# Electromagnetic Waves

From Maxwell’s equations it follows that $E=cB$, $B=\epsilon_0\mu_0cE$, $c=\frac1{\sqrt{\epsilon_0\mu_0}}$, where $\mu_0=4\pi\times 10^{-7}\frac Hm$ is the magnetic constant.

Sinusoidal electromagnetic waves traveling in vacuum in the $+x$-direction: $\vec{E}(x,t)=\vec{j}E_{\max}\cos(kx-\omega t)$$\vec{B}(x,t)=\vec{k}B_{\max}\cos(kx-\omega t)$, $E_{\max}=cB_{\max}$.

Electromagnetic waves in matter:  $v=\frac1{\sqrt{\epsilon\mu}}=\frac1{\sqrt{KK_m}}\frac1{\sqrt{\epsilon_0\mu_0}}=\frac{c}{\sqrt{KK_m}}$, where $\epsilon$ is the permittivity of the dielectric, $\mu$ is its permeability, $K$ is its dielectric constant, and $K_m$ is its relative permeability.

Energy flow rate (power per unit area): $\vec{S}=\frac1{\mu_0}\vec{E}\times\vec{B}$ (Poynting vector), the intensity $I=S_{av}=\frac{E_{\max}B_{\max}}{2\mu_0}=\frac{E_{\max}^2}{2\mu_0c}=\frac12\sqrt{\frac{\epsilon_0}{\mu_0}}E_{\max}^2=\frac12\epsilon_0cE_{\max}^2$.

Radiation pressure on a perpendicular surface: $p_{\mathrm{rad}}=\frac Ic$ for a totally absorbing surface, $p_{\mathrm{rad}}=\frac{2I}c$ for a perfect reflector.

Flow rate of electromagnetic momentum:  $\frac1A\frac{dp}{dt}=\frac Sc=\frac{EB}{\mu_0c}$.

Standing electromagnetic waves:  If a perfect reflecting surface is placed at $x=0$, the incident and reflected waves form a standing wave.  Nodal planes for $\vec{E}$ occur at $kx=0,\pi,2\pi,\ldots$, and nodal planes for $\vec{B}$ are at $kx=\frac{\pi}2,\frac{3\pi}2,\frac{5\pi}2,\ldots$