The Nature and Propagation of Light

Light and its properties.  Light is an electromagnetic wave.  When emitted or absorbed, it also shows particle properties.  It is emitted by accelerated electric charges.

A wave front is a surface of constant phase; wave fronts move with a speed equal to the propagation speed of the wave.  A ray is a line along the direction of propagation, perpendicular to the wave fronts.

When light is transmitted from one material to another, the frequency of the light is unchanged, but the wavelength and the wave speed can change.  The index of refraction of a material n=\frac cv, \lambda=\frac{\lambda_0}n.

Reflection and refraction.  \theta_r=\theta_a (law of reflection), n_a\sin\theta_a=n_b\sin\theta_b (law of refraction).

Total internal reflection.  When a ray travels in a material of index of refraction n_a toward a material of index n_b<n_a, total internal reflection occurs at the interface when the angle of incidence equals or exceeds a critical angle \theta_{\mathrm{crit}}, \sin\theta_{\mathrm{crit}}=\frac{n_b}{n_a}.

Polarization of light.  The direction of polarization of a linearly polarized electromagnetic wave is the direction of the \vec{E} field.

Malus’s law.  When polarized light of intensity I_{\max} is incident on a polarizing filter used as an analyzer, I=I_{\max}\cos^2\phi, I is intensity of the light transmitted through the analyzer, \phi is the angle between the polarization direction of the incident light and the polarizing axis of the analyzer.

Polarization by reflection.  When unpolarized light strikes an interface between two materials, Brewster’s law states that the reflected light is completely polarized perpendicular to the plane of incidence (parallel to the interface) if the angle of incidence is \theta_p=\arctan\frac{n_b}{n_a}.

Huygens’s principle.  If the position of a wave front at one instant is known, then the position of the front at a later time can be constructed by imagining the front as a source of secondary wavelets.

Электромагнитные волны

Поток энергии (energy flux): количество энергии, переносимое через некоторую произвольную площадку в единицу времени, \Pi=\frac{dW}{dt}.

Плотность потока энергии (energy flow density/rate, power per unit area): физическая величина, численно равная потоку энергии через малую площадку единичной площади, перпендикулярную направлению потока, J=\frac{d^2W}{dt\,dS}.

Магнитная индукция (magnetic field) \vec{B}: сила Лоренца \vec{F}, действующая со стороны магнитного поля на заряд q, движущийся со скоростью \vec{v}: \vec{F}=q[\vec{v}\times\vec{B}].

Интенсивность (intensity): скалярная физическая величина, количественно характеризующая мощность, переносимую волной в направлении распространения. Численно интенсивность равна усреднённой за период колебаний волны мощности излучения, проходящей через единичную площадку, расположенную перпендикулярно направлению распространения энергии.

Плотность импульса (momentum density) электромагнитной волны: \frac{dp}{dV}=\frac{EB}{\mu_0c^2}=\frac{S}{c^2}.

Плотность потока импульса (flow rate of electromagnetic momentum): \frac1A\frac{dp}{dt}=\frac Sc=\frac{EB}{\mu_0c}.

Electromagnetic Waves

From Maxwell’s equations it follows that E=cB, B=\epsilon_0\mu_0cE, c=\frac1{\sqrt{\epsilon_0\mu_0}}, where \mu_0=4\pi\times 10^{-7}\frac Hm is the magnetic constant.

Sinusoidal electromagnetic waves traveling in vacuum in the +x-direction: \vec{E}(x,t)=\vec{j}E_{\max}\cos(kx-\omega t)\vec{B}(x,t)=\vec{k}B_{\max}\cos(kx-\omega t), E_{\max}=cB_{\max}.

Electromagnetic waves in matter:  v=\frac1{\sqrt{\epsilon\mu}}=\frac1{\sqrt{KK_m}}\frac1{\sqrt{\epsilon_0\mu_0}}=\frac{c}{\sqrt{KK_m}}, where \epsilon is the permittivity of the dielectric, \mu is its permeability, K is its dielectric constant, and K_m is its relative permeability.

Energy flow rate (power per unit area): \vec{S}=\frac1{\mu_0}\vec{E}\times\vec{B} (Poynting vector), the intensity I=S_{av}=\frac{E_{\max}B_{\max}}{2\mu_0}=\frac{E_{\max}^2}{2\mu_0c}=\frac12\sqrt{\frac{\epsilon_0}{\mu_0}}E_{\max}^2=\frac12\epsilon_0cE_{\max}^2.

Radiation pressure on a perpendicular surface: p_{\mathrm{rad}}=\frac Ic for a totally absorbing surface, p_{\mathrm{rad}}=\frac{2I}c for a perfect reflector.

Flow rate of electromagnetic momentum:  \frac1A\frac{dp}{dt}=\frac Sc=\frac{EB}{\mu_0c}.

Standing electromagnetic waves:  If a perfect reflecting surface is placed at x=0, the incident and reflected waves form a standing wave.  Nodal planes for \vec{E} occur at kx=0,\pi,2\pi,\ldots, and nodal planes for \vec{B} are at kx=\frac{\pi}2,\frac{3\pi}2,\frac{5\pi}2,\ldots

How to Remove Git History

  1. Checkout
    git checkout --orphan latest_branch
  2. Add all the files
    git add -A
  3. Commit the changes
    git commit -am "commit message"
  4. Delete the branch
    git branch -D master
  5. Rename the current branch to master
    git branch -m master
  6. Finally, force update your repository
    git push -f origin master

From here:

Alternating Current

Voltage, current, and phase angle.  In general, the instantaneous voltage v=V\cos(\omega t+\phi) between two points in an ac circuit is not in phase with the instantaneous current i=I\cos\omega t passing through those points.

Resistance and reactance.  The voltage across a resistor is in phase with the current, V_R=IR.  The voltage across an inductor leads the the current by \frac\pi 2, V_L=IX_L, inductive reactance X_L=\omega L.  The voltage across a capacitor lags the the current by \frac\pi 2, V_C=IX_C, capacitive reactance X_C=\frac1{\omega C}.

Impedance and the L-R-C series circuit.  In general ac circuit, the voltage and current amplitutes are related by the circuit impedance Z, V=IZ.  In an L-R-C series circuit, Z=\sqrt{R^2+(\omega L-\frac1{\omega C})^2}, \tan\phi=\frac{\omega L-\frac1{\omega C}}R.

Power in ac circuits.  The average power input to an ac circuit: P_{av}=\frac12VI\cos\phi=V_{\mathrm{rms}}I_{\mathrm{rms}}\cos\phi, where \phi is the phase angle of the voltage relative to the current.  The factor \cos\phi is called the power factor of the circuit.

Resonance angular frequencey.  \omega_0=\frac1{\sqrt{LC}}.

Transformers.  \frac{V_2}{V_1}=\frac{N_2}{N_1}, V_1I_1=V_2I_2.