Difference between revisions of "Faraday rotation"

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[[Radiative Processes in Astrophysics|Course Home]]
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===Short Topical Videos===
 
===Short Topical Videos===
 
* [https://www.youtube.com/watch?v=TNUjhwzW2Pc Faraday Rotation (Polin, UC Berkeley)]
 
* [https://www.youtube.com/watch?v=TNUjhwzW2Pc Faraday Rotation (Polin, UC Berkeley)]
  
 
===Reference Material===
 
===Reference Material===
*  
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===Need to Review?===
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* [[Electromagnetic Plane Waves]]
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* [[Plasma Frequency]]
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* [[Polarization]]
  
 
<latex>
 
<latex>
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k_{R,L}\approx \frac{\omega}{c}\left[1-\frac{\omega_p^2}{2\omega^2}\left(1\mp\frac{\omega_B}{\omega}\right)\right]
 
k_{R,L}\approx \frac{\omega}{c}\left[1-\frac{\omega_p^2}{2\omega^2}\left(1\mp\frac{\omega_B}{\omega}\right)\right]
 
\end{align}
 
\end{align}
where $\omega_p=\sqrt{4\pi n_ee^2}{m_e}$ is the plasma frequency. After substituting in $k_{R,L}$ for the phase angles and substituting $\omega_p$ and $\omega_B$, we find
+
where $\omega_p=\sqrt{\frac{4\pi n_ee^2}{m_e}}$ is the plasma frequency. After substituting in $k_{R,L}$ for the phase angles and substituting $\omega_p$ and $\omega_B$, we find
 
\begin{align}
 
\begin{align}
 
\Delta\theta = \frac{2\pi e^3}{m_e^2c^2\omega^2}\int_0^d n_e B_{||}\,ds
 
\Delta\theta = \frac{2\pi e^3}{m_e^2c^2\omega^2}\int_0^d n_e B_{||}\,ds
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We are able to derive lower limits on the mean magnetic field, since
 
We are able to derive lower limits on the mean magnetic field, since
 
\begin{align}
 
\begin{align}
\langle B_{||}\rangle_ = \frac{\int_0^d n_e B_{||}\,ds}{\int_0^d n_e\,ds}
+
\langle B_{||}\rangle_{n_e} = \frac{\int_0^d n_e B_{||}\,ds}{\int_0^d n_e\,ds}
 
\end{align}
 
\end{align}
 
The numerator is found by measuring the rotation measure, and the denominator is simply the dispersion measure. Note that this is a lower limit since it measures only the line-of-sight component of the magnetic field.
 
The numerator is found by measuring the rotation measure, and the denominator is simply the dispersion measure. Note that this is a lower limit since it measures only the line-of-sight component of the magnetic field.
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Latest revision as of 10:35, 27 October 2020

Course Home

Short Topical Videos[edit]

Reference Material[edit]

Need to Review?[edit]

Faraday Rotation

1 Overview

When a polarized electromagnetic wave propagates through a magnetized plasma, the group velocity depends on whether it is a left- or right-hand circularly polarized wave. Since a plane polarized wave is a linear superposition of a right- and left-hand circular wave, this manifests itself as a rotation of the plane of polarization. The polarization angle rotates by an amount

where

is the rotation measure. This process is called Faraday rotation.

2 Derivation

Suppose we have a circularly polarized wave

where and correspond to right and left circular polarization, respectively. Additionally, the wave propagates through a magnetic field . The equation of motion for an electron is thus

It can be shown that the velocity of the electron is

where is the cyclotron frequency. From this, it is apparent that right and left circular polarized waves will propagate through the plasma at different velocities. For a linearly polarized wave (a superposition of a right- and left- circular wave), the plane of polarization rotates by an amount

where is the phase angle, and (assuming and )

where is the plasma frequency. After substituting in for the phase angles and substituting and , we find

Note that the magnetic field that appears in the expression is the component along the line of sight.

3 Measuring magnetic fields

We are able to derive lower limits on the mean magnetic field, since

The numerator is found by measuring the rotation measure, and the denominator is simply the dispersion measure. Note that this is a lower limit since it measures only the line-of-sight component of the magnetic field.