# Electromagnetic Wave – Physics School – What kind of wave can propagate through a vacuum?

May 25, 2022

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< /span>

$\lambda$

propagates in the direction that

${\vec {E}}$

(in blue) and the

${\vec {B}}$

(in red) are at right angles to each other and to the direction of propagation, forming in in this order a legal system.

Linear polarized electromagnetic wave in vacuum. The monochromatic wave with wavelength propagates in direction, the electric field strength (in blue) and the magnetic flux density (in red) are at right angles to each other and to the direction of propagation and form a legal system in this order.

An electromagnetic wave, also electromagnetic radiation, is a wave of coupled electric and magnetic fields. Sometimes radiation is also briefly spoken of, although there is a risk of confusion with other particle radiation. Examples of electromagnetic waves are Radio waves, Microwaves, Thermal radiation, Light, X-rays and Gamma radiation (listed in increasing frequency). Electromagnetic waves in a vacuum are shear waves. The interaction of electromagnetic waves with matter depends on their frequency, which can vary over many orders of magnitude.

Unlike sound waves, for example, electromagnetic waves do not require a medium to propagate.[ 1] They can therefore also spread over the longest distances in space. They move in a vacuum at Speed ​​of Light, regardless of their frequency. However, electromagnetic waves can also propagate in matter (e.g. a gas or a liquid), but their speed is reduced. The index of refraction indicates the ratio by which the phase velocity of electromagnetic waves in matter is less than the speed of light in a vacuum.

As transverse waves, electromagnetic waves show the phenomenon of Polarization. In free space, the vectors of the electric and magnetic fields are perpendicular to each other and to the direction of propagation. Transversality may be violated if – as in the case of plasma oscillations (Plasmons) – Carrier of chemical properties, e.g. B. metallic or bound electrons are involved. Accordingly, the sources, propagation properties and effects of the radiation differ in the different areas of the electromagnetic spectrum.

## Origin

Electromagnetic waves can be caused by different reasons:

• Spontaneous Emission when the energy of an atom decreases. Energy changes in the atomic shell are usually orders of magnitude smaller than energy changes in the atomic nucleus. If the atom is “left alone” (as in rarefied gases) during the period of energy radiation, a sharp line spectrum is produced. This is not the case at high pressure, such as in high-pressure lamps and xenon light, or with atoms in solids. Well-defined spectral lines can no longer be measured there because of Pressure widening.
• Radiation: Electromagnetic waves also occur when charge carriers are accelerated. This happens, for example, in the plasma of the sun or in the X-ray tube.
• Molecular vibrations (periodic movements of neighboring atoms in a molecule)
• Larmor precession of a particle with a magnetic dipole moment about the direction of an externally applied magnetic field.
• A movement of electrically charged particles at high speed through a medium. If the speed of the particles is greater than the Phase Velocity of electromagnetic waves in this medium, then Cherenkov Radiation.
• A time-varying electric current emits electromagnetic waves. In radio technology, this is used with transmitting antennas for the wireless transmission of signals.
• In pair annihilation, matter is converted into electromagnetic radiation. The energy of the radiation results from the mass and the kinetic energy of the particles.

## Properties

### Detect and measure existing electromagnetic waves

Receivers of electromagnetic radiation are called sensors or detectors, in living things photoreceptors. Radio waves can be detected by antennas.

The Wave speed can be measured on an electromagnetic wave, on the one hand the universal constant Light speed $c$, as well as deviating values ​​for the Phase Velocity $c_{\mathrm {med} }$ in a permeable (transparent) medium. The intensity can also be measured, equivalent to the power, or to the energy transported through a specific cross-section per unit of time. span>.

There are different methods to measure the wavelength, depending on whether the wavelengths are shorter or longer. Wavelength $\lambda$ and frequency $\nu$ let through

$\nu ={\frac {c_{\mathrm {med} }}{\lambda }}$

convert into each other.

### wave character

From a physical point of view, electromagnetic waves are propagating oscillations of the electromagnetic field. Here electric and magnetic field in linearly polarized waves perpendicular to each other and have a fixed size ratio, which is given by the wave impedance. In particular, electric and magnetic fields vanish in the same places at the same time, so the commonly read notion that electric and magnetic energy cyclically interconvert is correct in the far field is. However, it is correct, for example, for the near field of an electrical dipole or resonant circuit that generates electromagnetic waves.

The formation of electromagnetic waves can be explained by the Maxwell’s equations: The temporal change in the electric field is always linked to a spatial change in the magnetic field. Likewise, the temporal change in the magnetic field is in turn linked to a spatial change in the electric field. For periodically (particularly sinusoidally) changing fields, these effects combine to produce a progressive wave.

• Phenomena such as coherence and interference can only be explained with the wave model, because the phase of the wave is needed for this.
• Antennas for the radiation emitted by radio transmitters are tuned to the size of the wavelength. For example, an efficient dipole antenna is half the wavelength. A description of the radiation as a very large number of photons offers no advantage, since there is no measuring device to individually detect such low-energy photons.

### particle character

For certain properties of electromagnetic waves (e.g. photoelectric effect), the wave model described above is not sufficient to describe all observable phenomena; rather, the particle properties of individual Photons, the quanta of the electromagnetic field, in the foreground. However, the wave character (e.g. Interference) is fully retained. One therefore speaks of the dualism of particles and waves.

Within the framework of this particle conception of light, each frequency $f$ is assigned the energy of a single photon $h\cdot f$, where $h$ is the Planck’s quantum of action. On the other hand, particles, such as electrons moving across several atoms, also have wave properties (see also Electric current).
Both aspects of electromagnetic waves can be explained within the framework of Quantum Electrodynamics.

• With the Compton effect, an electromagnetic wave with a wavelength of about 20 pm hits an electron whose cross section is about three orders of magnitude smaller. To explain the physical process of the interaction, the particle character of the light must be used. Any attempt to explain the observed change in wavelength with the wave model fails.
• With the photoelectric effect, the kinetic energy does not depend on the amplitude of the radiation, but increases linearly with the frequency. This can only be explained by the particle character.
• The production of laser light relies on the properties of individual atoms, each of which is smaller than the wavelength produced. Therefore, to explain production, one has to fall back on the photon model.

Photons with sufficient energy (about a few electronvolts upwards) have a ionizing effect on matter and can trigger chemical () effects if the binding energies are exceeded (photochemistry). This chemical effectiveness is also referred to as actinity.

### Waves in the medium

The phase velocity $c_{\text{med}}$ with which a monochromatic wave travels in a medium is typically lower than in a vacuum. It depends in a linear approximation on the permittivity $\varepsilon$ and the permeability $\mu$ of the substance,

$c_{\text{med}}={\frac {1}{\sqrt {\mu \varepsilon }}}\,,$

and is therefore dependent on the Frequency of the wave (see Dispersion) and with birefringent media also from their polarisation and direction of propagation. The influencing of the optical properties of a medium by static fields leads to electro-optics or magneto-optics.

A direct force (e.g. change of direction) on a propagating electromagnetic wave can only occur through the propagation medium (see Refraction, Reflection, Scattering and Absorption) or (see Nonlinear Optics and Acoustooptic Modulator).

### spectrum

Electromagnetic waves are classified according to wavelength in the electromagnetic spectrum. A list of frequencies and examples of electromagnetic waves can be found in the relevant article.

Visible light represents only a small part of the entire spectrum and, with the exception of infrared radiation (heat), is the only range that can be perceived by humans without technical aids. At lower frequencies, the energy of the Photons is too low to trigger chemical processes. At higher frequencies, on the other hand, the range of ionizing radiation (Radioactivity) begins, where a single photon can destroy molecules. This effect already occurs with ultraviolet radiation and is responsible for the formation of skin cancer from excessive exposure to the sun.

In the case of light, the frequency determines the color of the light and not, as is often wrongly assumed, the wavelength in a medium during propagation. Unlike the wavelength, the frequency is not affected by the transition to optically denser media. However, since the color does not change when passing through a medium, only the frequency is characteristic of the color of the light. For historical reasons, however, the wavelength is specified as a characteristic property in spectra. The relationship between color and wavelength only applies in a vacuum and, to a good approximation, in air. Monochromatic light, i.e. light with only a single wavelength, always has a spectral colour.

## Biological and chemical effect

Sensitivity distribution of the three types of cones in humans: The sensitivity of the rods is shown in black. The curves are each scaled so that their maximum is 100%.

In the interaction of electromagnetic radiation with biological matter, ionizing radiation (greater than 5 eV) and non-ionizing radiation. With ionizing radiation, the energy is sufficient to ionize atoms or molecules, i. H. knock out electrons. This creates free radicals that cause biologically harmful reactions.
If the energy of photons equals or exceeds the binding energy of a molecule, each photon can destroy a molecule, which can lead to accelerated aging of the skin or skin cancer, for example. Chemical binding energies of stable molecules are above about 3 eV per bond. If molecular changes are to occur, photons must have at least this energy, which corresponds to violet light or higher-frequency radiation.

In the interaction of non-ionizing radiation, a distinction is made between thermal effects[2] (radiation has a heating effect because it passes through the is absorbed), direct field effects (induced dipole moments, change of membrane potentials), quantum effects[3] and resonance effects (synchronization with oscillation of the cell structure).[4 ]

A photon with a wavelength of 700 nm or shorter can cause a conformation change in the rhodopsin molecule. This change is recorded in the eye and further processed as a signal by the nervous system. The sensitivity to a specific wavelength changes with modifications of the rhodopsin. This is the biochemical basis of color sense. Photons of light with a wavelength longer than 0.7 µm have an energy below 1.7 eV. These waves cannot cause chemical reactions on molecules that are stable at room temperature. Because of this, animal eyes typically cannot see infrared or thermal radiation. However, in 2013, researchers discovered that the cichlid can see in the near-infrared range.[5] There are also other infrared-sensing organs, such as the pit organ in snakes.

Photons can excite vibrations in molecules or in the crystal lattice of a solid body. These vibrations are noticeable in the material as thermal energy. Additional vibrations excited by electromagnetic waves increase the temperature of the material. Unlike the effect of individual photons on chemical bonds, it is not the energy of the individual photons that matters here, but the sum of the energy of all photons, i.e. the intensity of the radiation. Long-wave electromagnetic radiation can indirectly alter biological substances through heat denaturation.

## Speed ​​of light and special theory of relativity

It has been known since 1676 how fast light travels roughly . However, until 1865 there was no connection to other physical phenomena. James Clerk Maxwell was able to produce this in the years 1861 to 1862 using the Maxwell equations he found >[6], which predict the existence of electromagnetic waves. Their speed corresponded so well with the speed of light known at the time that a connection was immediately established. Heinrich Hertz was able to prove these waves experimentally in the 1880s.

In classical mechanics, waves (in the direction of propagation $x$) are represented by the wave equation

${\frac {\partial ^{2}}{\partial t^{2}}} {\vec {f}}=c^{2}{\frac {\partial ^{2}}{\partial x^{2}}}{\vec {f}}$

described. Here ${\vec {f}}$ denotes the deflection of the wave and $c$ denotes its phase velocity, here referred to as propagation velocity of the wave can be interpreted.

From Maxwell’s equations, the electric field strength ${\vec {E}}$ in a vacuum can be calculated as follows:

${\frac {\partial ^{2}}{\partial t^{2}}}{\vec {E} }={\frac {1}{\varepsilon _{0}\mu _{0}}}{\frac {\partial ^{2}}{\partial x^{2}}}{\vec {E} }$

derive (in SI units; see section Mathematical Description). In this respect, the electric field strength behaves like a wave; the size

$c={\frac {1}{\sqrt {\varepsilon _{0}\mu _ {0}}}}$

appears as the speed of propagation. This speed $c$ is composed exclusively of natural constants that are independent of the observer’s reference system, which is consequently transferred to the quantity $c$.

is the same for both observers.

Situation at the pond: The moving observer sees the propagation speed of a water wave reduced by its own speed. For electromagnetic waves, Maxwell predicts that the propagation speed is the same for both observers.

The basis of classical mechanics is the Galilean Principle of Relativity, which states that the laws of nature in all inertial systems – those reference systems in which bodies that are not acted on by a straight line move forward – have the same shape (). A frame of reference moving to an inertial frame with constant speed is also an inertial frame.

According to this principle of relativity, it would now be expected that an observer moving at a constant speed relative to the electromagnetic wave would measure a different propagation speed, such as a person walking at a constant speed on the edge of a pond would measure a different speed speed of propagation of a water wave on the pond than a stationary observer. However, Maxwell’s equations predict the same propagation speed for both observers – they are not Galileo-invariant.

This contradiction to classical mechanics is resolved in favor of Maxwell’s equations: The fact that electromagnetic waves propagate in all inertial systems with the same speed – the much-cited one – forms a postulate in Einstein Special Theory of Relativity published in 1905. The so-called .

## mathematical description< /p>

The electromagnetic wave equation results directly from Maxwell’s equations and the non-divergence of electromagnetic waves and reads in a vacuum

$\left(\nabla ^{2}-{\frac {1}{c^{2 }}}{\frac {\partial ^{2}}{\partial t^{2}}}\right){\vec {E}}({\vec {r}},t)=0$.

If one considers the propagation of electromagnetic waves in polarizable media, then the polarization ${\vec {P}}$ must also be considered:

$\left(\nabla ^{2}-{\frac {1}{c^{2 }}}{\frac {\partial ^{2}}{\partial t^{2}}}\right){\vec {E}}({\vec {r}},t)={\frac { 1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}{\vec {P}}$

< /dd>

### Derivation of the electromagnetic wave equation

The mathematical relationships associated with wave propagation can be understood on the basis of Maxwell’s equations. In particular, one can derive the same form of the wave equation that propagates other types of waves, such as sound waves.

In a vacuum, i.e. in a charge-free space excluding dielectric, dia and paramagnetic effects are the Material equations of electrodynamics ${\vec {D}}=\varepsilon _{0}{\vec {E} }$ and ${\vec {B}}=\mu _{0}{\vec {H}}$. Also, the current density ${\vec {\jmath }}$ and Charge density $\varrho$ zero.

Starting from Maxwell’s third equation

$\nabla \times {\vec {E}}=-{\frac {\partial {\vec {B }}}{\partial t}}$

(1)

applies the rotation operator to both sides. This gives you:

$\nabla \times (\nabla \times {\vec {E}})=-\nabla \times \left({\frac {\partial {\vec {B}}}{\partial t}}\right)=-\mu _{0}{\frac {\partial }{\partial t}}\ left(\nabla \times {\vec {H}}\right)$

.

If you insert Maxwell’s fourth equation (with ${\vec {\jmath }}=0$) into it,

$\nabla \times {\vec {H}}={\frac {\partial {\vec {D}}}{\partial t}}$

,

arises

$\nabla \times (\nabla \times {\vec {E}})=-\mu _{0 }{\frac {\partial }{\partial t}}\left({\frac {\partial {\vec {D}}}{\partial t}}\right)=-\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}{\vec {E}}}{\partial t^{2}}}$

.

(2)

The vector analytic relationship applies in general

$\nabla \times (\nabla \times {\vec {A}})=\nabla ( \nabla \cdot {\vec {A}})-\Delta {\vec {A}}$

.

$\Delta {\vec {A}}$ means the application of the vectorial Laplace operator to the vector field ${\vec {A}}$ . In Cartesian coordinates, the vectorial Laplace operator acts like the scalar Laplace operator $\textstyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^ {2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$ on each component of ${\vec {A}}$ .

Applying this relation to ${\vec {E}}$ and taking into account that the charge-free space is considered, in which according to Maxwell’s first equation the divergence of ${\vec {D}}$ is zero, then:

$\nabla \times (\nabla \times {\vec {E}})=\nabla (\nabla \ cdot {\vec {E}})-\Delta {\vec {E}}=-\Delta {\vec {E}}$

.

(3)

If you now put (2) and (3) together, you get the following Wave equation:

$\Delta {\vec {E}}=\mu _{0}\varepsilon _{0}{\ frac {\partial ^{2}{\vec {E}}}{\partial t^{2}}}$

.

(4)

Almost all waves can be represented by equations of the form

${\frac {\partial ^{2}f}{\partial t^{2}} }=v^{2}\Delta f$

, where $v$ is the propagation speed of the wave.
For the propagation speed of electromagnetic waves, the Speed ​​of light $c$, therefore applies:

$c^{2}={\frac {1}{\mu _{0}\varepsilon _{0}}}$

.

This gives the equation from (4).

${\frac {\partial ^{2}{\vec {E}}}{\partial t^{2}}}=c^{2}\Delta {\vec {E}}$

.

In the same way, for the magnetic flux density ${\vec {B}}$ the relation

${\frac {\partial ^{2}{\vec {B}}}{\partial t^{2}}}=c^{2}\Delta {\vec {B}}$

derive. The solutions to these equations describe waves that propagate in a vacuum with the speed of light $c$. Propagates the electromagnetic wave in isotropic material with the dielectric constant $\varepsilon$ and the permeability $\mu$ is the propagation speed

$c_{\text{med}}={\frac {1}{\sqrt {\mu \varepsilon }}}$

.

In general, the material constants are not linear, but can depend on the field strength or the frequency. While light propagates in air almost at the vacuum speed of light $c$ (the material constants are a good approximation 1), this does not apply to propagation in water, which among other things causes the Cherenkov effect< /span> enabled.

The ratio of the vacuum speed of light to the speed in the medium is called the Index of Refraction $n$.

$n={\sqrt {\frac {\mu \varepsilon }{\mu _{0} \varepsilon _{0}}}}={\sqrt {\mu _{r}\varepsilon _{r}}}$

,

where $\mu _{r}$ and $\varepsilon _{r}$ denote the relative permeability and relative permittivity of the medium.

### Propagation of electromagnetic waves

With the help of Maxwell’s equations, further conclusions can be drawn from the wave equation. Consider a general plane wave for the electric field

${\vec {E}}={\vec {E}}_{0}f( {\hat {k}}\cdot {\vec {x}}-ct)$

,

where ${\vec {E}}_{0}$ is the (constant) amplitude, $f$ is an arbitrary C2 function, ${\hat {k}}$ is a unit vector pointing in the direction of propagation, and ${\vec {x}}$ a position vector. First of all, by inserting it into the wave equation, one sees that $f({\hat {k}}\cdot {\vec {x}}-ct)$ satisfies the wave equation, i.e. that

$\Delta f({\hat {k}}\cdot {\vec {x}}- ct)={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}f({\hat {k}}\cdot { \vec {x}}-ct)$

.

For ${\vec {E}}$ to describe an electromagnetic wave, it not only has to satisfy the wave equation, but also Maxwell’s equations. That means

$\nabla \cdot {\vec {E}}={\hat {k}}\cdot {\vec {E}}_{0}{\frac {\partial f({\hat {k}}\cdot {\vec {x}}-ct)}{\partial ({\hat {k}} \cdot {\vec {x}})}}=0$

,

${\vec {E}}\cdot {\hat {k}}=0$

.

The electric field is always perpendicular to the direction of propagation, so it is a transversal wave.
Substituting ${\vec {E}}$ into another Maxwell equation yields

$\nabla \times {\vec {E}}={\hat {k}}\times {\vec {E}}_{0}{\frac {\partial f({\hat {k}}\cdot {\vec {x}}-ct)}{\partial ({\hat {k}} \cdot {\vec {x}})}}=-{\frac {\partial {\vec {B}}}{\partial t}}$

and since $\textstyle -{\frac {\partial f({\hat {k}}\cdot {\vec {x}}-ct)}{\partial ({\hat {k}}\cdot {\vec {x}})}}={\frac {\partial f({\hat {k}}\cdot {\vec {x}}-ct)}{\partial (ct)}}$, follows from this

${\vec {B}}={\frac {1}{c}}{\hat {k}}\times {\vec {E}}$

.

The magnetic flux density in the electromagnetic wave is also perpendicular to the direction of propagation and also perpendicular to the electric field. In addition, their amplitudes are in a fixed ratio. Their quotient is the speed of light $c$

${\frac {E_{0}}{B_{0}}}\;=\; c$

.

In natural units ($c=1$) both amplitudes have the same value.

With this relationship, a statement can be made about the energy density

$w_{\mathrm {em} }={\frac {1}{2}}\varepsilon _{0}(E^{2}+c^{2}B^{2})$

of the electromagnetic field for the case of the electromagnetic wave:

$w_{\mathrm {em} }=\varepsilon _{0}E^{2}= {\frac {1}{\mu _{0}}}B^{2}$

.

Not every electromagnetic wave has the property that its propagation direction and the directions of the electric and magnetic fields are orthogonal to each other in pairs, so the wave is a pure transversal wave, too TEM wave called. The plane waves demonstrated here are of this type, but there are also waves in which only one of the two field vectors is perpendicular to the direction of propagation, but the other has a component in the direction of propagation
(TM and TE waves). An important application for such not purely transverse electromagnetic waves are cylindrical waveguides. However, what has been said applies above all in crystals with birefringence.[7] However, there are no purely longitudinal > electromagnetic waves.

## Literature

• John David Jackson: 4th Edition. de Gruyter, Berlin and others 2006, ISBN 3-11-018970-4.
• Karl Küpfmüller, Wolfgang Mathis, Albrecht Reibiger: 16th edition. Springer, Berlin et al. 2005, ISBN 3-540-20792-9.
• Claus Müller: (= 88, ). Springer, Berlin et al. 1957.
• Eduard Rhein: , edition 69.-80. Thsd., German publisher d. Ullstein A.G., Berlin-Tempelhof 1954. DNB
• Károly Simonyi: 10th edition. Barth, Leipzig and others 1993, ISBN 3-335-00375-6.

– Collection of images, videos and audio files

– Collection of images, videos and audio files

## Individual evidence

1. 22nd, completely re-edited edition. Springer, Berlin et al. 2004, ISBN 3-540-02622-3, p. 177.

22nd, completely revised edition. Springer, Berlin et al. 2004, ISBN 3-540-02622-3, p. 177.

2. In: Vol. 162, No. 2, 2004, pp. 219-225, JSTOR

Kenneth R. Foster, Michael H. Repacholi In:Vol. 162, No. 2, 2004, pp. 219-225, JSTOR 3581139

3. In: Vol. 18, No. 2, 1997, pp. 187-189, doi:10.1002/(SICI)1521-186X(1997)18:2<187: :AID-BEM13>3.0.CO;2-O

Henrik Bohr, Søren Brunak, Jakob Bohr:In:Vol. 18, No. 2, 1997, pp. 187-189, doi:

4. Walter Hoppe, Wolfgang Lohmann, Hubert Markl, Hubert Ziegler (eds.): 2nd, completely revised edition. Springer, Berlin et al. 1982, ISBN 3-540-11335-5.

5. First detection in animals: infrared while catching prey. In: Frankfurter Allgemeine Zeitung. February 4, 2013 (

Reinhard Wandtner: In: February 4, 2013 ( faz.net ).

6. Two Millennia of Light: The Long Path to Maxwell’s Waves. In: IEEE Industrial Electronics Magazine. 9, No. 2, 2015, pp. 54–56+60. doi:10.1109/MIE.2015.2421754

M. Guarnieri:. In:. 9, No. 2, 2015, pp. 54–56+60. doi:

7. More about crystal optics ( birefringence and others) in: W. Döring, Göschen-Bädchen zur Theoretische Physik, volume “Optics”.

< /li>

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