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: http://stackoverflow.com/questions/13716658/how-to-delete-all-commit-history-in-github.

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.