PHYS 208 Lecture 23

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Electromagnetic Waves

Wave equation and solution:

$\frac{{\partial }^{2}E(x,t)}{\partial x^2}={\mu }_0{ϵ}_0\frac{{\partial }^{2}E(x,t)}{\partial t^2}$

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

For waves traveling in the $+x$ direction:

$E=E_{max}\sin (\omega t-kx)\hat{ȷ}$

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

The speed of propagation is $v=c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$

Amplitudes are related by:

$E=cB$

Energy in a Wave

Energy density:

$u=u_E+u_B={ϵ}_0{E}^2$

Poynting Vector and Intensity

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

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

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

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

Example

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

$\begin{array}{rl}{S}_{av}=I& =\frac{E_{max}{B}_{max}}{2{\mu }_0}=\frac{1}{2{\mu }_{0}c}{E}_{max}^2\\ & =\frac{\mathrm{P}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}{\mathrm{A}\mathrm{r}\mathrm{e}\mathrm{a}}=\frac{P}{4\pi R^2}\\ E_{max}& =\sqrt{\frac{2P{\mu }_{0}c}{A}}=7.8\mathrm{V}/\mathrm{m}\\ B_{max}& =\frac{E_{max}}{c}=2.5\times 10^{-7}\mathrm{T}\end{array}$

Momentum and Pressure

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

$U=|\overrightarrow{p}|c$
Therefore $|\overrightarrow{p}|=\frac{U}{c}$

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

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

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

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

Speed of light in vacuum is $c$. When traveling through other materials,

$v=\frac{c}{\sqrt{k_m{k}_e}}$

Index of Refraction

$n=\frac{v}{c}=\frac{1}{\sqrt{k_m{k}_e}}$

Some measured index of refraction:

Water	$n=1.33$
Air	$n=1.0003$
Diamond	$n=2.42$

Reflection by a Plane Mirror

Angle of ray to normal when traveling toward the mirror is the same as the angle when traveling away from the mirror.

For a curved surface, take the normal of the tangent plane at the point of reflection

The image is what the brain percieves as the reflection. The object is the actual thing that is reflected.