PHYS 208 Lecture 23

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Electromagnetic Waves

Wave equation and solution:

${\frac {\partial ^{2}E(x,t)}{\partial x^{2}}}=\mu _{0}\epsilon _{0}{\frac {\partial ^{2}E(x,t)}{\partial t^{2}}}$

$f(x,t)=A\sin(kx-\omega t)\,\!$

For waves traveling in the $+x$ direction:

$E=E_{max}\sin(\omega t-kx){\hat {\jmath }}$

$B=B_{max}\sin(\omega t-kx){\hat {k}}$

The speed of propagation is $v=c={\frac {1}{\sqrt {\mu _{0}\epsilon _{0}}}}$

Amplitudes are related by:

E=cB\,\!

Energy in a Wave

Energy density:

u=u_{E}+u_{B}=\epsilon _{0}E^{2}\,\!

Poynting Vector and Intensity

"Points" along the direction of travel, with magnitude equal to the power per unit area: [W/m²]

{\vec {S}}={\frac {{\vec {E}}\times {\vec {B}}}{\mu _{0}}}

Most often work with the time-averaged poynting vector or the intensity of radiation (still same units)

$I=S_{av}={\frac {E_{max}B_{max}}{2\mu _{0}}}={\frac {1}{2}}\epsilon _{0}cE_{max}^{2}={\frac {1}{2}}{\frac {E_{max}^{2}}{\mu _{0}c}}$

Example

Power for a 100 Watt point-source light bulb: Find $E_{max}$ and $B_{max}$ at 1 meter.

${\begin{aligned}S_{av}=I&={\frac {E_{max}B_{max}}{2\mu _{0}}}={\frac {1}{2\mu _{0}c}}E_{max}^{2}\\&={\frac {\mathrm {Power} }{\mathrm {Area} }}={\frac {P}{4\pi R^{2}}}\\E_{max}&={\sqrt {\frac {2P\mu _{0}c}{A}}}=7.8\ \mathrm {V/m} \\B_{max}&={\frac {E_{max}}{c}}=2.5\times 10^{-7}\ \mathrm {T} \end{aligned}}$

Momentum and Pressure

Energy of a light "particle" (quantum; photon) = momentum × $c$

U=|{\vec {p}}|c

Therefore

|{\vec {p}}|={\frac {U}{c}}

Note:  Changing momentum is force:  $F={\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}$

Replacing energy with momentum results in the following (units af [N/m²], which is pressure)

{\frac {1}{A}}{\frac {\mathrm {d} {\vec {p}}}{\mathrm {d} t}}={\frac {1}{c}}{\frac {\mathrm {d} U}{\mathrm {d} vol}}={\frac {S}{c}}={\frac {EB}{\mu _{0}c}}

Intro to Geometrical Optics

ray: Light travels in a straight line outward from its source
wavefront: A collection of points in space that have the same phase ( $kx-\omega t$ ); From a point source, these shells are spherical and move out at the speed of light