Electromagnetic wave

electromagnetic waves are us in the everyday life beside water waves and acoustic waves most frequently meeting kinds of waves. To them among other things the visible light and the broadcast waves belong. Contrary to acoustic waves it actselectromagnetic waves, like with water waves, i.e. around transverse waves. Direction of propagation and oscillation direction stand perpendicularly to each other, which at the phenomenon of the polarization becomes noticeable.

Physically regarded it concerns with electromagnetic waves spreading oscillations of the electromagnetic field. Here stand electrical and magnetic field perpendicularly one on the other and have a firm relative importance (in SI - this straight by the speed of light C is given units). In particular it disappears electrical and magnetic field at the same places at the same time so that the frequently read representation thatelectrical and magnetic energy cyclically into one another are converted, is not completely correct. It is however e.g. correct. for the near field electromagnetic waving of a producing electrical dipole or resonant circuit.

The emergence of electromagnetic waves explains itself from the Maxwell equations: The temporalChange of the electrical field is always linked with a spatial change of the magnetic field. Likewise again the temporal change of the magnetic field is linked with a spatial change of the electrical field. For periodically (in particular sinusoidally) changing fields these result inEffects together a progressive wave.

The special at the electromagnetic wave is that no medium must be present; such a wave can reproduce itself thus in the absolutely empty area. In response to it the subject waves stand, like z. B. thatSound, which need a medium for transmission.

In the vacuum an electromagnetic wave with the vacuum speed of light spreads <math> c_0 = 299. \, 792. \, 458 \; \ mathrm {\ frac {m} {s}}< /math> out. This value is accurate, since the unit meter is defined by the speed of light C, and applies independently ofthe frequency of the wave.

In a medium (thus in subject) the speed is reduced dependent on the Permittivität and the permeability of the material. It applies then: <math> c= \ frac {1} {\ sqrt {\ mu \ varepsilon}}< /math>. Besides it becomes dependent on the frequency <math> \ omega< /math> the wave (dispersion), as well as (depending upon medium) dependent on their polarization and their direction of propagation broken. A direct application of force (e.g. Change of direction) on a spreading electromagnetic wave can take place only via the propagation medium (delimitations included such as mirrors) or the gravitation strength.

Electromagnetic waves are in the electromagnetic spectrum sorted according to the wavelength (a list of frequencies and examples of electromagnetic waves give it in the there article).

The best well-known and by most studied example of an electromagnetic wave is the visible light. With the light determinesthe frequency and/or the wavelength the color of the light. Monochromatic light, thus light only one wavelength, has always a Spektralfarbe.


With electromagnetic waves or with the short-wave manifestations of the electromagnetic waves (for example gamma radiation) meets extremely small intensity, in order to describe all observable phenomena, on the contrary the particle characteristics of individual photons, the quanta of the electromagnetic field do not step the wave model no more described above, into the foreground. The wave character (for instance interference) withdraws against it. In the context of this particle conception of theLight becomes each frequency <math> \ nu< /math> the energy of an individual photon <math> h \ cdot \ nu< /math> assigned. Both aspects of electromagnetic jets are discussed theoretically in the context of quantum electrodynamics.

Some newer theories, for example the loop quantum gravitation, say a small frequency response of the speed of light C in the vacuumahead.

mathematical description

the existence of electromagnetic waves follows from the Maxwell equations. They were theoretically postulated 1865 by James Clerk Maxwell, before Heinrich Rudolf Hertz could prove them to 1888 experimentally.

Here are first electromagnetic wavesin the vacuum to be regarded, thus waves in the charge-free area under exclusion of dielectric, dia. and paramagnetic effects (<math> \ vec D = \ varepsilon_0 \ vec E< /math> and <math> \ vec to B = \ mu_0 \ vec H< /math>, see material equations of electrodynamics). Current density j and charge densityρ are zero.

One goes first from the third Maxwell equation (with j=0):

<math>

(1) \ \ operator name {red} \ vec E = - {\ partial \ vec B \ more over \ partial t} </math>

and applies to both sides the rotation operator. Toone receives one thereby

< math>

\ operator name {red} \ \ operator name {red} \ vec E = - \ operator name {red} \ left ({\ partial \ vec B \ more over \ partial t} \ right) </math>

<math>

= - \ mu_0 {\ partial \ more over \ partial t} \ left (\ operator name {red} \ vec H \ right), </math>

and the fourth Maxwell equation begins,

< math>

= - \ mu_0 {\ partial \ more over \ partial t} \ left ({\ partial \ vec D \ more over \ partial t} \ right) </math>

<math>

(2) \ = - \ mu_0 \ varepsilon_0 {\ partial^2 \ vec E \ more over\ partial t^2} </math>

On the other hand the vector-analytic relationship completely generally applies

< math>

\ operator name {red} \ \ operator name {red} \ vec A = \ operator name {degrees} \ \ operator name {div} \ vec A \ delta \ vec A </math>

with the Laplace operator Δ

< math>

\ Delta = \ partial^2/\ partial x^2+ \ partial^2/\ partial y^2 + \ partial^2/\ partial z^2 </math>.

One applies this relationship to (1), and considers one that the charge-free area is regarded, in which after the first Maxwell equation the divergence of D zero is,thus arises

< math>

\ operator name {red} \ \ operator name {red} \ vec E = \ operator name {degrees} \ \ operator name {div} \ vec E - \ delta \ vec E </math>

<math>

= \ operator name {degrees} \ 0 \ delta \ vec E </math>

<math>

(3) \ = - \ delta \ vec E </math>.

One setsnow (2) and (3) together the following wave equation results

< math> (4) \ \ delta \ vec E = \ mu_0 \ varepsilon_0 {\ partial^2 \ vec E \ more over \ partial t^2}

< /math>.

Nearly all waves leave themselves by equations of the form

< math>

(5) \ {\ partial^2f \ more over \ partial t^2} = v^2 f </math>

describe, whereby v is the propagation speed of the wave. The propagation speed of electromagnetic waves is the speed of light C. To it applies therefore

< math>

c^2 = {1 \ more over \ mu_0 \ varepsilon_0} </math>.

Thus one receives the equation thus from (4)

< math>

{\ partial^2 \ vec E \ more over \ partial t^2} = c^2 \ delta \ vec E, </math>

for each component the wave equation of the form (5) represents. Their solutions are waves, itself alsoSpeed of light C spread.

If the wave in linear materials with that spreads to dielectric constant ε and the permeability μ, then the speed of light C is somewhat lower, i.e.

< math>

c={ 1 \over \sqrt{\mu_0 \mu \varepsilon_0 \varepsilon} }, </math>

whereby in howeverthe general material constants not linear even e.g. are, but. depend on the field strength or the frequency.

While the light spreads in air still nearly with vacuum speed of light C (the material constants are in good approximation 1), applies toWater already no more that among other things the Tscherenkow effect makes possible.

Further also a mathematical description is possible by Potenzialen, because left

< because of> math (1) \ quad \ nabla \ cdot \ (\ nabla \ times \ vec A \ right) = 0< /math>

and

< math> (2) \ quad \ nabla \ \ vec B cdot= 0< /math>

the field vector of the magnetic flow density can be understood also as rotation of a vector field A. A is called therefore the Vektorpotenzial by B and it applies:

<math> (3) \ quad \ vec B = \ nabla \ times \ vec A< /math>

This relationship can nowfurther to be used. The rotation of the electrical field is certainly through

< math> (4) \ quad \ nabla \ times \ vec E = - {{\ partial \ vec B} \ more over {\ partial t}}< /math>

If one uses now the evenly won relationship out (3) in (4), then one receives

<math> (5) \ quad \ nabla \ times \ vec E = - {{\ partial} \ more over {\ partial t}} \ nabla \ times \ vec A

< /math>

and from it follows

< math>

(6) \ quad \ nabla \ times \ left \ lbrack \ vec E + {{\ partial \ vec A} \ more over {\ partial t}} \ right \ rbrack = 0</math>

Now disappears however the rotation each to gradients, so that the internal expression of (6) a function scalar as gradient can be understood:

<math>

(7) \ quad - \ nabla \ phi = \ vec E + {{\ partial \ vec A} \ more over {\ partialt}} </math>

<math>

(8) \ quad \ vec E = - \ nabla \ phi - {{\ partial \ vec A} \ more over {\ partial t}} </math>

This can be used now again in the original Maxwell equations. With

<math>

(9) \ quad \ nabla \ \ vec E = {cdot \ rho \ more over\ epsilon} </math>

<math>

(10) \ quad \ nabla \ times \ vec B = \ mu \ kappa \ vec E + \ mu \ epsilon {{\ partial \ vec E} \ more over {\ partial t}} </math>

and (8) and the relationship

< math> \ quad \ nabla \ times \ nabla \ times \ vec A = \ nabla (\ nabla \ cdot\ vec A) - \ nabla^2 \ vec A

< /math>

one receives

< math>

(11) \ quad \ nabla^2 \ phi + {{\ partial} \ more over {\ partial t}} \ nabla \ cdot \ vec A = - {\ rho \ more over \ epsilon} </math>

<math>

(12) \ quad \ nabla^2 \ vec A \ mu \ epsilon {{\ partial^2 \ vecA} \ more over {\ partial t^2}} - \ mu \ kappa {{\ partial \ vec A} \ more over {\ partial t}} - \ nabla \ left \ lbrack \ nabla \ cdot \ vec A + \ mu \ epsilon {{\ partial \ phi} \ more over {\ partial t}} + \ mu \ kappa \ phi \ right \ rbrack = 0 </math>

Around theseEquations (11) and (12) from each other to decouple, it is required that the term under that disappears to gradients in (12) (see calibration transformation), thus

< math>

(13) \ quad \ nabla \ \ vec A cdot + \ mu \ epsilon {{\ partial \ phi} \ more over {\ partial t}} + \ mu \ kappa\ phi = 0 </math>

If the condition is out (13) fulfilled, then the wave equation for the Vektorpotenzial results automatically A with math from (12

<)>

(14) \ quad \ nabla^2 \ vec A \ mu \ epsilon {{\ partial^2 \ vec A} \ more over {\ partial t^2}} - \ mu\ kappa {{\ partial \ vec A} \ more over {\ partial t}} = 0 </math>

and out (11) and (13) the wave equation of the scalar Potenzialfunktion with

<math> (15) \ quad \ nabla^2 \ phi - \ mu \ epsilon {{\ partial^2 \ phi} \ more over {\ partial t^2}} - \ mu \ kappa {{\ partial \ phi} \ more over {\ partialt}} = - {\ rho \ more over \ epsilon}

< /math>

In the pour-free vacuum follows

<math>

(16) \ quad \ nabla^2 \ vec A \ mu_0 \ epsilon_0 {{\ partial^2 \ vec A} \ more over {\ partial t^2}} = 0 </math>

<math>

(17) \ quad \ nabla^2 \ phi - \ mu_0 \ epsilon_0 {{\ partial^2 \ phi} \ more over{\ partial t^2}} = 0 </math>

<math>

(18) \ quad \ nabla^2= \ delta </math>

This description of electromagnetic phenomena can be adapted by calibration transformation to different problems around these to simplify. In quantum mechanics the Vektorpotenzial of the magnetic field a more fundamental role than often becomesthe field size attributed. The Vektorpotenzial is even present if the magnetic field disappears. This phenomenon is well-known under the name Aharonov Bohm effect. Experimentally the Vektorpotenzial can be proven by interference by electron beams, which run by a shielded magnetic field.The electrons are thus not affected by the magnetic field. The interference samples are changed nevertheless by the condition of the field. As a cause the Vektorpotenzial is accepted, which can exist also with missing B-field. This opinion is however disputed.

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