EM Waves - ch. 32

ucla | PHYSICS 1C | 2023-02-13T10:50

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Table of Contents

• Definitions
• Big Ideas
  • Maxwell and Light
  • Simple Plane EM Wave
    • Gauss’ Laws
    • Faraday’s Law
    • Ampere’s Law
  • Properties of EM Waves
  • Sinusoidal EM Waves
  • EM Waves in Matter
  • Energy in EM Waves (Poynting)
  • Radiation Pressure
  • Standing EM Waves
• Resources

Definitions

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Big Ideas

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Maxwell and Light

• Maxwell proved that an EM wave should propagate in a vacuum (free space) at the speed of light → thus light is an EM wave → Maxwell Equations
• these imply an accelerating electric charge must produce EM waves e.g. power lines have AC which create EM waves → buzzing sound

• the EM spectrum - a description of the from of EM waves (visible light at 80-750 nm) based on wavelength and frequencies such that

$\lambda f=c$

Simple Plane EM Wave

• divide space s.t a plane propagates orth. (transverse) to the electric and magnetic field vectors behind it and 0 in front - wave front → resulting wave: plane wave

Gauss’ Laws

• create a volume from the wave front and behind to the origin

• there is no enclosed charge and magnetic flux through the volume is 0 → satisfies Gauss’ laws if the wave is a transverse wave moving orthogonal to $\overrightarrow{E}$ and $\overrightarrow{B}$

Faraday’s Law

• drawing a loop of height $a$ and width $\mathrm{\Delta }x$ parallel to $\overrightarrow{E}$ s.t. the wave front passes through it gives

$\oint \overrightarrow{E}\cdot d\overrightarrow{l}=-Ea$

• for the iterative magnetic flux for iterative time step $dt$ (by Faraday’s Law) is

$\frac{d{\mathrm{\Phi }}_B}{dt}=Bac⟹E=cB$

Ampere’s Law

• drawing a loop of height $a$ parallel to $\overrightarrow{B}$ s.t. that the wave front passes through it gives

$\oint \overrightarrow{B}\cdot d\overrightarrow{l}=Ba$

• Ampere’s law gives

$\frac{d{\mathrm{\Phi }}_E}{dt}=Eac⟹B={ϵ}_0{\mu }_{0}cE$

Properties of EM Waves

• EM waves are transverse (in direction $\overrightarrow{E}\times \overrightarrow{B}$)
• $E=cB⟹\frac{E}{B}=c$
• EM waves in a vacuum travel at the speed of light

$c=\frac{1}{\sqrt{{ϵ}_0{\mu }_0}}=2.99\times {10}^8$

• EM waves don’t require a medium to propagate through

Sinusoidal EM Waves

• EM waves by an oscillating point charge are an example of sinusoidal waves but for a small space far enough away, the waves can be approximately modeled by plane waves
• given and amplitude of oscillation we can find the displacement using the wavenumber $k=2\pi /\lambda$

$y(x,t)=\cos (kx-\omega t)$

• Similarly, given a sinusoidal EM wave traveling on the x-axis with electric fields on y-axis and magnetic fields of z-axis

${\overrightarrow{E}}_y(x,t)=E_{max}\cos (kx-\omega t)\hat{j}{\overrightarrow{B}}_z(x,t)=B_{max}\cos (kx-\omega t)\hat{k}$

• the characteristics are

$A\sim E_{max}=cB_{max}$

EM Waves in Matter

• EM waves can travel through vacuum and matter → when they travel through dielectric materials, the speed is not the same as the speed of light in a vacuum
• given a material with permittivity $ϵ=K{ϵ}_0$ and permeability $\mu =K_m{\mu }_0$ the wave travels with speed $v$ s.t.

$E=vBB=ϵ\mu vE$

$v=\frac{1}{\sqrt{ϵ\mu }}=\frac{1}{\sqrt{K{K}_m}}\frac{1}{\sqrt{ϵ\mu }}=\frac{c}{\sqrt{K{K}_m}}$

• the index of refraction of the material is the ratio of speed of light in vacuum to speed in the material

$n=\frac{c}{v}=\sqrt{K{K}_m}$

Energy in EM Waves (Poynting)

• The energy density $u$ in a region of space containing field vectors (given $B=E/c$) is

$u=\frac{1}{2}{ϵ}_0{E}^2+\frac{1}{2{\mu }_0}{B}^2={ϵ}_0{E}^2$

• then the energy into a volume swept out by a propagating plane wave is

$dU=udV=({ϵ}_0{E}^2)(Acdt)$

• then, the energy flow per unit time per unit area $S$ is, the vector form of this is the Poynting vector which shows the direction of energy flow rate

$S=\frac{1}{A}\frac{dU}{dt}={ϵ}_{0}c{E}^2=\frac{EB}{{\mu }_0}$

$\overrightarrow{S}=\frac{1}{{\mu }_0}\overrightarrow{E}\times \overrightarrow{B}$

• the total energy through any closed surface is

$P=\oint \overrightarrow{S}\cdot d\overrightarrow{A}$

• for sinusoidal waves with E in y-dir and B in z-dir

$\overrightarrow{S}(x,t)=\frac{1}{{\mu }_0}\overrightarrow{E}(x,t)\times \overrightarrow{B}(x,t)=\frac{E_{max}{B}_{max}}{2{\mu }_0}{\cos }^2(kx-\omega t)\hat{i}$

• the average of the magnitude of the Poynting vector is the intensity of the wave

$I=S_{avg}=\frac{E_{max}{B}_{max}}{2{\mu }_0}=\frac{1}{2}{ϵ}_{0}c{E}_{max}^2$

Radiation Pressure

• EM waves carry energy AND momentum $p$ with a density

$\frac{dp}{dV}=\frac{EB}{{\mu }_0{c}^2}=\frac{S}{c^2}$

• the momentum flow rate for the iterative volume $dV=Acdt$ s.t.

$\frac{1}{A}\frac{dp}{dt}=\frac{S}{c}=\frac{EB}{{\mu }_{0}c}$

• the momentum is responsible for the radiation pressure that can be absorbed or reflected by a surface

$p_{rad}=\frac{S_{avg}}{c}=\frac Ic\quad \text{(Totally Absorbed)}\space$

• Power

$I=\frac{P}{A}[W/m^2]$

Standing EM Waves

• a standing EM wave is a superposition of an incident and reflected wave using conductors/dielectrics as reflection surfaces s.t.

$E_y(x,t)=-2E_{max}\sin kx\space\sin\omega t=E_{max}\cos(kx-\omega t+\phi)$

• at the boundaries of the standing wave, the electric field is always 0
• the nodes for the standing wave occur at nodal planes where $\sin kx=0$ s.t.

$x=0,\frac \lambda2,\frac{2\lambda}2,\frac{3\lambda}2,…\quad \text{for $\overrightarrow{E}$}\space$

• the standing wave on distance $L$ has allowed wavelengths and frequencies

${\lambda }_n=\frac{2L}{n}{f}_n=\frac{c}{{\lambda }_n}=\frac{nc}{2L}(n\in {\text{Z}}^{+})$

Resources

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**SUMMARY
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