Geometric optics
law of reflection: $\theta _1=\theta _2$
law of refraction: $n_1\sin \theta _1=n_2\sin \theta _2$
total internal refraction, critical angle: $\theta _c=\sin^{-1}(\frac{n_2}{n_1})$ (incident from greater n(1))
Brewster angle: $\theta =\tan^{-1}(\frac{n_2}{n_1})$ (incident from n1)
spherical mirror (Real side is where reflected)
focus: $f=\frac{r}{2}$ (+: concave, -: convex)
relationship of object, image distance: $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$ (+: real, R side, upright; -: virtual, V side, inverted) (p is +)
lateral magnification: $|m|=\frac{h_{\textrm{image}}}{h_{\textrm{obj}}}$, $m=-\frac{i}{p}$ (+: same orientation; -: opposite)
spherical refracting surface (Real side is where refracted)
relationship: $\frac{n_1}{p}+\frac{n_2}{i}=\frac{n_2-n_1}{r}$ (p is +)
thin lens
relation 1: $\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$
relation 2: $\frac{1}{f}=(n-1)(\frac{1}{r_1}-\frac{1}{r_2})$ (n=nlens/nmedium, r1: first side light goes thru)
angular magnification, simple magnifer: $m_\theta =\frac{15\textrm{ cm}}{f}$
angular magnification, refracting telescope: $m_\theta =-\frac{f_{ob}}{f_{ey}}$
magnification, compound microscope: $M=-\frac{s}{f_{ob}}\frac{15\textrm{ cm}}{f_{ey}}$ (s: distance of two focuses)
Interference and diffraction
index of refraction: $n=\frac{c}{v}$
wavelength in medium: $\lambda _n=\frac{\lambda }{n}$
two mediums, same light, # of wavelength difference: $N_2-N_1=\frac{L}{\lambda }(n_2-n_1)$
double-slit interference
fully constructive: $d\sin\theta =m\lambda,m=0,1,2,...$
fully destructive: $d\sin\theta =(m+\frac{1}{2})\lambda,m=0,1,2,...$
illumination intensity: $I=4I_0\cos ^2(\frac{1}{2}\phi)$, $\phi=\frac{2 \pi d}{\lambda} \sin \theta $ (I0: intensity of one slit when the ohter covered, d: seperation of slits), $\bar{I}=2I_0$
thin film, n1,n3>n2 (incident at n1) (every larger n of refraction side causes phase change of λ/2)
fully constructive: $2n_2L=(m+\frac{1}{2})\lambda ,m=0,1,2,...$
fully destructive: $2n_2L=m\lambda ,m=0,1,2,...$
single-slit diffraction
intensity minima: $a\sin \theta =m\lambda,m=1,2,3,...$ (maximum Im at centre)
intensity: $I=I_m(\frac{\sin \alpha }{\alpha })^2$, $\alpha =(\frac{\pi a }{\lambda })\sin \theta $ (a: width)
minimum angle to distinguish two sources/Rayleigh's criteria: $\theta _R=1.22\frac{\lambda }{d}$ (d: len's diameter)
seperation of two sources: $\Delta x\approx f\theta $ (f: may be viewing distance)
real double-slit
intensity: $I=I_m\underbrace{(\cos ^2 \beta )}_{\textrm{intfr}}\underbrace{(\frac{\sin \alpha }{\alpha })^2}_{\textrm{diffr}}$, $\beta =(\frac{\pi d}{\lambda })\sin \theta $, $\alpha =(\frac{\pi a}{\lambda})\sin \theta $
multiple slits (N slits)
grating maxima: $d\sin\theta =m\lambda,m=0,1,2,...$
line width: $\Delta \theta =\frac{\lambda }{Nd\cos \theta }$
dispersion/seperation of lines: $D=\frac{\Delta \theta }{\Delta \lambda }=\frac{m}{d\cos \theta }$
resolving power: $R=Nm=\frac{\bar{\lambda} }{\Delta \lambda }$
Special relativity
speed parametre: $\beta =v/c$
Lorentz factor: $\gamma =\frac{1}{\sqrt{1-\beta ^2}}$
time dilation: $\Delta t =\gamma \Delta t _0$
length contraction: $L=\frac{L_0}{\gamma }$
Lorentz transformation (S, S'):
$\left\{\begin{array}{l}
x'=\gamma (x-vt)\\ y'=y \\ z'=z \\ t'=\gamma (t-vx/c^2)
\end{array}\right.$, $\left\{\begin{array}{l}
x=\gamma (x'+vt')\\ y=y' \\ z=z' \\ t=\gamma (t'+vx'/c^2)
\end{array}\right.$
$\Delta x=\gamma (\Delta x'+v\Delta t')$, $\Delta x'=\gamma (\Delta x-v\Delta t)$
$\Delta t=\gamma (\Delta t'+v\Delta x'/c^2)$, $\Delta t'=\gamma (\Delta t-v\Delta x/c^2)$
relativistic velocity law: $v=\frac{v'+u}{1+uv'/c^2}$ (v: obj in S, v': obj in S', u: S' in S)
Doppler effect: $f=f_0\sqrt{\frac{1\pm \beta }{1\mp \beta }}$
relativistic momentum: $\mathbf{p}=\gamma m\mathbf{v}$
relativistic kinetic energy: $K=mc^2(\gamma -1)$
total energy: $E=\gamma mc^2=\underbrace{mc^2}_{\textrm{rest E}}+\underbrace{K}_{\textrm{kinetic E}}$
relations: $(pc)^2=K^2+2Kmc^2$, $E^2=(pc)^2+(mc^2)^2$