Electromagnetic Fields and Waves

${\vec{E}}_B={\vec{v}}_{BA}\times {\vec{B}}_A$

Speed of light: $v_{em}=c=\frac{1}{\sqrt{{ϵ}_0{\mu }_0}}$

Field Transformations

Fields measured in frame A to be ${\vec{E}}_A$ and ${\vec{B}}_A$ are found in frame B to be

${\vec{E}}_B={\vec{E}}_A+{\vec{v}}_{BA}\times {\vec{B}}_A$

$\vec{B}={\vec{B}}_A-\frac{1}{c^2}{\vec{v}}_{BA}\times {\vec{E}}_A$

Maxwell's Equations

Other Equations

Lorentz force law: $\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$

Wave equation: $\frac{{\partial }^2{E}_y}{\partial t^2}=\frac{1}{{ϵ}_0{\mu }_0}\frac{{\partial }^2{E}_y}{\partial x^2}$

Poynting vector: $\vec{S}=\frac{1}{{\mu }_0}\vec{E}\times \vec{B}$

Wave intensity: $I=\frac{c{ϵ}_0}{2}{E}_0^2$

Classical intensity spread: $E=\frac{P_{source}}{4\pi r^2}$

Radiation Pressure $\Delta p=\frac{\text{energy absorbed}}{c}=\frac{I}{c}$

Incident light polarized: $I_{transmitted}=I_0{\cos }^2(\theta )$

Unpolarized light transmitted: $I_{transmitted}=\frac{1}{2}{I}_0{\cos }^2(\theta )$

Electric Field/Magnetic Field relationship: $E=cB$

Light relationship: $c=\lambda v$

Constants

Elementary Charge: $e=1.602\times 10^{-19}C$

Mass of Electron: $m=9.109\times 10^{-31}kg$

Mass of Proton: $m=1.672\times 10^{-27}$

Vacuum Permittivity: ${ϵ}_0=8.854\times 10^{-12}$

Coulomb Constant $k=9.0\times 10^9\frac{N{m}^2}{C^2}$

Vacuum Permittivity Magnetism: ${\mu }_0=4\pi \times 10^{-7}$

Interference

Double Slit

Angles of bright fringes: ${\theta }_m=m\frac{\lambda }{d}m=0,1,2,3,...$

Position of fringes: $y_m=\frac{m\lambda L}{d}m=0,1,2,3,...$

Ideal double slit pattern: $I_{double}=4I_1{\cos }^2(\frac{\pi d}{\lambda L}y)$

Complete intensity: $I_{double}=I_0(\frac{\sin (\pi ay/\lambda L)}{\pi ay/\lambda L}{)}^2{\cos }^2(\pi dy/\lambda L)$

Diffraction Grating

Bright and narrow fringes are at:

$d\sin (\theta {)}_m=m\lambda$

$y_m=L\tan (\theta {)}_m$

Single Slit

Intensity as a function of y:

$I=I_0\frac{{\sin }^2(\frac{\pi ay}{\lambda D})}{(\frac{\pi ay}{\lambda D}{)}^2}$

$I_{slit}=I_0(\frac{\sin (\pi a\sin (\theta /\lambda ))}{\pi a\sin (\theta /\lambda )}{)}^2$

Dark fringes: ${\theta }_p=p\frac{\lambda }{a}p=1,2,3,...$

Positions of dark fringes: $y_p=\frac{p\lambda L}{a}p=1,2,3,...$

Width of central maximum: $w=\frac{2\lambda L}{a}$

Ray Optics

$\text{Angle of incidence}=\text{Angle of reflection}$

Pinhole Camera Relationship

$\frac{h_i}{h_0}=\frac{d_i}{d_0}$

Where i is image or box dimensions.

Snell's law of refraction

$n_1\sin ({\theta }_1)=n_{2}sin({\theta }_2)$

Where $\theta$ is angle from normal and $n=\frac{c}{v_{medium}}$.

Critical angle for total internal reflection: ${\theta }_c={\sin }^{-1}(\frac{n_2}{n_1})$

Thin Lens

$\frac{1}{s}+\frac{1}{s^{\prime }}=\frac{1}{f}$

s is the length to lens, s' is the length to image.

Focal length equation

$\frac{1}{f}=(n-1)(\frac{1}{R_1}-\frac{1}{R_2})$

Magnification

$m=-\frac{s^{\prime }}{s}$

(m is + for upright)