Difference between revisions of "Electromagnetic Plane Waves"

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(Created page with '===Short Topical Videos=== * [http://www.youtube.com/watch?v=xkG86pwaOH0 Plane Waves (kridnix, Bucknell U.)] ===Reference Material=== * [http://en.wikipedia.org/wiki/Plane_wave…')
 
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===Short Topical Videos===
 
* [http://www.youtube.com/watch?v=xkG86pwaOH0 Plane Waves (kridnix, Bucknell U.)]
 
 
===Reference Material===
 
* [http://en.wikipedia.org/wiki/Plane_wave Plane Waves (Wikipedia)]
 
* [http://web.mit.edu/sahughes/www/8.022/lec20.pdf Wave equation & electromagnetic radiation (Scott Hughes, MIT)]
 
 
 
===Prerequisites===
 
===Prerequisites===
 
* [[Maxwell Equations]] for E&M waves
 
* [[Maxwell Equations]] for E&M waves
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\usepackage{graphicx}
 
\usepackage{graphicx}
 
\usepackage{natbib}
 
\usepackage{natbib}
 +
 +
\def\pp#1#2{\frac{\partial {#1}}{\partial {#2}}
 +
\def\ppt#1{\pp{#1}{t}}
  
 
\begin{document}
 
\begin{document}
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\begin{eqnarray}
 
\begin{eqnarray}
\vec E &= E_0e^{i(\vec k\cdot\vec x - \omega t)}\\
+
\vec E &= \vec E_0e^{i(\vec k\cdot\vec x - \omega t)}\\
\vec B &= B_0e^{i(\vec k\cdot\vec x - \omega t)}\\
+
\vec B &= \vec B_0e^{i(\vec k\cdot\vec x - \omega t)}\\
 +
\end{eqnarray}
 +
where $E_0$ is the magnitude of the electric field, $B_0$ is the magnitude of the magnetic field, $\vec k$ is a ``wavevector'', and $\omega$ is the angular frequency.  Note that $\vec k$ is a 3D spatial analog of $\omega$ --- it encodes a direction-dependent spatial frequency, with units of inverse length.  The larger $k\equiv|\vec k|$ is, the shorter the distance you have to travel to trace out a period of the sine wave.  The directionality of $\vec k$ means that you walk through a period fastest along the direction of $\vec k$, and if you walk perpendicular to $\vec k$, you stay at the same phase.
 +
 
 +
Plane waves are useful because they are the 3D analog of sine waves.  If you take a 3D Fourier transform, you decompose any function in 3D into a sum of plane waves.  Hence, as a basis for functions in 3D, plane waves can be used to describe many things of interest, including light propagation.
 +
 
 +
\section*{Light Propagation}
 +
 
 +
Speaking of light propagation, plane waves can be useful for teasing a few more properties about electromagnetic plane waves.  Firstly, it's worth pointing out that $\vec k$ and $\omega$ can't be independent.  We know how fast light travels ($c$), so it must be true that as light propagates through the spatial distance that corresponds to a period, the time that elapses must also correspond to a period for the given frequency.  Mathematically, the spatial distance corresponding to one period is $2\pi/k$, and the time corresponding to one period is $2\pi/\omega$, so we have
 +
\begin{eqnarray}
 +
\frac{2\pi}{k}\frac1c&=\frac{2\pi}{\omega}\\
 +
ck&=\omega\\
 
\end{eqnarray}
 
\end{eqnarray}
where $E_0$ is the magnitude of the electric field, $B_0$ is the magnitude of the magnetic field, $\vec k$ is a ``wave-vector'', and $\omega$ is the angular frequency.
+
This is really just another way of saying
 +
\begin{equation}
 +
\nu\lambda=c
 +
\end{equation}
 +
Moreover, light must be propagating in the direction $\vec k$.
 +
 
 +
\subsection*{Relative Orientation of $\vec E$, $\vec B$, and $\vec k$}
 +
 
 +
Plane waves are great for working out the mathematical constraints of Maxwell's equations in free space, which we only sketched out in the section on that topic.  To begin, the $\vec E$ plane wave that we wrote down must satisfy the following equation:
 +
\begin{eqnarray}
 +
\nabla\cdot \vec E &= 0\\
 +
\nabla\cdot \vec E_0e^{i(\vec k\cdot\vec x - \omega t)}&=0\\
 +
\vec E_0\cdot (i\vec k)e^{i(\vec k\cdot\vec x - \omega t)}&=0\\
 +
\vec E_0\cdot \vec k&=0.\\
 +
Similarly, $\nabla\cdot\vec B=0$ implies that $\vec B_0\cdot\vec k=0$.  Since $\vec k$ encodes the direction of propagation, this means that the directions of the $\vec E$ and $\vec B$ fields are orthogonal to the direction of propogation.
 +
 
 +
Using another one of Maxwell's equations, we can figure out the relative orientation of $\vec E$ and $\vec B$:
 +
\begin{eqnarray}
 +
\nabla\times\vec B &= \frac1c\ppt{\vec E}\\
 +
(\pp{B_z}{y}-\pp{B_y}{z})\hat x + (\pp{B_x}{z}-\pp{B_z}{x})\hat y + (\pp{B_y}{z}-\pp{B_z}{y}) &= -\frac{\omega}{c}\vec E_0e^{i(\vec k\cdot\vec x - \omega t)}\\
 +
\end{eqnarray}
 +
 
  
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Revision as of 18:10, 21 August 2013

Prerequisites

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.