Editing Faraday effect (section)

=== In plasma ===
In [[Plasma (physics)|plasma]], the effect is caused by free [[electrons]] and can be characterized as a difference in the [[refractive index]] seen by the two circularly polarized propagation modes. Hence, in contrast to the Faraday effect in solids or liquids, the Faraday rotation angle (β) has a simple dependence on the wavelength of light (λ), namely:

:<math> \beta = R_M \, \lambda^2 </math>,

where the overall strength of the effect is characterized by <math>R_M</math>, the '''rotation measure'''. This in turn depends on the projection of the magnetic field along the line of sight <math>B_\parallel</math> and the number density of electrons ''n<sub>e</sub>'', both of which vary along the propagation path. In the [[ideal plasma|ideal-plasma]] approximation, the rotation measure is given (in the [[Gaussian units|Gaussian CGS units]]) by

:<math>R_M = \frac{e^3}{2\pi m^2c^4}\int_0^d n_e(s) B_\parallel(s) \;\mathrm{d}s</math>

or, in the [[SI]] units, by
:<math>R_M = \frac{e^3}{8\pi^2 \varepsilon_0 m^2 c^3} \int_0^d n_e(s) B_{||}(s) \;\mathrm{d}s \approx
  (2.62 \times 10^{-13}\, \mathrm T^{-1})\times\, \int_0^d n_e(s) B_\parallel(s)\; \mathrm{d}s
</math>

where
:''n<sub>e</sub>(s)'' is the density of free electrons at each point ''s'' along the path;
:''B<sub>‖</sub>(s)'' is the component of the magnetic field in the direction of propagation at each point ''s'' along the path;
:''e'' is the [[elementary charge]];
:''c'' is the [[speed of light|speed of light in vacuum]];
:''m'' is the [[electron mass]];
:<math>\scriptstyle\epsilon_0</math> is the [[vacuum permittivity]],
and the integral is taken over the entire path from the source to the observer.

Electron Coulomb collisions and plasma instabilities may significantly alter this simple expression, however.<ref>{{Cite journal| doi = 10.1063/5.0159061| issn = 1070-664X| volume = 30| issue = 7| pages = 072109| last1 = Keenan| first1 = Brett D.| last2 = Stark| first2 = David J.| title = Faraday effect in collisional magnetized plasmas| journal = Physics of Plasmas| access-date = 2025-03-18| date = 2023-07-18| url = https://doi.org/10.1063/5.0159061}}</ref>

==== Interstellar medium ====
Faraday rotation is an important tool in [[astronomy]] for the measurement of magnetic fields, which can be estimated from rotation measures given a knowledge of the electron number density in the [[interstellar medium]].<ref>{{cite book|last1=Longair|first1=Malcolm | author-link=Malcolm Longair|title=High Energy Astrophysics|publisher=Cambridge University Press|date = 1992|isbn=978-0-521-43584-0 }}</ref> In the case of [[radio pulsar]]s, the [[dispersion (optics)|dispersion]] caused by these electrons results in a time delay between pulses received at different wavelengths, which can be measured in terms of the electron column density, or [[dispersion measure]]. A measurement of both the dispersion measure and the rotation measure therefore yields the weighted mean of the magnetic field along the line of sight.  The same information can be obtained from objects other than pulsars, if the dispersion measure can be estimated based on reasonable guesses about the propagation path length and typical electron densities. In particular, Faraday rotation measurements of polarized radio signals from extragalactic radio sources occulted by the solar corona can be used to estimate both the electron density distribution and the direction and strength of the magnetic field in the coronal plasma.<ref>{{cite journal |last1=Mancuso |first1=S. |last2=Spangler |first2=S. R. |title=Faraday Rotation and Models for the Plasma Structure of the Solar Corona |date=2000 |journal=[[The Astrophysical Journal]] |volume=539 |issue=1 |pages=480–491 |doi=10.1086/309205 |bibcode = 2000ApJ...539..480M |doi-access=free }}</ref>

==== Ionosphere ====
[[Radio wave]]s passing through the Earth's [[ionosphere]] are likewise subject to the Faraday effect. The ionosphere consists of a plasma containing free electrons which contribute to Faraday rotation according to the above equation, whereas the positive ions are relatively massive and have little influence. In conjunction with the Earth's magnetic field, rotation of the polarization of radio waves thus occurs. Since the density of electrons in the ionosphere varies greatly on a daily basis, as well as over the [[sunspot cycle]], the magnitude of the effect varies. However the effect is always proportional to the square of the wavelength, so even at the UHF television frequency of 500&nbsp;MHz (λ = 60&nbsp;cm), there can be more than a complete rotation of the axis of polarization.<ref>Larry Wolfgang, Charles Hutchinson, (ed), ''The ARRL |Handbook for Radio Amateurs, Sixty Eighth Edition '', American Radio Relay League, 1990 {{ISBN|0-87259-168-9}}, pages 23-34, 23-25,</ref>  A consequence is that although most radio transmitting antennas are either vertically or horizontally polarized, the polarization of a medium or short wave signal after [[Skywave|reflection by the ionosphere]] is rather unpredictable. However the Faraday effect due to free electrons diminishes rapidly at higher frequencies (shorter wavelengths) so that at [[microwave]] frequencies, used by [[satellite communications]], the transmitted polarization is maintained between the satellite and the ground.