of magnetic phenomena ultimately traced back to the structure of space and time, as described in the theory of relativity. From this perspective, the structure of the basic equations for static magnetic fields with their cross products appear less surprising.

In the manifestly Lorentz - forminvariante description of the electrodynamics of the scalar potential and the vector potential form a four-vector, analogous to the four-vector of space and time, so that the Lorentz transformations can be applied by analogy to the electromagnetic potentials. In a special Lorentz transformation with velocity in - direction apply to the fields in the usual SI unit system, the transformation equations:

(In cgs units, these equations are modified only slightly: You have to formally substitute only or by or. )

Extensions

However, the classical electrodynamics does not provide a consistent description of moving charges, on small scales, problems arise such as the Abraham - Lorentz equation. Quantum electrodynamics ( QED) combines the electrodynamics therefore with quantum mechanical concepts. The theory of the electroweak interaction combines the QED and weak interaction and is part of the Standard Model of elementary particle physics. The structure of QED is also the starting point for the quantum chromodynamics ( QCD), which describes the strong interaction. However, the situation is even more complicated ( for example, three types of cargo, see color charge ).

A unification of electrodynamics with the general theory of relativity ( gravitation) is known as the Kaluza - Klein theory, and represents an early attempt to unify the fundamental interactions dar.