Difference between revisions of "Electromagnetic Plane Waves"

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\usepackage{graphicx}
 
\usepackage{graphicx}
 
\usepackage{natbib}
 
\usepackage{natbib}
 
+
\def\ppt#1{\frac{\partial #1}{\partial t}}
\def\pp#1#2{\frac{\partial {#1}}{\partial {#2}}
+
\def\pp#1#2{\frac{\partial #1}{\partial #2}}
\def\ppt#1{\pp{#1}{t}}
 
 
 
 
\begin{document}
 
\begin{document}
 
\title{Electromagnetic Plane Waves}
 
\title{Electromagnetic Plane Waves}
Line 45: Line 43:
 
\vec E_0\cdot (i\vec k)e^{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.\\
 
\vec E_0\cdot \vec k&=0.\\
 +
\end{eqnarray}
 
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.
 
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.
  
Line 50: Line 49:
 
\begin{eqnarray}
 
\begin{eqnarray}
 
\nabla\times\vec B &= \frac1c\ppt{\vec E}\\
 
\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)}\\
+
\left(\pp{B_z}{y}-\pp{B_y}{z}\right)\hat x + \left(\pp{B_x}{z}-\pp{B_z}{x}\right)\hat y + \left(\pp{B_y}{z}-\pp{B_z}{y}\right)\hat z &= -\frac{i\omega}{c}\vec E_0e^{i(\vec k\cdot\vec x - \omega t)}\\
 +
\left[(iB_{0,z}k_y-iB_{0,y}k_z)\hat x + (iB_{0,x}k_z-iB_{0,z}k_x)\hat y + (iB_{0,y}k_z-iB_{0,z}k_y)\hat z\right] e^{i(\vec k\cdot\vec x - \omega t)} &= -ik\vec E_0e^{i(\vec k\cdot\vec x - \omega t)}\\
 +
i(\vec B_0\times\vec k)e^{i(\vec k\cdot\vec x - \omega t)}&= -ik\vec E_0e^{i(\vec k\cdot\vec x - \omega t)}\\
 +
\vec B_0\times\vec k&= -k\vec E_0\\
 +
\vec B_0\times\hat k&= -\vec E_0\\
 
\end{eqnarray}
 
\end{eqnarray}
 
+
So that settles it.  $\vec B$ and $\vec E$ must be orthogonal, of equal magnitude ($\hat k$ is just a unit vector), and reordering some terms, we have
 +
\begin{equation}
 +
\vec E\times\vec B=|\vec E|^2\hat k
 +
\end{equation}
 +
This is called the Poynting vector (the most appropriately named vector in physics).  Crossing the E and B fields gives the energy, transferred in the direction of $\hat k$.
  
 
\end{document}
 
\end{document}
 
</latex>
 
</latex>

Revision as of 18:40, 21 August 2013

Prerequisites

Electromagnetic Plane Waves

where is the magnitude of the electric field, is the magnitude of the magnetic field, is a “wavevector”, and is the angular frequency. Note that is a 3D spatial analog of — it encodes a direction-dependent spatial frequency, with units of inverse length. The larger is, the shorter the distance you have to travel to trace out a period of the sine wave. The directionality of means that you walk through a period fastest along the direction of , and if you walk perpendicular to , 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.

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 and can’t be independent. We know how fast light travels (), 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 , and the time corresponding to one period is , so we have

This is really just another way of saying

Moreover, light must be propagating in the direction .

Relative Orientation of , , and

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 plane wave that we wrote down must satisfy the following equation:

Similarly, implies that . Since encodes the direction of propagation, this means that the directions of the and fields are orthogonal to the direction of propogation.

Using another one of Maxwell’s equations, we can figure out the relative orientation of and :

So that settles it. and must be orthogonal, of equal magnitude ( is just a unit vector), and reordering some terms, we have

This is called the Poynting vector (the most appropriately named vector in physics). Crossing the E and B fields gives the energy, transferred in the direction of .