<blockquote>
<p>What is the mathematical difference between their refraction indices?</p>
</blockquote>
<p>The conventions on the signs of the direction of magnetic field vectors and of charges.</p>
<hr />
<p><a href="https://i.sstatic.net/POzma.png" rel="noreferrer"><img src="https://i.sstatic.net/POzma.png" alt="screen shot from Abigail Polin's Faraday Rotation video" /></a></p>
<p><a href="https://i.sstatic.net/djmZX.png" rel="noreferrer"><img src="https://i.sstatic.net/djmZXm.png" alt="screen shot from Abigail Polin's Faraday Rotation video" /></a></p>
<p>The tutorial specifies several things that can have a sign;</p>
<ol>
<li>direction of the light propagation</li>
<li>direction of the magnetic field</li>
<li>sign of the particle's charge (electrons have negative charge)</li>
</ol>
<p>Let's look at WIkipedia's <a href="https://en.wikipedia.org/wiki/Faraday_effect#Interstellar_medium" rel="noreferrer">Faraday effect; Interstellar medium</a> where an equation for <span class="math-container">$RM$</span> or rotation measure is given:</p>
<p><span class="math-container">$$RM = \frac{e^3}{2 \pi m^2 c^4} \int_0^d n_e(s) B_{\parallel}(s) ds$$</span></p>
<p>This expression includes <span class="math-container">$e^3$</span> and <span class="math-container">$B_{\parallel}$</span>. Notice that the power of <span class="math-container">$e$</span> is odd. The assumption is that the interstellar medium is negative electrons, so they've replaced <span class="math-container">$q^3$</span> with <span class="math-container">$e^3$</span> but if it were positrons they'd have to write</p>
<p><span class="math-container">$$q^3=-e^3$$</span>.</p>
<p>Also remember that <span class="math-container">$B_{\parallel}$</span> can be positive or negative since the field could point either way relative to the direction of propagation.</p>
<p>The tutorial shows some of the intermediate steps where we get the idea that both the sign of the charge and the directions of the field will be important.</p>
<p><a href="https://i.sstatic.net/L9JCZ.png" rel="noreferrer"><img src="https://i.sstatic.net/L9JCZ.png" alt="screen shot from Abigail Polin's Faraday Rotation video" /></a></p>
<blockquote>
<p>Why right handed circular polarization gets lagged when going through ionized plasma?</p>
</blockquote>
<p>It could be either the RCP or LCP that is delayed more than the other, depending on the direction of the magnetic field relative to the direction of propagation, and on the sign of the lightest charged particles in the plasma that can respond most quickly to the electromagnetic wave.</p>
<p>In this example, the field just happened to be drawn parallel to the propagation, and normal plasma with negative electrons. In this case RCP is delated more than LHP. Had either one of those been opposite in the example, the result would have been oposite.</p>

CC BY-SA 4.0
2020-11-29T02:51:16.993Z
2020-11-29T02:51:16.993Z
uhoh
<p>I have a question related to <a href="https://en.wikipedia.org/wiki/Faraday_effect" rel="nofollow noreferrer">Faraday rotation</a>. In <a href="https://youtu.be/TNUjhwzW2Pc" rel="nofollow noreferrer">Abigail Polin's Faraday rotation video</a> I have seen that a linearly polarized wave can be decomposed in a left-handed circular wave and a right-handed circular wave and that when passing through ionized-magnetized plasma one of them gets lagged due to a difference on their refraction index.</p>
<p>Moreover, the video shows that the wave that gets lagged most is the right-hand circularly polarized.</p>
<p>Why is not the left-hand circularly polarized? What is the mathematical difference between their refraction indices?</p>

CC BY-SA 4.0
2020-11-29T00:52:44.797Z
https://youtu.be/TNUjhwzW2Pc in this Faraday Rotation video on youtube
Miguel Cárcamo
CC BY-SA 4.0
2020-11-29T04:06:10.890Z
[...and one of the beams will be slowed more than the other, causing a phase difference between the left- and right-polarized beam.](https://en.wikipedia.org/wiki/Faraday_effect#Physical_interpretation)
uhoh
CC BY-SA 4.0
2020-11-29T04:32:09.823Z
The answer to Physics SE's [Is there a simple model explaining Faraday effect?](https://physics.stackexchange.com/a/87743/83380) for example is either to read a thorough treatment in a book, or to go through a few not so simple equations. I suppose you could post a new question and cite that answer as a starting point, then explain that you're looking for *an even simpler explanation*. They are sometimes nicer in Physics SE if you describe how far you've gotten and where you're stuck.
uhoh
CC BY-SA 4.0
2020-11-29T04:34:40.070Z
But other than saying "the electrons try to respond to the Electric field of the incoming wave in a normal way by tending to radiate themselves, but as soon as they start moving the external magnetic field alters their trajectories in a way that depends on the direction fo the field." I don't think it's really easy to explain everything without equations.
uhoh
CC BY-SA 4.0
2020-11-29T04:39:18.783Z
for example there's no easy answers here: http://hyperphysics.phy-astr.gsu.edu/hbase/ligcon.html#c1 and if I'm not mistaken Feynman's Lectures on Physics doesn't cover it either. https://www.feynmanlectures.caltech.edu/II_toc.html Plasma physics is always fairly "mathy" and full of vectors, there's no easy way around it.
uhoh
<magnetic-field><galaxy-cluster><polarization>
Why does a right-handed circular polarized wave get lagged when going through ionized plasma?
Miguel Cárcamo
# Why does a right-handed circular polarized wave get lagged when going through ionized plasma?

(Asked by: username_0 on 2020-11-28)

I have a question related to [Faraday rotation](https://en.wikipedia.org/wiki/Faraday_effect). In [Abigail Polin's Faraday rotation video](https://youtu.be/TNUjhwzW2Pc) I have seen that a linearly polarized wave can be decomposed in a left-handed circular wave and a right-handed circular wave and that when passing through ionized-magnetized plasma one of them gets lagged due to a difference on their refraction index.

Moreover, the video shows that the wave that gets lagged most is the right-hand circularly polarized.

Why is not the left-hand circularly polarized? What is the mathematical difference between their refraction indices?

**Tags:** magnetic-field, galaxy-cluster, polarization
> **Comments:**
> **username_0**: https://youtu.be/TNUjhwzW2Pc in this Faraday Rotation video on youtube  
> **username_1**: [...and one of the beams will be slowed more than the other, causing a phase difference between the left- and right-polarized beam.](https://en.wikipedia.org/wiki/Faraday_effect#Physical_interpretation)  
> **username_1**: The answer to Physics SE's [Is there a simple model explaining Faraday effect?](https://physics.stackexchange.com/a/87743/83380) for example is either to read a thorough treatment in a book, or to go through a few not so simple equations. I suppose you could post a new question and cite that answer as a starting point, then explain that you're looking for *an even simpler explanation*. They are sometimes nicer in Physics SE if you describe how far you've gotten and where you're stuck.  
> **username_1**: But other than saying "the electrons try to respond to the Electric field of the incoming wave in a normal way by tending to radiate themselves, but as soon as they start moving the external magnetic field alters their trajectories in a way that depends on the direction fo the field." I don't think it's really easy to explain everything without equations.  
> **username_1**: for example there's no easy answers here: http://hyperphysics.phy-astr.gsu.edu/hbase/ligcon.html#c1 and if I'm not mistaken Feynman's Lectures on Physics doesn't cover it either. https://www.feynmanlectures.caltech.edu/II_toc.html Plasma physics is always fairly "mathy" and full of vectors, there's no easy way around it.  

---

## 1 answers

### Answer by username_1 on 2020-11-29. Score: 5 | Accepted answer

> What is the mathematical difference between their refraction indices?

The conventions on the signs of the direction of magnetic field vectors and of charges.

---

[![screen shot from Abigail Polin's Faraday Rotation video](https://i.sstatic.net/POzma.png)](https://i.sstatic.net/POzma.png)

[![screen shot from Abigail Polin's Faraday Rotation video](https://i.sstatic.net/djmZXm.png)](https://i.sstatic.net/djmZX.png)

The tutorial specifies several things that can have a sign;

1. direction of the light propagation
2. direction of the magnetic field
3. sign of the particle's charge (electrons have negative charge)

Let's look at WIkipedia's [Faraday effect; Interstellar medium](https://en.wikipedia.org/wiki/Faraday_effect#Interstellar_medium) where an equation for $RM$ or rotation measure is given:

$$RM = \frac{e^3}{2 \pi m^2 c^4} \int\_0^d n\_e(s) B\_{\parallel}(s) ds$$

This expression includes $e^3$ and $B\_{\parallel}$. Notice that the power of $e$ is odd. The assumption is that the interstellar medium is negative electrons, so they've replaced $q^3$ with $e^3$ but if it were positrons they'd have to write

$$q^3=-e^3$$.

Also remember that $B\_{\parallel}$ can be positive or negative since the field could point either way relative to the direction of propagation.

The tutorial shows some of the intermediate steps where we get the idea that both the sign of the charge and the directions of the field will be important.

[![screen shot from Abigail Polin's Faraday Rotation video](https://i.sstatic.net/L9JCZ.png)](https://i.sstatic.net/L9JCZ.png)

> Why right handed circular polarization gets lagged when going through ionized plasma?

It could be either the RCP or LCP that is delayed more than the other, depending on the direction of the magnetic field relative to the direction of propagation, and on the sign of the lightest charged particles in the plasma that can respond most quickly to the electromagnetic wave.

In this example, the field just happened to be drawn parallel to the propagation, and normal plasma with negative electrons. In this case RCP is delated more than LHP. Had either one of those been opposite in the example, the result would have been oposite.


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