PHYS 208 Lecture 23

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Electromagnetic Waves

Wave equation and solution:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^2 E(x,t)}{\partial x^2} = \mu_0 \epsilon_0 \frac{\partial^2 E(x,t)}{\partial t^2}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,t) = A\sin(kx-\omega t)\,\!}

For waves traveling in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +x} direction:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = E_{max} \sin(\omega t - k x) \hat\jmath}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = B_{max} \sin(\omega t - k x) \hat{k}}

The speed of propagation is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}}

Amplitudes are related by:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = cB\,\!}

Energy in a Wave

Energy density:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = S_{av} = \frac{E_{max} B_{max}}{2 \mu_0} = \frac{1}{2} \epsilon_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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{max}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{max}} at 1 meter.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{2 P \mu_0 c}{A}} = 7.8\ \mathrm{V/m} \\ B_{max} &= \frac{E_{max}}{c} = 2.5 \times 10^{-7}\ \mathrm{T} \end{align}}


Momentum and Pressure

Energy of a light "particle" (quantum; photon) = momentum × Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = |\vec{p}| c}
Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\vec{p}| = \frac{U}{c}} 
Note:  Changing momentum is force: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = \frac{\mathrm{d} \vec{p}}{\mathrm{d} t}}

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} . When traveling through other materials,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v = \frac{c}{\sqrt{k_m k_e}}}

Index of Refraction

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \frac{v}{c} = \frac{1}{\sqrt{k_m k_e}}}

Some measured index of refraction:

Water Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=1.33}
Air Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 1.0003}
Diamond Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.

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