# Faraday rotation

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# 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.