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EM Waves and Light Wave

EM Waves and Light Wave

Electromagnetic Wave Spectrum

  • Visible Light: 400nm ~ 700nm
  • Infrared: 0.7um ~ 1mm
  • Microwave: 1mm ~ 1m
  • Radio: 1m ~ 100km
  • Ultraviolet: 10nm ~ 400nm
  • X-ray: 0.01nm ~ 10nm
  • Gamma Ray: < 0.01nm


The Properties Of Electromagnetic Wave

  • \(E \perp H\)
  • \(E\) and \(H\) are in phase
  • The direction of propagation is parallel to \(E \times H\), obey Right-Hand rule
  • The speed of electromagnetic wave is \(v = \dfrac{1}{\sqrt{\kappa_e\epsilon_0\kappa_m\mu_0}}\)


The Energy Flux Density of Electromagnetic Wave

The rate of the electromagnetic energy changing is \(\(\frac{dU}{dt} = \frac{d}{dt} \iiint \left( \frac{1}{2} \vec{D} \bullet \vec{E} + \frac{1}{2} \vec{B} \bullet \vec{H} \right) dv\)\)

Beacuse

\[ \begin{aligned} & \frac{d}{dt} \left( \frac{1}{2} \vec{D} \bullet \vec{E} + \frac{1}{2} \vec{B} \bullet \vec{H} \right) \\ =& \frac{1}{2} \kappa_e\epsilon_0 \frac{\partial}{\partial t} (\vec{E} \bullet \vec{E}) + \kappa_m\mu_0 \frac{\partial}{\partial t} (\vec{H} \bullet \vec{H}) \\ =& \kappa_e\epsilon_0 \vec{E} \bullet \frac{\partial \vec{E}}{\partial t} + \kappa_m\mu_0 \vec{H} \bullet \frac{\partial \vec{H}}{\partial t} \\ =& \vec{E} \bullet \frac{\partial \vec{D}}{\partial t} + \vec{H} \bullet \frac{\partial \vec{B}}{\partial t} \\ =& \vec{E} \bullet (\nabla \times \vec{H} - \vec{j}_0) - \vec{H} \bullet (\nabla \times \vec{E}) \\ =& \vec{E} \bullet (\nabla \times \vec{H}) - \vec{H} \bullet (\nabla \times \vec{E}) - \vec{j}_0 \bullet \vec{E} \\ =& - \nabla \bullet (\vec{E} \times \vec{H}) - \vec{j}_0 \bullet \vec{E} \\ \end{aligned} \]

So

\[ \begin{aligned} \frac{dU}{dt} &= - \iiint \nabla \bullet (\vec{E} \times \vec{H}) dv - \iiint (\vec{j}_0 \bullet \vec{E}) dv \\ &= - \oiint (\vec{E} \times \vec{H}) d\vec{A} - \iiint (\vec{j}_0 \bullet \vec{E}) dv \\ \end{aligned} \]

For the latter term, we have

\[ \begin{gathered} \vec{j}_0 = \sigma (\vec{E} + \vec{K}) \\\\ \begin{aligned} \iiint (\vec{j}_0 \bullet \vec{E}) dv &= \vec{j}_0 \bullet (\rho \vec{j}_0 - \vec{K}) \Delta A \Delta l \\ &= \rho \vec{j}_0^2 \Delta A \Delta l - (\vec{j}_0 \Delta A) (\vec{K} \Delta l) \\ &= i_0^2R - i_0 \Delta \epsilon \\ &= Q - P \end{aligned} \end{gathered} \]

For the former term, we introduce Poynting Vector \(\vec{S} = \vec{E} \times \vec{H}\). Then we have the Electromagnetic Energy Flux: \(\(\frac{dU}{dt} = - \oiint \vec{S} \bullet d\vec{A} - Q + P\)\)

Define \(Z_0 = \mu_0c = 377 \Omega\), then we have \(\(S = \frac{EB}{\mu_0} = \frac{E^2}{\mu_0c} = \frac{E^2}{Z_0}\)\)

Then the Intensity of a wave (\(W/m^2\)) is \(\(I = \left< S \right> = \frac{\left< E^2 \right>}{Z_0} = \frac{1}{2} \frac{E^2_{\max}}{377 \Omega} = \frac{E^2_{\text{rms}}}{377 \Omega}\)\)

We can also define the Intensity of a wave as average energy density times wave velocity \(\(I = c\left< u \right> = c\epsilon_0 \left< E^2 \right> = c\epsilon_0 E^2_{\text{rms}} = \frac{E^2_{\text{rms}}}{\mu_0c}\)\)


Momentum and Pressure of Radiation

For the force on the area \(\Delta A\) metal plate

\[ \begin{gathered} \Delta F \cdot c \Delta t = (S_{in} - S_{out}) \Delta A \Delta t \\\\ \Rightarrow \Delta \vec{F} = \frac{1}{c} (\vec{S}_{in} - \vec{S}_{out}) \Delta A \end{gathered} \]

So the pressure of radiation is \(\(P = \frac{|\Delta F|}{\Delta A} = \frac{1}{c} (|\vec{S}_{in}| + |\vec{S}_{out}|)\)\)

For white body (reflectivity 100%), we have \(P = \dfrac{2}{c} |\vec{S}_{in}| = \dfrac{2}{c} I\).

For black body (reflectivity 0%), we have \(P = \dfrac{1}{c} |\vec{S}_{in}| = \dfrac{1}{c} I\).

The change of momentum of EMW is

\[ \begin{gathered} \Delta G = - \Delta \vec{F} \cdot \Delta t = \frac{1}{c} (\vec{S}_{out} - \vec{S}_{in}) \Delta A \Delta t \\\\ \Rightarrow \Delta \vec{g} = \frac{\Delta G}{\Delta V} = \frac{1}{c^2} (\vec{S}_{out} - \vec{S}_{in}) \end{gathered} \]

So the momentum density of EMW is \(\(\vec{g} = \frac{1}{c^2} \vec{S} = \frac{1}{c^2} (\vec{E} \times \vec{H})\)\)


The Doppler effect for light wave

The Dopper effect for light wave is \(\(f = f_0 \frac{\sqrt{1 - \dfrac{u^2}{c^2}}}{1 + \dfrac{u}{c} \cos\theta}\)\)

where \(f_0\) is measured in the frame the source is fixed

At \(\theta = \pi/2\), the light traveling perpendicular to the relative motion of the frames, \(\(f = f_0 \sqrt{1 - \frac{u^2}{c^2}}\)\)

At \(\theta = 0\), the source is leaving (redshift), \(\(f = f_0 \sqrt{\frac{1 - u/c}{1 + u/c}}\)\)

At \(\theta = \pi\), the source is approaching (blueshift), \(\(f = f_0 \sqrt{\frac{1 + u/c}{1 - u/c}}\)\)


Shorthands

  • Electromagnetic Energy Flux
  • Intensity of a wave
  • Pressure of radiation
  • Redshift is leaving or approaching?