The previous post (Part (1)) assumed that there was no motion in the medium other than harmonic motion of the charged particles within it. Now, however, we consider a layer in a conducting medium in which an applied magnetic field is moved or displaced.

We begin by looking at *Faraday rotation*, which involves the rotation of the position angle q of the linear polarization of radiation passing through a magnetized (B) plasma. If B and the angle f between B and the direction of wave propagation, are known then a measurement of q at a wavelength l permits a determination of the total number of charged particles in a column of 1 m ^{2}^{ }cross section between source and observer. This is given by:

*N*** **_{t}* *= ò^{r}_{ o}_{ } Ndr

Where *N*** **** _{t }** is the total number of charged particles in a column of length r and cross section 1 m

^{2}.

We note that the natural modes for this polarization are circular as shown in the diagram below, i.e. for 'wave 1' and 'wave 2' with respective position angles q1 and q2.

Then the 2 circularly polarized waves (or opposite rotation) represent the resolution of the linearly polarized wave. If the two circularly polarized waves have different phase constants (b and - b) then the plane of polarization of the resultant linearly polarized wave rotates as the wave propagates.Then, in the absence of a magnetic field we can write for the index of refraction of the plasma:

n ^{2} = 1 - w_{ e} ^{2} / w ^{2}

And w_{ e} = [n_{e} e^{2}/ m_{e} ε_{o}] ^{½}

is the electron plasma frequency. We can then write the equation applicable for a magnetic field in the following abbreviated form:

n ^{2} = 1 - X / ( 1 __+__ Y)

And for the *ordinary* wave: n _{o} ^{2} = 1 - X / ( 1 + Y)

And for the extra *ordinary* wave: n _{x} ^{2} = 1 - X / ( 1 - Y)

(Where: X = w_{ e} ^{2} / w ^{2} and Y = w_{ H} ^{2} / w ^{2})

Where w_{ H} denotes the *hybrid frequency*, which is a mix of the electron plasma and electron cyclotron frequencies, e.g.

w_{ H} ^{2} = w_{ e} ^{2} + w _{c} ^{2 } ( w _{c}** ** = qB/ m

_{e})

Note the *ordinary* wave occurs when the EM wave components **E** _{1} and **B** _{o} are parallel, e.g. **E** _{1} ∥ **B** _{o }

The extra*ordinary* wave occurs when an EM wave (e.g. radio wave) propagates partly transverse, partly longitudinal. I.e. propagates perpendicular to **B** _{o }_{ }with **E** _{1 }perpendicular to **B** _{o}.

It is important to point out that in dealing with magnetically-affected radio waves we will expect *cutoffs* and *resonance*s. I.e. as a radio wave propagates through a region in which w_{ e} and w _{c} are changing it may encounter cutoffs and resonances. A cutoff occurs in a plasma when the index of refraction, n, goes to zero. That is, when the wavelength becomes infinite. So if the wave number vector k = 2 p/ l and the mean index of refraction is: __n__ = c k / w

Then when n -> 0:

0 = c ( 2 p/ l) / w

Or: 2 p/ l = w (0)/ c or: l = 2 p c / 0 = _{¥}

Conversely, a resonance occurs when the index of refraction becomes infinite, i.e. when the *wavelength* becomes zero, so that:

n = c ( 2 p **/ **0 ) **/** w

Or: n w = c ( 2 p **/ **0 ) = 2 p c / 0 = _{¥}

Thus, for any finite w, k -> _{¥ }implies w -> w_{ H}

So that the resonance occurs at a point in the plasma where:

w_{ H} ^{2} = w_{ e} ^{2} + w _{c} ^{2 }= w ^{2 }

The conditions for cutoff and resonance can be written in terms of the ordinary frequency, f (i.e. w = 2 p f) referenced to electron and cyclotron frequencies. So that:

The extraordinary wave is cut off when:

f_{ e} ^{2} / f_{ x }( f_{ x } **- **f_{ H} ) = 1

The ordinary wave is cut off when:

f_{ e} ^{2} / f_{ o }( f_{ o } + ** **f_{ H} ) = 1

Referring back to the plane of polarization being Faraday-rotated (as illustrated above) we note the measurment can be done by means of a microwave 'horn' - a component of many radio telescopes. This horn gives a value for w_{ e} ^{2} and hence of the number density of the plasma. (Refer back to Problem (2) of Part (1)

Other inferences which can be made regarding ordinary and extraordinary waves:

1) If f > f_{ x } then both indices of refraction are real and both modes are propagated.

2) If f_{ x } > f > f _{o } then only the ordinary wave propagates.

3) If f > f _{o } then no waves are propagated. The magnitude of rotation in radians will then be equal to:

2.38 x 10 ^{6 } [ 1 / f ^{2 }ò n_{ }_{e} H dz ]

Where dz is the thickness of the plasma through which the radio wave is moving,

n_{ }_{e} is the number of electrons per cubic centimeter

H is the magnetic field in gauss. Faraday rotation has been found important at the very low densities of the interstellar medium and has been used to explain the polarization of microwave radiation generated by maser action, e.g. on OH and H2O molecular clouds. (See also my Dec. 16 post on *Cosmic Masers.*)

**Suggested Problems**:

1) A space plasma with electron number density 10 ^{12} /cm ^{3} features a magnetic field of 0.0001 T (Tesla). Find: a) the electron plasma frequency, b) the cyclotron frequency and c) the hybrid frequency.

Thence or otherwise find the refractive index of the plasma and whether ordinary or extraordinary wave propagation can be expected.

2) For the plasma in (1) and assuming condition (3) (f > f _{o} ) find the amount of rotation of the plane of linear polarization expected, assuming a plasma thickness dz = 0.15 cm.

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