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


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