Formulas

• (doc) conditions.dynamics.equilibrium

  • (doc) Total torque is zero

    \[\tau = 0\]

• (doc) conditions.thermodynamics.dielectrics

  • (doc) Equation of state

    \[D = f{\left(E,T,\rho \right)}\]

• (doc) definitions

  • (doc) Acceleration is speed derivative

    \[a{\left(t \right)} = \frac{d}{d t} v{\left(t \right)}\]

  • (doc) Admittance is inverse impedance

    \[Y = \frac{1}{Z}\]

  • (doc) Angular acceleration is angular speed derivative

    \[\alpha{\left(t \right)} = \frac{d}{d t} \omega{\left(t \right)}\]

  • (doc) Angular speed is angular distance derivative

    \[\omega{\left(t \right)} = \frac{d}{d t} \theta{\left(t \right)}\]

  • (doc) Angular wavenumber is inverse wavelength

    \[k = \frac{2 \pi}{\lambda}\]

  • (doc) Boltzmann factor via state energy and temperature

    \[f = \exp{\left(- \frac{E_{i}}{k_\text{B} T} \right)}\]

  • (doc) Compliance is inverse stiffness

    \[c = \frac{1}{k}\]

  • (doc) Compressibility factor is deviation from ideal gas

    \[Z = \frac{p V}{n R T}\]

  • (doc) Current is charge derivative

    \[I{\left(t \right)} = \frac{d}{d t} q{\left(t \right)}\]

  • (doc) Damped harmonic oscillator equation

    \[\frac{d^{2}}{d t^{2}} x{\left(t \right)} + 2 \zeta \omega \frac{d}{d t} x{\left(t \right)} + \omega^{2} x{\left(t \right)} = 0\]

  • (doc) Density from mass and volume

    \[\rho = \frac{m}{V}\]

  • (doc) Electrical conductance is inverse resistance

    \[G = \frac{1}{R}\]

  • (doc) Harmonic oscillator is a second order derivative equation

    \[\frac{d^{2}}{d t^{2}} x{\left(t \right)} = - \omega^{2} x{\left(t \right)}\]

  • (doc) Heat capacity ratio

    \[\gamma = \frac{C_{p}}{C_{V}}\]

  • (doc) Impedance is resistance and reactance

    \[Z = R + i X\]

  • (doc) Impulse is integral of force over time

    \[J = \int\limits_{t_{0}}^{t_{1}} F{\left(t \right)}\, dt\]

  • (doc) Intensity of sound wave is rate of energy transfer over area

    \[I = \frac{P}{A}\]

  • (doc) Linear coefficient of thermal expansion

    \[\alpha_{l} = \frac{\frac{\partial}{\partial T} l{\left(T,p \right)}}{l{\left(T,p \right)}}\]

  • (doc) Lorentz factor

    \[\gamma = \frac{1}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]

  • (doc) Mass flow rate

    \[\mu{\left(t \right)} = \frac{d}{d t} m{\left(t \right)}\]

  • (doc) Mass fraction of mixture component

    \[w_{i} = \frac{m_{i}}{m}\]

  • (doc) Mechanical energy is kinetic and potential energy

    \[E = K + U\]

  • (doc) Momentum is mass times speed

    \[p = m v\]

  • (doc) Net force is sum of individual forces

    \[F = \sum_i {F}_{i}\]

  • (doc) Number density is number of objects per unit volume

    \[n = \frac{N}{V}\]

  • (doc) Period from angular frequency

    \[T = \frac{2 \pi}{\omega}\]

  • (doc) Power is energy derivative

    \[P{\left(t \right)} = \frac{d}{d t} E{\left(t \right)}\]

  • (doc) Quality factor is ratio of energies

    \[Q = \frac{\omega E}{P}\]

  • (doc) Radiant exitance is radiant flux emitted per unit area

    \[M_\text{e} = \frac{d}{d A} \Phi_\text{e}{\left(A \right)}\]

  • (doc) Relative refractive index is ratio of wave speeds

    \[n = \frac{v_\text{incident}}{v_\text{refracted}}\]

  • (doc) Rotational inertia is mass times squared radius

    \[I = m r^{2}\]

  • (doc) Sound level in decibels

    \[L_{I} = L_{I0} \log_{10} \left( \frac{I}{I_0} \right)\]

  • (doc) Speed is distance derivative

    \[v{\left(t \right)} = \frac{d}{d t} s{\left(t \right)}\]

  • (doc) Temporal frequency from period

    \[f = \frac{1}{T}\]

  • (doc) Temporal frequency is number of events per unit time

    \[f = \frac{N}{t}\]

  • (doc) Thermal de Broglie wavelength

    \[\lambda = \hbar \sqrt{\frac{2 \pi}{m k_\text{B} T}}\]

  • (doc) Thermal resistance to conduction

    \[R_\text{val} = \frac{h}{k}\]

  • (doc) Thermodynamic compressibility

    \[\beta = - \frac{\frac{\partial}{\partial p} V{\left(p,q \right)}}{V{\left(p,q \right)}}\]

  • (doc) Volumetric coefficient of thermal expansion

    \[\alpha_{V} = \frac{\frac{\partial}{\partial T} V{\left(T,q \right)}}{V{\left(T,q \right)}}\]

  • (doc) Wave equation in one dimension

    \[\frac{\partial^{2}}{\partial x^{2}} u{\left(x,t \right)} = \frac{\frac{\partial^{2}}{\partial t^{2}} u{\left(x,t \right)}}{v^{2}}\]

• (doc) definitions.vector

  • (doc) Acceleration is velocity derivative

    \[{\vec a} \left( t \right) = \frac{d}{d t} {\vec v} \left( t \right)\]

  • (doc) Angular momentum is position cross linear momentum

    \[{\vec L} = \left[ {\vec r}, {\vec p} \right]\]

  • (doc) Damping force is proportional to velocity

    \[{\vec F} = - b {\vec v}\]

  • (doc) Momentum is mass times velocity (Vector)

    \[{\vec p} = m {\vec v}\]

  • (doc) Net force vector is sum of forces

    \[{\vec F} = \sum_i {{\vec F}}_{i}\]

  • (doc) Vector area is unit normal times scalar area

    \[{\vec A} = {\vec n} A\]

  • (doc) Velocity is position vector derivative

    \[{\vec v} \left( t \right) = \frac{d}{d t} {\vec d} \left( t \right)\]

• (doc) laws.astronomy

  • (doc) Absolute magnitude from apparent magnitude and distance

    \[M = m - 5 \log_{10} \left( \frac{d}{d_0} \right)\]

  • (doc) Angular altitude in upper culmination

    \[h = 90^\circ - \phi + \delta\]

  • (doc) Approximate lifetime of stars located on the main sequence

    \[t = t_\odot \frac{m}{M_\odot} \frac{L_\odot}{L}\]

  • (doc) Change in apparent magnitude from distance

    \[m_{2} - m_{1} = - 2.5 \log_{10} \left( \frac{E_{\text{e}2}}{E_{\text{e}1}} \right)\]

  • (doc) Latitude from zenith angle and declination

    \[\phi = \frac{\theta_\text{S} - \theta_\text{N} + \delta_\text{S} + \delta_\text{N}}{2}\]

  • (doc) Lifetime of star on main sequence

    \[t = 10 \, \text{Gyr} \left(\frac{m}{M_\odot}\right)^{1 - n}\]

  • (doc) Luminosity of star from absolute magnitude

    \[\log_{10} \left( \frac{L}{L_0} \right) = - 0.4 M\]

  • (doc) Luminosity of Sun in future from luminosity in present

    \[L_{1} = L \left(\frac{5.59}{\frac{t}{1 \, \text{Gyr}}} - 1.41 + 0.26 \frac{t}{1 \, \text{Gyr}}\right)\]

  • (doc) Luminosity of Sun in past from luminosity in present

    \[L_{0} = \frac{L}{1 + 0.4 \left(1 - \frac{t}{1 \, \text{Gyr}} \frac{1}{4.6}\right)}\]

  • (doc) Radius of planetary orbits from number

    \[r = a + b 2^{N}\]

  • (doc) Ratio of luminosities from ratio of masses of stars

    \[\frac{L_{2}}{L_{1}} = \left(\frac{m_{2}}{m_{1}}\right)^{4}\]

  • (doc) Speed of galaxy from distance to galaxy

    \[v = H d\]

• (doc) laws.astronomy.relativistic

  • (doc) Relative rocket speed from mass change and effective exhaust speed

    \[\frac{m_{1}}{m_{0}} = \left(\frac{1 - \frac{v}{c}}{1 + \frac{v}{c}}\right)^{\frac{c}{2 v_\text{e}}}\]

• (doc) laws.chemistry

  • (doc) Avogadro constant is particle count over amount of substance

    \[N_\text{A} = \frac{N}{n}\]

  • (doc) Boundary of thermalization zone of atomized atoms in magnetron

    \[l = N \lambda\]

  • (doc) Distance of greatest convergence of particles in magnetron

    \[d = - d_0 \left(Z_{1}^{0.0387} + Z_{2}^{0.0387}\right) \log \left( \frac{V}{V_0 \left(Z_{1} Z_{2}\right)^{1.4883}} \right)\]

  • (doc) Electron distribution function in gas plasma per Druyvestein

    \[f = \frac{E_0 \sqrt{e V}}{E^{\frac{3}{2}}} \exp{\left(- \frac{0.55 \left(e V\right)^{2}}{E^{2}} \right)}\]

  • (doc) Electron distribution function in gas plasma per Maxwell

    \[f = \frac{E_0 \sqrt{e V}}{E^{\frac{3}{2}}} \exp{\left(- \frac{1.55 e V}{E} \right)}\]

  • (doc) Electron current in probe circuit in gas plasma

    \[I = 0.25 A e n \sqrt{\frac{8 k_\text{B} T}{\pi m_\text{e}}} \exp{\left(- \frac{e \left(U_\text{f} - U_{\vec E}\right)}{k_\text{B} T} \right)}\]

  • (doc) Energy transfer coefficient for elastic scattering in magnetron

    \[x = \frac{2 m_{1} m_{2}}{\left(m_{1} + m_{2}\right)^{2}}\]

  • (doc) Etch rate of target in magnetron

    \[v = \frac{j M Y}{e \rho N_\text{A}}\]

  • (doc) Interaction cross section in Coulomb’s interaction model

    \[\sigma = \frac{e^{2}}{2 \pi \varepsilon_0^{2} E_\text{i}^{2}}\]

  • (doc) Interaction cross section in elastic interaction model

    \[\sigma = \pi D^{2} \left(1 + \frac{S}{T}\right)\]

  • (doc) Interaction cross section in model of hard spheres

    \[\sigma = \pi d^{2}\]

  • (doc) Interaction cross section in recharge model

    \[\sigma = \pi a_0^{2} \frac{\mathrm{IE}_\text{H}}{E_\text{i}} \log \left( \sqrt{\frac{3 k_\text{B} T}{m}} \sqrt{\frac{E_\text{i}}{\mathrm{IE}_\text{H}}} \frac{\sigma p m}{2 k_\text{B} T e E} \right)^{2}\]

  • (doc) Ionization cross section of atom by electrons per Granovsky

    \[\sigma_\text{eff} = \sigma_\text{max} \frac{E - \sigma_\text{i}}{E_\text{max} - \sigma_\text{i}} \exp{\left(\frac{E_\text{max} - E}{E_\text{max} - \sigma_\text{i}} \right)}\]

  • (doc) Ionization cross section of atom by electrons per Lotz-Drevin

    \[\sigma = \frac{2.66 \pi a_0^{2} N \mathrm{IE}_\text{H}^{2}}{E_\text{i}^{2}} \frac{A \left(\frac{E}{E_\text{i}} - 1\right)}{\left(\frac{E}{E_\text{i}}\right)^{2}} \log \left( 1.25 B \frac{E}{E_\text{i}} \right)\]

  • (doc) Mass of film deposited during electrolysis

    \[m = \frac{I M B t}{v \mathfrak{F}}\]

  • (doc) Mean free path of particles in gaseous medium

    \[\lambda = \frac{k_\text{B} T}{\sqrt{2} p \sigma}\]

  • (doc) Molar mass via molecular mass

    \[M = m_{0} N_\text{A}\]

  • (doc) Number density via volumetric density and molar mass

    \[n = \frac{\rho N_\text{A}}{M}\]

  • (doc) Number of collisions of particle with gas in magnetron

    \[N = \frac{\log \left( \frac{E}{E_{0}} \right)}{\log \left( 1 - x \right)}\]

  • (doc) Reaction equilibrium constant via standard Gibbs energy

    \[K = \exp{\left(- \frac{\Delta G}{R T} \right)}\]

  • (doc) Speed of charged particles in gas via mobility

    \[v = \frac{\mu E}{p}\]

  • (doc) Volumetric ionization coefficient of neutral particles by electrons

    \[\alpha = A p \exp{\left(- \frac{B p}{E} \right)}\]

• (doc) laws.chemistry.electrochemistry

  • (doc) Electrochemical equivalent from molar mass and valence

    \[Z = \frac{M}{\mathfrak{F} v}\]

  • (doc) Energy of electron in hydrogen atom per Bohr

    \[E = \frac{e^{2}}{8 \pi \varepsilon_0 r}\]

  • (doc) Mass of substance deposited on electrode

    \[m = Z I t\]

• (doc) laws.chemistry.potential_energy_models

  • (doc) Lennard-Jones potential

    \[U = 4 \varepsilon \left(\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right)\]

• (doc) laws.condensed_matter

  • (doc) Concentration of intrinsic charge carriers

    \[n = \sqrt{N_\text{c} N_\text{v}} \exp{\left(- \frac{E_\text{g}}{2 k_\text{B} T} \right)}\]

  • (doc) Current density from mobility

    \[j = e \left(- n_\text{e} \mu_\text{e} + n_\text{h} \mu_\text{h}\right) E\]

  • (doc) Current density in thermionic emission per Richardson

    \[j = a T^{2} \exp{\left(- \frac{W}{k_\text{B} T} \right)}\]

  • (doc) Current density via number density and drift velocity

    \[j = q n u\]

  • (doc) Diffusion coefficient from energy and temperature

    \[D = D_{0} \exp{\left(- \frac{E_\text{A}}{k_\text{B} T} \right)}\]

  • (doc) Drift velocity of charge carriers

    \[u = \mu E\]

  • (doc) Effective mass of electron via energy

    \[m_\text{eff} = \frac{\hbar^{2}}{\frac{d^{2}}{d k^{2}} E{\left(k \right)}}\]

  • (doc) Equilibrium voltage difference in p-n junction via concentrations

    \[\Delta V = \frac{k_\text{B} T}{q} \log \left( \frac{n_\text{d} n_\text{a}}{n^{2}} \right)\]

  • (doc) Resistance from temperature

    \[R = R_{0} \left(1 + a \left(T - T_\text{std}\right)\right)\]

• (doc) laws.conservation

  • (doc) Abbe invariant of two optical environments is constant

    \[n_{0} \left(\frac{1}{d_\text{o}} - \frac{1}{r}\right) = n \left(\frac{1}{d_\text{i}} - \frac{1}{r}\right)\]

  • (doc) Amount of mixture is sum of amounts of components

    \[n = \sum_i {n}_{i}\]

  • (doc) Charge is constant

    \[\frac{d}{d t} q{\left(t \right)} = 0\]

  • (doc) Initial mass equals final mass

    \[m{\left(t_{1} \right)} = m{\left(t_{0} \right)}\]

  • (doc) Initial mechanical energy equals final mechanical energy

    \[E{\left(t_{1} \right)} = E{\left(t_{0} \right)}\]

  • (doc) Initial momentum equals final momentum

    \[p{\left(t_{1} \right)} = p{\left(t_{0} \right)}\]

  • (doc) Mass is constant

    \[\frac{d}{d t} m{\left(t \right)} = 0\]

  • (doc) Mechanical energy is constant

    \[\frac{d}{d t} E{\left(t \right)} = 0\]

  • (doc) Mixture mass is sum of component masses

    \[m = \sum_i {m}_{i}\]

  • (doc) Momentum is constant

    \[\frac{d}{d t} p{\left(t \right)} = 0\]

• (doc) laws.dynamics

  • (doc) Acceleration is force over mass

    \[a = \frac{F}{m}\]

  • (doc) Braking path via speed and friction force

    \[s = \frac{m v^{2}}{2 F_\text{fr}}\]

  • (doc) Buoyant force from density and volume

    \[F_\text{A} = \rho g V\]

  • (doc) Coefficient of stiffness from area and length

    \[k = \frac{E A}{l}\]

  • (doc) Displacement in forced non-resonant oscillations

    \[q{\left(t \right)} = \frac{\frac{F}{m} \cos{\left(\omega t + \varphi \right)}}{\omega_{0}^{2} - \omega^{2}}\]

  • (doc) Force is derivative of momentum

    \[\frac{d}{d t} p{\left(t \right)} = F{\left(t \right)}\]

  • (doc) Forced oscillations equation

    \[\frac{d^{2}}{d t^{2}} x{\left(t \right)} + \omega_{0}^{2} x{\left(t \right)} = \frac{F}{m} \cos{\left(\omega t + \varphi \right)}\]

  • (doc) Friction force from normal force

    \[F_\text{fr} = \mu N\]

  • (doc) Instantaneous power is force times speed

    \[P = F v \cos{\left(\varphi \right)}\]

  • (doc) Kinetic energy from mass and speed

    \[K = \frac{m v^{2}}{2}\]

  • (doc) Kinetic energy from rotational inertia and angular speed

    \[K = \frac{I \omega^{2}}{2}\]

  • (doc) Kinetic energy via momentum

    \[K = \frac{p^{2}}{2 m}\]

  • (doc) Maximum height from initial speed

    \[h = \frac{v^{2}}{2 g}\]

  • (doc) Mechanical work is force times distance

    \[W = F s\]

  • (doc) Momentum derivative of kinetic energy is speed

    \[\frac{d}{d p{\left(v \right)}} K{\left(p{\left(v \right)} \right)} = v\]

  • (doc) Period of ideal pendulum from length

    \[T = 2 \pi \sqrt{\frac{l}{g}}\]

  • (doc) Period of physical pendulum

    \[T = 2 \pi \sqrt{\frac{I}{m g d}}\]

  • (doc) Period of spring from mass

    \[T = 2 \pi \sqrt{\frac{m}{k}}\]

  • (doc) Period of torsion pendulum from rotational inertia

    \[T = 2 \pi \sqrt{\frac{I}{\kappa}}\]

  • (doc) Elastic potential energy from displacement

    \[U = \frac{k d^{2}}{2}\]

  • (doc) Potential energy from mass and height

    \[U = m g h\]

  • (doc) Pressure from force and area

    \[p = \frac{F}{A}\]

  • (doc) Reaction force equals action force

    \[F_{21} = - F_{12}\]

  • (doc) Reduced mass of a two-body system

    \[\mu = \frac{1}{\frac{1}{m_{1}} + \frac{1}{m_{2}}}\]

  • (doc) Displacement in resonant oscillations

    \[x{\left(t \right)} = \frac{F}{2 m \omega_{0}} t \sin{\left(\omega_{0} t + \varphi \right)}\]

  • (doc) Rocket thrust is rocket mass times acceleration

    \[R v_\text{rel} = m a\]

  • (doc) Rotational work is torque times angular distance

    \[W = \tau \theta\]

  • (doc) Rocket speed from mass and impulse

    \[\Delta v = v_\text{e} \log \left( \frac{m_{0}}{m_{1}} \right)\]

  • (doc) Torque via force and radial distance

    \[\tau = r F \sin{\left(\varphi \right)}\]

  • (doc) Torque via rotational inertia and angular acceleration

    \[\tau = I \alpha\]

  • (doc) Total work is change in kinetic energy

    \[W = K{\left(t_{1} \right)} - K{\left(t_{0} \right)}\]

  • (doc) Work is integral of force over distance

    \[W = \int\limits_{x_{0}}^{x_{1}} F{\left(x \right)}\, dx\]

• (doc) laws.dynamics.damped_oscillations

  • (doc) Energy of underdamped oscillator

    \[E = \frac{m \omega_{0}^{2} A^{2} \exp{\left(- 2 \lambda t \right)}}{2}\]

  • (doc) Quality factor via bandwidth

    \[Q = \frac{f_\text{r}}{\Delta f}\]

  • (doc) Quality factor via damping ratio

    \[Q = \frac{1}{2 \zeta}\]

  • (doc) Quality factor via energy loss

    \[Q = \omega_\text{r} \frac{E_\text{stored}}{P_\text{loss}}\]

• (doc) laws.dynamics.deformation

  • (doc) Bulk modulus via Young modulus and Poisson ratio

    \[K = \frac{E}{3 \left(1 - 2 \nu\right)}\]

  • (doc) Elastic energy density of bulk compression via pressure

    \[w = \frac{p^{2}}{2 K}\]

  • (doc) Elastic energy density of compression via strain

    \[w = \frac{E e^{2}}{2}\]

  • (doc) Engineering normal strain is total deformation over initial dimension

    \[e = \frac{\Delta l}{l}\]

  • (doc) Poisson ratio is transverse to axial strain ratio

    \[\nu = - \frac{e_\text{transverse}}{e_\text{axial}}\]

  • (doc) Rotational stiffness is torque applied over angle

    \[\kappa = \frac{\tau}{\theta}\]

  • (doc) Shear stress is shear modulus times strain

    \[\tau = G \gamma\]

  • (doc) Superposition of small deformations

    \[e = e_{1} + e_{2}\]

  • (doc) Tensile stress is Young’s modulus times strain

    \[\sigma = E e\]

• (doc) laws.dynamics.fields

  • (doc) Conservative force is gradient of potential energy

    \[{\vec F} \left( {\vec r} \right) = - \text{grad} \, U{\left({\vec r} \right)}\]

• (doc) laws.dynamics.springs

  • (doc) Compliance of two serial springs

    \[c = c_{1} + c_{2}\]

  • (doc) Spring reaction is proportional to deformation

    \[F = - k \Delta l\]

  • (doc) Stiffness of two parallel springs

    \[k = k_{1} + k_{2}\]

• (doc) laws.dynamics.springs.vector

  • (doc) Spring reaction is proportional to deformation (vector)

    \[{\vec F} = - k {\vec s}\]

• (doc) laws.dynamics.vector

  • (doc) Acceleration from force and mass (vector)

    \[{\vec a} = \frac{{\vec F}}{m}\]

  • (doc) Force is derivative of momentum (vector)

    \[{\vec F} \left( t \right) = \frac{d}{d t} {\vec p} \left( t \right)\]

  • (doc) Instantaneous power is dot product of force and velocity

    \[P = \left( {\vec F}, {\vec v} \right)\]

  • (doc) Mechanical work from force and displacement

    \[W = \left( {\vec F}, {\vec s} \right)\]

  • (doc) Mechanical work is line integral of force

    \[W = \int_{C} \left( {\vec F} \left( {\vec r} \right), d \vec r \right)\]

  • (doc) Normal force via pressure and vector area

    \[{\vec F}_{n} = p {\vec A}\]

  • (doc) Relative acceleration from force

    \[{\vec a}_\text{rel} = \frac{{\vec F}}{m} + {\vec a}_\text{Cor} - {\vec a}_\text{tr}\]

  • (doc) Restoring torque due to twist of torsion pendulum

    \[{\vec \tau} = - \kappa {\vec \theta}\]

  • (doc) Torque is angular momentum derivative

    \[{\vec \tau} \left( t \right) = \frac{d}{d t} {\vec L} \left( t \right)\]

  • (doc) Torque of twisting force

    \[{\vec \tau} = \left[ {\vec r}, {\vec F} \right]\]

• (doc) laws.electricity

  • (doc) Absolute permittivity via relative permittivity

    \[\varepsilon = \varepsilon_0 \varepsilon_\text{r}\]

  • (doc) Admittance is conductance and susceptance

    \[Y = G + i B\]

  • (doc) Capacitance from charge and voltage

    \[C = \frac{q}{V}\]

  • (doc) Wave impedance from permeability and permittivity

    \[\eta = Z_0 \sqrt{\frac{\mu_\text{r}}{\varepsilon_\text{r}}}\]

  • (doc) Charge is quantized

    \[q = N e\]

  • (doc) Corona discharge current from voltage

    \[I = A \mu V \left(V - V_{0}\right)\]

  • (doc) Current is voltage over impedance

    \[I = \frac{V}{Z}\]

  • (doc) Current is voltage over resistance

    \[I = \frac{V}{R}\]

  • (doc) Electric charge is constant in isolated system

    \[q_{1} = q_{0}\]

  • (doc) Electric dipole moment is charge times distance

    \[p = q d\]

  • (doc) Electric displacement is permittivity times electric field

    \[D = \varepsilon E\]

  • (doc) Electric field due to dipole on the dipole axis

    \[E = \frac{1}{2 \pi \varepsilon_0} \frac{p}{d^{3}}\]

  • (doc) Electric field due to point charge

    \[E = \frac{1}{4 \pi \varepsilon_0} \frac{q}{d^{2}}\]

  • (doc) Electric field is force over test charge

    \[E = \frac{F}{q_{0}}\]

  • (doc) Electric field of uniformly charged plane

    \[E = \frac{\sigma}{2 \varepsilon_0}\]

  • (doc) Electric field outside charged sphere

    \[E = \frac{q}{4 \pi \varepsilon_0 d^{2}}\]

  • (doc) Electric flux through closed surface in linear material

    \[\Phi_{\vec E} = \frac{q_\text{free}}{\varepsilon}\]

  • (doc) Electric flux through closed surface via total charge

    \[\Phi_{\vec E} = \frac{q}{\varepsilon_0}\]

  • (doc) Electric field in gas gap between two electrodes

    \[E = \frac{3 \sqrt{\frac{x}{d}} \frac{V}{d}}{2}\]

  • (doc) Electromotive force induced in moving contour

    \[\mathcal{E} = - N \frac{d}{d t} \Phi_{\vec B}{\left(t \right)}\]

  • (doc) Electromotive force induced in rotating rod

    \[\mathcal{E} = \frac{B \omega l^{2}}{2}\]

  • (doc) Electromotive force induced in rotating coil

    \[\mathcal{E} = - N B A \omega \sin{\left(\omega t \right)}\]

  • (doc) Electrostatic force via charges and distance

    \[F = \frac{1}{4 \pi \varepsilon_0} \frac{q_{1} q_{2}}{d^{2}}\]

  • (doc) Electrostatic potential due to point charge

    \[U_{\vec E} = \frac{q}{4 \pi \varepsilon d}\]

  • (doc) Electrostatic potential energy of two charges via distance

    \[U_{\vec E} = \frac{q_{1} q_{2}}{4 \pi \varepsilon d}\]

  • (doc) Electrostatic potential is work to bring from reference point over charge

    \[U_{\vec E} = \frac{W}{q}\]

  • (doc) Energy density via permittivity and electric field

    \[w = \frac{\varepsilon E^{2}}{2}\]

  • (doc) Energy of magnetic field of coil

    \[E = \frac{\mu_0 \mu_\text{r} H^{2} V}{2}\]

  • (doc) Energy via constant power and time

    \[E = P t\]

  • (doc) Force between parallel wires

    \[F = \frac{\mu I_{1} I_{2} l}{2 \pi d}\]

  • (doc) Inductance is magnetic flux over current

    \[L = \frac{\Phi_{\vec B}}{I}\]

  • (doc) Inductance is proportional to turn count

    \[L = \frac{\mu N^{2} A}{l}\]

  • (doc) Inductance via number of turns and coil volume

    \[L = \mu n^{2} V\]

  • (doc) Instantaneous energy of magnetic field

    \[E = \frac{L I_\text{max}^{2}}{2} \cos^{2}{\left(\omega t + \varphi \right)}\]

  • (doc) Magnetic field due to current loop along axis

    \[B = \frac{\mu_0 I r^{2}}{2 \left(d^{2} + r^{2}\right)^{\frac{3}{2}}}\]

  • (doc) Magnetic field due to finite coil along axis

    \[B = \frac{\mu_0 I N}{2 \ell} \left(\cos{\left(\varphi_{1} \right)} + \cos{\left(\varphi_{2} \right)}\right)\]

  • (doc) Magnetic field due to infinite wire

    \[B = \frac{\mu I}{2 \pi r}\]

  • (doc) Magnetic field of coil

    \[B = \frac{\mu_0 I N}{l}\]

  • (doc) Magnetic flux from magnetic flux density and area

    \[\Phi_{\vec B} = B A \cos{\left(\varphi \right)}\]

  • (doc) Magnetic flux density from magnetic field strength

    \[B = \mu H\]

  • (doc) Magnetic flux density of linear conductor of finite length

    \[B = \frac{\mu I \left(\cos{\left(\varphi_{1} \right)} + \cos{\left(\varphi_{2} \right)}\right)}{4 \pi d}\]

  • (doc) Magnetic moment via current and contour area

    \[m = I A\]

  • (doc) Period of rotation of charged particle in magnetic field

    \[T = \frac{2 \pi m}{q B}\]

  • (doc) Power factor is real power over apparent power

    \[\mathrm{pf} = \frac{P}{S}\]

  • (doc) Power via current and resistance

    \[P = I^{2} R\]

  • (doc) Power via voltage and current

    \[P = I V\]

  • (doc) Power via voltage and resistance

    \[P = \frac{V^{2}}{R}\]

  • (doc) Radius of curvature of charged particle in magnetic field

    \[r = \frac{m v}{q B}\]

  • (doc) Resistance via resistivity and dimensions

    \[R = \frac{\rho l}{A}\]

  • (doc) Self-induced electromotive force via time derivative of current

    \[\mathcal{E}{\left(t \right)} = - L \frac{d}{d t} I{\left(t \right)}\]

  • (doc) Voltage is electric field times distance

    \[V = E d\]

  • (doc) Voltage is line integral of electric field

    \[V = - \int\limits_{s_{0}}^{s_{1}} E_{s}{\left(s \right)}\, ds\]

• (doc) laws.electricity.circuits

  • (doc) Admittance in parallel connection

    \[Y = \sum_i {Y}_{i}\]

  • (doc) Capacitance in parallel connection

    \[C = \sum_i {C}_{i}\]

  • (doc) Capacitance is proportional to plate area

    \[C = \frac{\varepsilon A}{d}\]

  • (doc) Capacitance of spherical capacitor

    \[C = \frac{4 \pi \varepsilon r_\text{in} r_\text{out}}{r_\text{out} - r_\text{in}}\]

  • (doc) Capacitor impedance from capacitance and frequency

    \[Z = - \frac{i}{\omega C}\]

  • (doc) Capacitor impedance from capacitor reactance

    \[Z = - i X\]

  • (doc) Coil impedance from inductive reactance

    \[Z = i X\]

  • (doc) Coil impedance via inductance and frequency

    \[Z = i \omega L\]

  • (doc) Energy stored in capacitor via capacitance and voltage

    \[W = \frac{C V^{2}}{2}\]

  • (doc) Energy stored in inductor via inductance and current

    \[W = \frac{L I^{2}}{2}\]

  • (doc) Impedance in serial connection

    \[Z = \sum_i {Z}_{i}\]

  • (doc) Impedance module of serial resistor-coil-capacitor circuit

    \[|Z| = \sqrt{R^{2} + \left(X_\text{L} - X_\text{C}\right)^{2}}\]

  • (doc) Inductance in serial connection

    \[L = \sum_i {L}_{i}\]

  • (doc) Input impedance of thin film resistor

    \[Z = \frac{R}{1 + \frac{i \omega R C}{3}}\]

  • (doc) Oscillation period of inductor-capacitor network

    \[T = 2 \pi \sqrt{L C}\]

  • (doc) Resistivity of serial resistors

    \[R = \sum_i {R}_{i}\]

  • (doc) Sum of currents through junction is zero

    \[\sum_k {I}_{k} = 0\]

  • (doc) Sum of voltages in loop is zero

    \[\sum_i {V}_{i} = 0\]

  • (doc) Time constant of resistor-capacitor circuit

    \[\tau = R C\]

  • (doc) Capacitance of p-n varactor junction

    \[C = \frac{C_{0}}{\left(1 - \frac{V}{V_{0}}\right)^{y}}\]

  • (doc) Voltage across charging capacitor in serial resistor-capacitor circuit

    \[V = V_{0} \left(1 - \exp{\left(- \frac{t}{\tau} \right)}\right)\]

• (doc) laws.electricity.circuits.couplers

  • (doc) Attenuation of three link microwave attenuator

    \[A = \exp{\left(\operatorname{acosh}{\left(1 + \frac{R_{1}}{R_{2}} \right)} \right)}\]

  • (doc) Admittance of rectangular loop coupler

    \[\begin{split}\begin{pmatrix} Y_{1} \\ Y_{2} \\ Y_{3} \\ Y_{4} \end{pmatrix} = \begin{pmatrix} \frac{Y_{0}}{\sqrt{k}} \\ Y_{0} \sqrt{\frac{k + 1}{k}} \\ Y_{0} \sqrt{\frac{k + 1}{k}} \\ \frac{Y_{0}}{\sqrt{k}} \end{pmatrix}\end{split}\]

  • (doc) Total gain of transistor amplifier

    \[\text{gain} = \text{gain}_\text{i} \text{gain}_\text{t} \text{gain}_\text{o}\]

  • (doc) Impedance of Wilkinson microstrip divider

    \[\begin{split}\begin{pmatrix} Z_{1} \\ Z_{2} \\ Z_{3} \\ Z_{4} \end{pmatrix} = \begin{pmatrix} Z_{0} \sqrt{k \left(1 + k^{2}\right)} \\ Z_{0} \sqrt{\frac{1 + k^{2}}{k^{3}}} \\ Z_{0} \sqrt{k} \\ \frac{Z_{0}}{\sqrt{k}} \end{pmatrix}\end{split}\]

  • (doc) Section length of multistage transformer

    \[l = \frac{\lambda_{1} \lambda_{2}}{2 \left(\lambda_{1} + \lambda_{2}\right)}\]

  • (doc) Relative operating bandwidth of quarter-wave transformer

    \[b = 2 - \frac{4}{\pi} \operatorname{acos}{\left(\frac{2 \Gamma \sqrt{R_\text{L} Z_\text{S}}}{\sqrt{1 - \Gamma^{2}} \left|{R_\text{L} - Z_\text{S}}\right|} \right)}\]

  • (doc) Resistor resistance in Wilkinson divider

    \[R = \frac{R_{0} \left(1 + k^{2}\right)}{k}\]

  • (doc) Transient attenuation of separate loop coupler with cascade connection

    \[A_{0} = 20 \log_{10} \left( \sin{\left(N \operatorname{asin}{\left(10^{\frac{A}{20}} \right)} \right)} \right)\]

  • (doc) Wave impedance of even mode of Lange coupler

    \[\eta_\text{e} = \frac{\eta_\text{o} \left(C + \sqrt{C^{2} + \left(1 - C^{2}\right) \left(N - 1\right)^{2}}\right)}{\left(N - 1\right) \left(1 - C\right)}\]

  • (doc) Wave impedance of odd mode of Lange coupler

    \[\eta_\text{o} = Z_\text{S} \sqrt{\frac{1 - C}{1 + C}} \frac{\left(N - 1\right) \left(1 + \sqrt{C^{2} + \left(1 - C^{2}\right) \left(N - 1\right)^{2}}\right)}{C + \sqrt{C^{2} + \left(1 - C^{2}\right) \left(N - 1\right)^{2}} + \left(N - 1\right) \left(1 - C\right)}\]

  • (doc) Wave impedance of Lange coupler

    \[\eta = \sqrt{\frac{\eta_\text{o} \eta_\text{e} \left(\eta_\text{o} + \eta_\text{e}\right)^{2}}{\left(\eta_\text{o} + \eta_\text{e} \left(N - 1\right)\right) \left(\eta_\text{e} + \eta_\text{o} \left(N - 1\right)\right)}}\]

• (doc) laws.electricity.circuits.diodes

  • (doc) Current from voltage and diode constant in vacuum diode

    \[I = g U_\text{a}^{\frac{3}{2}}\]

  • (doc) Current from voltage and triode constant in vacuum triode

    \[I = g \left(U_\text{a} + \text{gain}_{V} U_\text{g}\right)^{\frac{3}{2}}\]

  • (doc) Diode constant for parallel-plane vacuum diode

    \[g = \frac{4 \varepsilon_0}{9} \sqrt{\frac{2 e}{m_\text{e}}} \frac{A}{d^{2}}\]

  • (doc) Diode constant of cylindrical diode

    \[g = \frac{\frac{4 \varepsilon_0}{9} \sqrt{\frac{2 e}{m_\text{e}}} A_\text{a}}{r_\text{a}^{2} \left(1 - \frac{r_\text{c}}{r_\text{a}}\right)^{2}}\]

  • (doc) Direct permeability coefficient of triode with flat electrodes

    \[D = \frac{C_{1} d_{0}}{d C_{2}}\]

  • (doc) Internal resistance of vacuum diode

    \[R = \frac{2}{3 g \sqrt{V}}\]

  • (doc) Limit operating frequency of vacuum diode

    \[f = \frac{\sqrt{\frac{2 e V}{m_\text{e}}}}{6 d}\]

  • (doc) Charge density in diode

    \[\rho = \frac{\frac{4 \varepsilon_0}{9} V}{d^{2}}\]

  • (doc) Steepness of volt-ampere characteristic of vacuum diode

    \[S = \frac{3 g}{2} \sqrt{V}\]

  • (doc) Voltage of equivalent diode for pentode

    \[V = \frac{V_{1} + V_{2} D_{1} + V_{3} D_{1} D_{2} + U_\text{a} D_{1} D_{2} D_{3}}{1 + D_{1} \left(\frac{d_\text{a}}{d_{1}}\right)^{\frac{4}{3}}}\]

  • (doc) Equivalent diode voltage for tetrode

    \[V = \frac{V_{1} + V_{2} D_{1} + U_\text{a} D_{1} D_{2}}{1 + D_{1} \left(\frac{d_\text{a}}{d_{1}}\right)^{\frac{4}{3}}}\]

  • (doc) Equivalent diode voltage for triode

    \[V = \frac{U_\text{g} + \frac{U_\text{a}}{\text{gain}_{V}}}{1 + \frac{\left(\frac{d_\text{a}}{d_\text{g}}\right)^{\frac{4}{3}}}{\text{gain}_{V}}}\]

• (doc) laws.electricity.circuits.filters

  • (doc) Transmission coefficient approximation of low-pass filter

    \[H = \frac{1}{1 + e^{2} F^{2}}\]

  • (doc) Band pass Chebyshev filter order from distortion and frequency

    \[N = \frac{\operatorname{acosh}{\left(\frac{e}{e_{1}} \right)}}{\operatorname{acosh}{\left(\frac{f_{1}^{2} - f_{0}^{2}}{\Delta f f_{1}} \right)}}\]

  • (doc) Butterworth filter order from distortion and frequency

    \[N = \frac{\log \left( \frac{e_{1}}{e} \right)}{\log \left( \frac{f_{1}}{f_{0}} \right)}\]

  • (doc) Filter order from distortion and frequency

    \[F = \frac{e_{1}}{e}\]

  • (doc) High pass Chebyshev filter from distortion and frequency

    \[N = \frac{\operatorname{acosh}{\left(\frac{e_{1}}{e} \right)}}{\operatorname{acosh}{\left(\frac{f_{0}}{f_{1}} \right)}}\]

  • (doc) Low-pass Chebyshev filter order from distortion and frequencies

    \[N = \frac{\operatorname{acos}{\left(\frac{e_{1}}{e} \right)}}{\operatorname{acos}{\left(\frac{f_{1}}{f_{0}} \right)}}\]

• (doc) laws.electricity.circuits.resonators

  • (doc) Coupling parameter of resonator from quality factor

    \[g = \frac{Q_{0}}{Q_\text{e}}\]

  • (doc) Coupling parameter of resonator from resistance

    \[g = \frac{R_{0}}{R_\text{L}}\]

  • (doc) Instantaneous energy of resonator

    \[E = E_{0} \exp{\left(- \frac{\omega t}{Q} \right)}\]

  • (doc) Quality factor of loaded resonator from circuit parameters

    \[Q_{1} = \frac{R_\text{L} R_{0}}{\omega L \left(R_\text{L} + R_{0}\right)}\]

  • (doc) Quality resonator of loaded resonator from quality factors

    \[\frac{1}{Q_{1}} = \frac{1}{Q_{0}} + \frac{1}{Q_\text{e}}\]

  • (doc) Quality factor of resonator

    \[Q = \frac{R}{\omega L}\]

  • (doc) Quality factor of empty rectangular resonator for traverse electric waves

    \[Q = \frac{\omega \mu l_{3} l_{2} l_{1} \left(l_{2}^{2} + l_{1}^{2}\right)}{2 R_\text{s} \left(l_{2}^{3} \left(l_{1} + 2 l_{3}\right) + l_{1}^{3} \left(l_{2} + 2 l_{3}\right)\right)}\]

  • (doc) Quality factor of filled rectangular resonator

    \[Q_{1} = \frac{1}{\frac{1}{Q_{0}} + \tan \delta}\]

  • (doc) Resonant frequency of rectangular resonator

    \[f_\text{r} = \frac{c}{2 \sqrt{\varepsilon_\text{r} \mu_\text{r}}} \sqrt{\left(\frac{m}{l_{1}}\right)^{2} + \left(\frac{n}{l_{2}}\right)^{2} + \left(\frac{p}{l_{3}}\right)^{2}}\]

  • (doc) Resonant frequency of ring resonator

    \[f = \frac{N c}{l \sqrt{\varepsilon_\text{r}}}\]

• (doc) laws.electricity.circuits.transmission_lines

  • (doc) Standing wave ratio from reflection coefficient

    \[\text{SWR} = \frac{1 + \left|{\Gamma}\right|}{1 - \left|{\Gamma}\right|}\]

  • (doc) Standing wave ratio from voltage

    \[\text{SWR} = \frac{\min{|V|}}{\max{|V|}}\]

  • (doc) Hybrid parameters matrix equation

    \[\begin{split}\begin{pmatrix} V_\text{i} \\ I_\text{o} \end{pmatrix} = \begin{pmatrix} H_\text{ii} & H_\text{io} \\ H_\text{oi} & H_\text{oo} \end{pmatrix} \begin{pmatrix} I_\text{i} \\ V_\text{o} \end{pmatrix}\end{split}\]

  • (doc) Impedances of π-type circuit of transmission line

    \[\begin{split}\begin{pmatrix} Z_{1} \\ Z_{2} \\ Z_{3} \end{pmatrix} = \begin{pmatrix} Z_\text{S} \coth{\left(\frac{l \gamma}{2} \right)} \\ Z_\text{S} \coth{\left(\frac{l \gamma}{2} \right)} \\ Z_\text{S} \sinh{\left(l \gamma \right)} \end{pmatrix}\end{split}\]

  • (doc) Impedances of T-type circuit of transmission line

    \[\begin{split}\begin{pmatrix} Z_{1} \\ Z_{2} \\ Z_{3} \end{pmatrix} = \begin{pmatrix} Z_\text{S} \tanh{\left(\frac{l \gamma}{2} \right)} \\ Z_\text{S} \tanh{\left(\frac{l \gamma}{2} \right)} \\ \frac{Z_\text{S}}{\sinh{\left(l \gamma \right)}} \end{pmatrix}\end{split}\]

  • (doc) Input impedance from transmission matrix

    \[Z_\text{in} = \frac{A Z_\text{L} + B}{C Z_\text{L} + D}\]

  • (doc) Input impedance of lossless transmission line

    \[Z_\text{in} = \frac{Z_\text{S} \left(Z_\text{L} + i Z_\text{S} \tan{\left(\beta l \right)}\right)}{Z_\text{S} + i Z_\text{L} \tan{\left(\beta l \right)}}\]

  • (doc) Input impedance of lossy transmission line

    \[Z_\text{in} = \frac{\cosh{\left(\gamma l \right)} Z_\text{L} + Z_\text{S} \sinh{\left(\gamma l \right)}}{\frac{Z_\text{L} \sinh{\left(\gamma l \right)}}{Z_\text{S}} + \cosh{\left(\gamma l \right)}}\]

  • (doc) Reflection coefficient from ratio of average power to incident power

    \[\frac{\langle P \rangle}{P_\text{incident}} = 1 - \left|{\Gamma}\right|^{2}\]

  • (doc) Standing wave ratio from ratio of average power to incident power

    \[\frac{\langle P \rangle}{P_\text{incident}} = \frac{4 \text{SWR}}{\left(\text{SWR} + 1\right)^{2}}\]

  • (doc) Reflection coefficient from load and surge impedance

    \[\Gamma = \frac{Z_\text{L} - Z_\text{S}}{Z_\text{L} + Z_\text{S}}\]

  • (doc) Scattering matrix equation

    \[\begin{split}\begin{pmatrix} b_\text{i} \\ b_\text{o} \end{pmatrix} = \begin{pmatrix} S_\text{ii} & S_\text{io} \\ S_\text{oi} & S_\text{oo} \end{pmatrix} \begin{pmatrix} a_\text{i} \\ a_\text{o} \end{pmatrix}\end{split}\]

  • (doc) Scattering matrix to transmission matrix

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \frac{\left(1 + S_\text{ii}\right) \left(1 - S_\text{oo}\right) + S_\text{io} S_\text{oi}}{2 S_\text{oi}} & \frac{Z_\text{S} \left(\left(1 + S_\text{ii}\right) \left(1 + S_\text{oo}\right) - S_\text{io} S_\text{oi}\right)}{2 S_\text{oi}} \\ \frac{\left(1 - S_\text{ii}\right) \left(1 - S_\text{oo}\right) - S_\text{io} S_\text{oi}}{Z_\text{S}} \frac{1}{2 S_\text{oi}} & \frac{\left(1 - S_\text{ii}\right) \left(1 + S_\text{oo}\right) + S_\text{io} S_\text{oi}}{2 S_\text{oi}} \end{pmatrix}\end{split}\]

  • (doc) Admittance matrix equation

    \[\begin{split}\begin{pmatrix} I_\text{i} \\ I_\text{o} \end{pmatrix} = \begin{pmatrix} Y_\text{ii} & Y_\text{io} \\ Y_\text{oi} & Y_\text{oo} \end{pmatrix} \begin{pmatrix} V_\text{i} \\ V_\text{o} \end{pmatrix}\end{split}\]

  • (doc) Impedance matrix equation

    \[\begin{split}\begin{pmatrix} V_\text{i} \\ V_\text{o} \end{pmatrix} = \begin{pmatrix} Z_\text{ii} & Z_\text{io} \\ Z_\text{oi} & Z_\text{oo} \end{pmatrix} \begin{pmatrix} I_\text{i} \\ I_\text{o} \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix equation

    \[\begin{split}\begin{pmatrix} V_\text{i} \\ I_\text{i} \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} V_\text{o} \\ I_\text{o} \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix for parallel load in line

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ \frac{1}{Z_\text{L}} & 1 \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix for serial load in line

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1 & Z_\text{L} \\ 0 & 1 \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix of lossless transmission line

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \cos{\left(\beta l \right)} & i Z_\text{S} \sin{\left(\beta l \right)} \\ \frac{i}{Z_\text{S}} \sin{\left(\beta l \right)} & \cos{\left(\beta l \right)} \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix of lossy transmission line

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} \cosh{\left(l \gamma \right)} & Z_\text{S} \sinh{\left(l \gamma \right)} \\ \frac{\sinh{\left(l \gamma \right)}}{Z_\text{S}} & \cosh{\left(l \gamma \right)} \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix of π-type matrix

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1 + \frac{Z_{3}}{Z_{2}} & Z_{3} \\ \frac{1}{Z_{1}} + \frac{1}{Z_{2}} + \frac{Z_{3}}{Z_{1} Z_{2}} & 1 + \frac{Z_{3}}{Z_{1}} \end{pmatrix}\end{split}\]

  • (doc) Transmission matrix of T-type circuit

    \[\begin{split}\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 1 + \frac{Z_{1}}{Z_{3}} & Z_{1} + Z_{2} + \frac{Z_{1} Z_{2}}{Z_{3}} \\ \frac{1}{Z_{3}} & 1 + \frac{Z_{2}}{Z_{3}} \end{pmatrix}\end{split}\]

  • (doc) Surge impedance of microstrip line when effective width is greater than substrate thickness

    \[Z_\text{S} = \frac{Z_0}{\sqrt{\varepsilon_\text{eff}}} \frac{1}{\frac{w_\text{eff}}{h} + 1.393 + 0.667 \log \left( \frac{w_\text{eff}}{h} + 1.444 \right)}\]

  • (doc) Surge impedance of microstrip line when effective width is less than substrate thickness

    \[Z_\text{S} = \frac{Z_0}{\sqrt{\varepsilon_\text{eff}}} \log \left( \frac{8 h}{w_\text{eff}} + \frac{w_\text{eff}}{4 h} \right)\]

• (doc) laws.electricity.circuits.transmission_lines.coplanar_lines

  • (doc) Effective permittivity of coplanar transmission line when distance is greater than thickness

    \[\varepsilon_\text{eff} = 1 + \frac{\varepsilon_\text{r} - 1}{2} \frac{\log \left( \frac{2 \left(1 + \left(1 - \left(\frac{l}{d}\right)^{2}\right)^{\frac{1}{4}}\right)}{1 - \left(1 - \left(\frac{l}{d}\right)^{2}\right)^{\frac{1}{4}}} \right)}{\log \left( \frac{2 \left(1 + \left(1 - \left(\frac{\sinh{\left(\frac{\pi l}{4 h} \right)}}{\sinh{\left(\frac{\pi d}{4 h} \right)}}\right)^{2}\right)^{\frac{1}{4}}\right)}{1 - \left(1 - \left(\frac{\sinh{\left(\frac{\pi l}{4 h} \right)}}{\sinh{\left(\frac{\pi d}{4 h} \right)}}\right)^{2}\right)^{\frac{1}{4}}} \right)}\]

  • (doc) Effective permittivity of coplanar transmission line when distance is less than thickness

    \[\varepsilon_\text{eff} = \frac{1 + \varepsilon_\text{r}}{2}\]

  • (doc) Wave impedance of coplanar line when hyperbolic sine ratio squared is between \(0\) and \(\frac{1}{2}\)

    \[\eta = \frac{R_0}{\sqrt{\varepsilon_\text{eff}}} \log \left( \frac{2 \left(1 + \left(1 - \left(\frac{l}{d}\right)^{2}\right)^{\frac{1}{4}}\right)}{1 - \left(1 - \left(\frac{l}{d}\right)^{2}\right)^{\frac{1}{4}}} \right)\]

  • (doc) Wave impedance of coplanar line when length to distance ratio squared is between \(\frac{1}{2}\) and \(1\)

    \[\eta = \frac{R_0}{\sqrt{\varepsilon_\text{eff}}} \frac{1}{\log \left( \frac{2 \left(1 + \sqrt{\frac{l}{d}}\right)}{1 - \sqrt{\frac{l}{d}}} \right)}\]

• (doc) laws.electricity.circuits.transmission_lines.microstrip_lines

  • (doc) Attenuation coefficient in dielectric substate of microstrip line

    \[\alpha = 27.3 \frac{\varepsilon_\text{r}}{\sqrt{\varepsilon_\text{eff}}} \frac{\varepsilon_\text{eff} - 1}{\varepsilon_\text{r} - 1} \frac{\tan \delta}{\lambda}\]

  • (doc) Attenuation coefficient in metal of microstrip line when width is greater than thickness

    \[\alpha = \frac{a R_\text{s} Z_\text{S} \varepsilon_\text{eff}}{h} \left(\frac{w_\text{eff}}{h} + \frac{0.667 \frac{w_\text{eff}}{h}}{\frac{w_\text{eff}}{h} + 1.444}\right) \left(1 + \frac{1 - \frac{1.25}{\pi} \frac{t}{h} + \frac{1.25}{\pi} \log \left( \frac{2 h}{t} \right)}{\frac{w_\text{eff}}{h}}\right)\]

  • (doc) Attenuation coefficient in microstrip metal when thickness is greater than width times \(2 \pi\)

    \[\alpha = \frac{1.38 R_\text{s}}{h Z_\text{S}} \frac{32 - \left(\frac{w_\text{eff}}{h}\right)^{2}}{32 + \left(\frac{w_\text{eff}}{h}\right)^{2}} \left(1 + \frac{h}{w_\text{eff}} \left(1 + \frac{1.25}{\pi} \frac{t}{w} + \frac{1.25}{\pi} \log \left( \frac{4 \pi w}{t} \right)\right)\right)\]

  • (doc) Attenuation coefficient in microstrip metal when thickness is less than width times \(2 \pi\)

    \[\alpha = \frac{1.38 R_\text{s}}{h Z_\text{S}} \frac{32 - \left(\frac{w_\text{eff}}{h}\right)^{2}}{32 + \left(\frac{w_\text{eff}}{h}\right)^{2}} \left(1 + \frac{h}{w_\text{eff}} \left(1 - \frac{1.25}{\pi} \frac{t}{h} + \frac{1.25}{\pi} \log \left( 2 \frac{h}{t} \right)\right)\right)\]

  • (doc) Effective permittivity of microstrip line when width is greater than thickness

    \[\varepsilon_\text{eff} = \frac{1 + \varepsilon_\text{r}}{2} + \frac{\varepsilon_\text{r} - 1}{2} \frac{1}{\sqrt{1 + \frac{12 h}{w}}} - \frac{\varepsilon_\text{r} - 1}{4.6} \frac{t}{h} \sqrt{\frac{h}{w}}\]

  • (doc) Effective permittivity of microstrip line when width is less than thickness

    \[\varepsilon_\text{eff} = \frac{1 + \varepsilon_\text{r}}{2} + \frac{\varepsilon_\text{r} - 1}{2} \left(\frac{1}{\sqrt{1 + \frac{12 h}{w}}} + 0.04 \left(1 - \frac{w}{h}\right)^{2}\right) - \frac{\varepsilon_\text{r} - 1}{4.6} \frac{t}{h} \sqrt{\frac{h}{w}}\]

  • (doc) Effective permittivity of microstrip line from frequency

    \[\varepsilon_\text{eff} = \left(\frac{\sqrt{\varepsilon_\text{r}} - \sqrt{\varepsilon_{\text{eff}, 0}}}{1 + \frac{4}{\left(4 h f \left(1 + 2 \log \left( 1 + \frac{w}{h} \right)\right)^{2} \sqrt{\varepsilon_\text{r} - 1} \frac{1}{2 c}\right)^{\frac{3}{2}}}} + \sqrt{\varepsilon_{\text{eff}, 0}}\right)^{2}\]

  • (doc) Effective width of microstrip line when width is greater than thickness

    \[\frac{w_\text{eff}}{h} = \frac{w}{h} + \frac{1.25}{\pi} \frac{t}{h} \left(1 + \log \left( 2 \frac{h}{t} \right)\right)\]

  • (doc) Effective width of microstrip line when width is less than thickness

    \[\frac{w_\text{eff}}{h} = \frac{w}{h} + \frac{1.25}{\pi} \frac{t}{h} \left(1 + \log \left( 4 \pi \frac{w}{t} \right)\right)\]

  • (doc) Inductance of microstrip line strip

    \[L = L_0 l \left(\log \left( \frac{l}{w + t} \right) + 1.193 + \frac{0.2235}{\frac{l}{w + t}}\right)\]

  • (doc) Resistance of microstrip line

    \[R = \left(1.4 + 0.217 \log \left( \frac{w}{5 t} \right)\right) \frac{R_\text{s} l}{2 \left(w + t\right)}\]

  • (doc) Short circuit inductance of microstrip line

    \[L = \frac{\mu_0}{2 \pi} \left(h \log \left( \frac{h + \sqrt{r^{2} + h^{2}}}{r} \right) + 1.5 \left(r - \sqrt{r^{2} + h^{2}}\right)\right)\]

  • (doc) Surge impedance of microstrip line from frequency

    \[Z_\text{S} = Z_{\text{S}, 0} \sqrt{\frac{\varepsilon_{\text{eff}, 0}}{\varepsilon_\text{eff}}} \frac{\varepsilon_\text{eff} - 1}{\varepsilon_{\text{eff}, 0} - 1}\]

• (doc) laws.electricity.circuits.waveguides

  • (doc) Attenuation coefficient in dielectric

    \[\alpha = \frac{\omega \sqrt{\varepsilon \mu} \tan \delta}{2}\]

  • (doc) Attenuation coefficient in dielectric in rectangular waveguide

    \[\alpha = \frac{\pi}{\lambda} \frac{\eta}{\eta_{0}} \tan \delta\]

  • (doc) Attenuation coefficient in metal

    \[\alpha = \frac{\sqrt{\frac{\varepsilon_\text{r}}{\mu_\text{r}}} \left(\frac{R_\text{i}}{d_\text{i}} + \frac{R_\text{o}}{d_\text{o}}\right)}{\pi R_0 \log \left( \frac{d_\text{o}}{d_\text{i}} \right)}\]

  • (doc) Attenuation coefficient in metal in rectangular waveguide for transverse electric waves

    \[\alpha = \frac{2 \frac{R_\text{s}}{R}}{a \sqrt{1 - \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2}}} \left(\left(1 + \frac{a}{b}\right) \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2} + \frac{\left(1 - \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2}\right) \frac{a}{b} \left(\frac{a}{b} n^{2} + m^{2}\right)}{\left(\frac{a}{b} n\right)^{2} + m^{2}}\right)\]

  • (doc) Attenuation coefficient in metal in rectangular waveguide for transverse magnetic waves

    \[\alpha = \frac{2 \frac{R_\text{s}}{R} \left(n^{2} \left(\frac{a}{b}\right)^{3} + m^{2}\right)}{a \sqrt{1 - \left(\frac{\lambda}{2 \lambda_\text{c}}\right)^{2}} \left(n^{2} \left(\frac{a}{b}\right)^{2} + m^{2}\right)}\]

  • (doc) Wave impedance in rectangular waveguide for transverse electric waves

    \[\eta = \frac{\eta_{0}}{\sqrt{1 - \left(\frac{\lambda}{\lambda_\text{c}}\right)^{2}}}\]

  • (doc) Characteristic resistance of rectangular waveguide for transverse magnetic waves

    \[\eta = \eta_{0} \sqrt{1 - \left(\frac{\lambda}{\lambda_\text{c}}\right)^{2}}\]

  • (doc) Critical wavelength in rectangular waveguide

    \[\lambda_\text{c} = \frac{2}{\sqrt{\left(\frac{m}{a}\right)^{2} + \left(\frac{n}{b}\right)^{2}}}\]

  • (doc) Group speed of wave in rectangular waveguide

    \[v_\text{g} = \frac{c \sqrt{1 - \left(\frac{\lambda}{\lambda_\text{c}}\right)^{2}}}{\sqrt{\varepsilon_\text{r} \mu_\text{r}}}\]

  • (doc) Maximum electric field strength of main wave in rectangular waveguide

    \[E = \frac{2 Z_0 a H}{\lambda \sqrt{\varepsilon_\text{r}}}\]

  • (doc) Maximum voltage in coaxial line

    \[V = \frac{E d_\text{o} \log \left( \frac{d_\text{o}}{d_\text{i}} \right)}{2}\]

  • (doc) Phase speed of wave in rectangular waveguide

    \[v = \frac{c}{\sqrt{\varepsilon_\text{r} \mu_\text{r} \left(1 - \left(\frac{\lambda}{\lambda_\text{c}}\right)^{2}\right)}}\]

  • (doc) Power carried by coaxial waveguide

    \[P = \frac{V^{2}}{Z_0} \sqrt{\frac{\varepsilon_\text{r}}{\mu_\text{r} \log \left( \frac{d_\text{o}}{d_\text{i}} \right)}}\]

  • (doc) Power carried by main wave of rectangular waveguide

    \[P = \frac{a b \sqrt{1 - \left(\frac{\lambda}{2 a}\right)^{2}} E^{2}}{4 R}\]

  • (doc) Specific capacitance of coaxial waveguide

    \[C = \frac{2 \pi \varepsilon}{\log \left( \frac{r_\text{o}}{r_\text{i}} \right)}\]

  • (doc) Specific conductivity of coaxial waveguide

    \[G = \omega C \tan \delta\]

  • (doc) Specific inductance of coaxial waveguide

    \[L = \frac{\mu}{2 \pi} \log \left( \frac{r_\text{o}}{r_\text{i}} \right)\]

  • (doc) Specific resistance of coaxial waveguide

    \[R = \frac{\sqrt{\frac{\omega \mu}{2 G}}}{2 \pi} \left(\frac{1}{r_\text{i}} - \frac{1}{r_\text{o}}\right)\]

  • (doc) Surface resistance of metal

    \[R = \sqrt{\frac{\omega \mu}{2 G}}\]

  • (doc) Wave resistance of coaxial waveguide

    \[R = \frac{\sqrt{\frac{\mu_0 \mu_\text{r}}{\varepsilon_0 \varepsilon_\text{r}}}}{2 \pi} \log \left( \frac{r_\text{o}}{r_\text{i}} \right)\]

  • (doc) Wavelength in rectangular waveguide

    \[\lambda_\text{g} = \frac{\lambda}{\sqrt{1 - \left(\frac{\lambda}{\lambda_\text{c}}\right)^{2}}}\]

• (doc) laws.electricity.maxwell_equations

  • (doc) Divergence of electric displacement field is volumetric charge density

    \[\text{div} \, {\vec D} \left( {\vec r} \right) = \rho{\left({\vec r} \right)}\]

  • (doc) Curl of magnetic field is free current density and electric displacement derivative

    \[\text{curl} \, {\vec H} \left( {\vec r}, t \right) = {\vec J}_\text{f} \left( {\vec r}, t \right) + \frac{\partial}{\partial t} {\vec D} \left( {\vec r}, t \right)\]

  • (doc) Curl of electric field is negative magnetic flux density derivative

    \[\text{curl} \, {\vec E} \left( {\vec r}, t \right) = - \frac{\partial}{\partial t} {\vec B} \left( {\vec r}, t \right)\]

• (doc) laws.electricity.vector

  • (doc) Current density is charge density times drift velocity

    \[{\vec j} = \rho {\vec u}\]

  • (doc) Electric dipole moment is charge times displacement

    \[{\vec p} = q {\vec d}\]

  • (doc) Electric dipole moment of electrically neutral system

    \[{\vec p} = \sum_i {q}_{i} {{\vec r}}_{i}\]

  • (doc) Electric field is force over test charge (Vector)

    \[{\vec E} = \frac{{\vec F}}{q_{0}}\]

  • (doc) Electric flux of uniform electric field

    \[\Phi_{\vec E} = \left( {\vec E}, {\vec A} \right)\]

  • (doc) Force acting on dipole in non-uniform electric field

    \[{\vec F} = p \frac{d}{d x} {\vec E} \left( x \right)\]

  • (doc) Lorentz force via electromagnetic field

    \[{\vec F} = q \left({\vec E} + \left[ {\vec v}, {\vec B} \right]\right)\]

  • (doc) Magnetic field due to constant filamentary current

    \[d \vec{B} = \frac{\mu}{4 \pi} \frac{I \left[ d \vec{\ell}, {\vec r} - \vec{\ell} \right]}{\left \Vert {\vec r} - \vec{\ell} \right \Vert^{3}}\]

  • (doc) Potential energy of electric dipole in uniform electric field

    \[U = - \left( {\vec p}, {\vec E} \right)\]

  • (doc) Torque due to electric dipole moment in uniform electric field

    \[{\vec \tau} = \left[ {\vec p}, {\vec E} \right]\]

• (doc) laws.geometry

  • (doc) Cross product is proportional to sine of angle between vectors

    \[\left \Vert \left[ \vec u, \vec v \right] \right \Vert = u v \sin{\left(\varphi \right)}\]

  • (doc) Dot product is proportional to cosine of angle between vectors

    \[\left( \vec u, \vec v \right) = u v \cos{\left(\varphi \right)}\]

  • (doc) Scalar projection is vector length times cosine of angle

    \[s = a \cos{\left(\varphi \right)}\]

• (doc) laws.geometry.vector

  • (doc) Dot product is proportional to cosine of angle between vectors (vector)

    \[\left( {\vec u}, {\vec v} \right) = \left \Vert {\vec u} \right \Vert \left \Vert {\vec v} \right \Vert \cos{\left(\varphi \right)}\]

• (doc) laws.gravity

  • (doc) Angle of rotation during gravitational maneuver

    \[\varphi = 2 \operatorname{atan}{\left(\frac{G m}{d v^{2}} \right)}\]

  • (doc) Area rate of change is proportional to angular momentum

    \[\frac{d}{d t} A{\left(t \right)} = \frac{L}{2 m}\]

  • (doc) Corrected planet period squared is proportional to cube of semimajor axis

    \[T^{2} = \frac{4 \pi^{2} a^{3}}{G \left(M + m\right)}\]

  • (doc) Easterly deviation from plumbline of falling bodies

    \[s_\text{east} = \frac{4 \pi}{3} \frac{t}{T} h \cos{\left(\phi \right)}\]

  • (doc) Eccentricity of orbit

    \[e = \sqrt{1 - \left(\frac{b}{a}\right)^{2}}\]

  • (doc) First escape speed

    \[v = \sqrt{\frac{G m}{r + h}}\]

  • (doc) Free fall acceleration from height

    \[a = \frac{G m}{\left(r + h\right)^{2}}\]

  • (doc) Gravitational potential energy

    \[U = - \frac{G m_{1} m_{2}}{d}\]

  • (doc) Gravitational radius of massive body

    \[r = \frac{2 G m}{c^{2}}\]

  • (doc) Gravitational force from mass and distance

    \[F = \frac{G m_{1} m_{2}}{d^{2}}\]

  • (doc) Kepler’s constant via attracting body mass

    \[\mathfrak{K} = \frac{G M}{4 \pi^{2}}\]

  • (doc) Semimajor axis of orbit via mass and speed

    \[a = \frac{G m}{v^{2}}\]

  • (doc) Maximum angle of rotation during gravitational maneuver

    \[\varphi = \operatorname{atan}{\left(\left(\frac{v_{1}}{v}\right)^{2} \right)}\]

  • (doc) Maximum height of body thrown at angle to horizon

    \[h = \frac{v_{0}^{2} \sin^{2}{\left(\varphi \right)}}{2 g}\]

  • (doc) Time of flight of a projectile via initial velocity

    \[t = \frac{2 v_{0} \sin{\left(\varphi \right)}}{g}\]

  • (doc) Time of flight of a projectile via maximum height

    \[t = \sqrt{\frac{2 h}{g}}\]

  • (doc) Orbital speed from semimajor axis and planet mass

    \[v = \sqrt{G m \left(\frac{2}{d} - \frac{1}{a}\right)}\]

  • (doc) Planet period squared is proportional to cube of semimajor axis

    \[T^{2} = \frac{4 \pi^{2}}{G m} a^{3}\]

  • (doc) Radius of geostationary orbit

    \[r = \sqrt[3]{\frac{G m}{\omega^{2}}}\]

  • (doc) Horizontal displacement of projectile

    \[d = \frac{v_{0}^{2} \sin{\left(2 \varphi \right)}}{g}\]

  • (doc) Second escape velocity

    \[v = \sqrt{\frac{2 G m}{r + h}}\]

  • (doc) Southerly deviation from plumbline of falling bodies

    \[s_\text{south} = \pi \frac{t}{T} s_\text{east} \sin{\left(\phi \right)}\]

  • (doc) Third cosmic speed from orbital and second cosmic speed

    \[v_{3} = \sqrt{\left(\sqrt{2} - 1\right)^{2} v^{2} + v_{2}^{2}}\]

• (doc) laws.gravity.radial_motion

  • (doc) Average potential energy via average kinetic energy

    \[\langle U \rangle = - 2 \langle K \rangle\]

  • (doc) Potential energy of radial planetary motion

    \[U_\text{tot} = U_\text{gr} + \frac{L^{2}}{2 m d^{2}}\]

  • (doc) Radial kinetic energy plus potential energy is constant

    \[\frac{m v_{r}^{2}}{2} + U = E\]

  • (doc) Semimajor axis via Kepler’s constant and total energy

    \[a = \frac{2 \pi^{2} \mathfrak{K}}{\left|{\varepsilon}\right|}\]

  • (doc) Semiminor axis of elliptical orbit via orbit parameters

    \[b = 2 \sigma \sqrt{\frac{a}{G M}}\]

  • (doc) Total energy is negative average kinetic energy

    \[E = - \langle K \rangle\]

• (doc) laws.gravity.vector

  • (doc) Acceleration due to gravity via gravity force and centripetal acceleration

    \[{\vec g} = \frac{{\vec F}}{m} - {\vec a}_\text{cp}\]

  • (doc) Falling body displacement

    \[{\vec s} = {\vec v}_{0} t + t^{2} \left(\frac{{\vec g}}{2} + \left[ {\vec v}_{0}, {\vec \omega} \right]\right) + \frac{t^{3}}{3} \left(\left[ {\vec g}, {\vec \omega} \right] + 2 \left[ \left[ {\vec v}_{0}, {\vec \omega} \right], {\vec \omega} \right]\right) + \frac{t^{4}}{6} \left[ \left[ {\vec g}, {\vec \omega} \right], {\vec \omega} \right]\]

  • (doc) Relative acceleration from force and acceleration due to gravity

    \[{\vec a}_\text{rel} = {\vec g} - {\vec a}_\text{Cor} + \frac{{\vec F}}{m}\]

• (doc) laws.hydro

  • (doc) Bulk stress is bulk modulus times strain

    \[\Delta p = K e_{V}\]

  • (doc) Capillary height via surface tension and contact angle

    \[h = \frac{2 \gamma \cos{\left(\varphi \right)}}{\rho r g}\]

  • (doc) Dynamic pressure from density and flow speed

    \[q = \frac{\rho u^{2}}{2}\]

  • (doc) Efficiency of hydraulic press from force and height

    \[\eta = \frac{F_{2} d_{2}}{F_{1} d_{1}}\]

  • (doc) Efflux speed via height

    \[u = \sqrt{2 g h}\]

  • (doc) Efflux speed via hydrostatic pressure and density

    \[u = \sqrt{\frac{2 p}{\rho}}\]

  • (doc) Excess pressure under curved surface of bubble

    \[\Delta p = \frac{4 \gamma}{r}\]

  • (doc) Force to area ratio in hydraulic press

    \[\frac{F_{1}}{A_{1}} = \frac{F_{2}}{A_{2}}\]

  • (doc) Froude number via flow speed and characteristic length

    \[\text{Fr} = \frac{u}{\sqrt{g l_\text{c}}}\]

  • (doc) Hydrostatic pressure via density and height

    \[p = \rho g h\]

  • (doc) Hydrostatic pressure via density, height and acceleration

    \[p = \rho a h\]

  • (doc) Inner pressure is constant

    \[\frac{d}{d t} p_\text{inner}{\left(t \right)} = 0\]

  • (doc) Inner pressure is sum of pressures

    \[p_\text{inner} = p_\text{static} + q + p\]

  • (doc) Laplace pressure is pressure difference

    \[p_\text{L} = p_\text{o} - p_\text{i}\]

  • (doc) Mach number is flow speed over speed of sound

    \[\text{M} = \frac{u}{c}\]

  • (doc) Nusselt number via thermal parameters and characteristic length

    \[\text{Nu} = \frac{h l_\text{c}}{k}\]

  • (doc) Pressure difference at pipe ends from dynamic viscosity and flow rate

    \[\Delta p = \frac{8 \mu l Q}{\pi r^{4}}\]

  • (doc) Pressure of liquid in vessel moving horizontally

    \[p = \rho \sqrt{g^{2} + a^{2}} h\]

  • (doc) Pressure of liquid in vessel moving vertically

    \[p = \rho \sqrt{\left(g + a\right)^{2}} h\]

  • (doc) Reynolds number via fluid parameters and characteristic length

    \[\text{Re} = \frac{\rho u l_\text{c}}{\mu}\]

  • (doc) Shear stress is proportional to speed gradient

    \[\tau = \mu \frac{d}{d x} u{\left(x \right)}\]

  • (doc) Submerged volume of floating body via density ratio

    \[\frac{V_\text{fl}}{V} = \frac{\rho}{\rho_\text{fl}}\]

  • (doc) Surface tension force via surface tension and length

    \[F = \gamma l\]

  • (doc) Volume flux is constant

    \[\frac{d}{d t} A{\left(t \right)} u{\left(t \right)} = 0\]

  • (doc) Apparent weight of a fully submersed body in fluid

    \[W_\text{fl} = W_\text{vac} \left(1 - \frac{\rho_\text{fl}}{\rho_\text{b}}\right)\]

• (doc) laws.kinematics

  • (doc) Angular momentum is rotational inertia times angular speed

    \[L = I \omega\]

  • (doc) Angular position is arc length over radius

    \[\theta = \frac{s}{r}\]

  • (doc) Angular position via constant angular acceleration and time

    \[\theta = \theta_{0} + \omega_{0} t + \frac{\alpha t^{2}}{2}\]

  • (doc) Angular position via constant angular speed and time

    \[\theta = \theta_{0} + \omega t\]

  • (doc) Angular speed via constant angular acceleration and time

    \[\omega = \omega_{0} + \alpha t\]

  • (doc) Average angular speed is angular distance over time

    \[\langle \omega \rangle = \frac{\theta}{t}\]

  • (doc) Centripetal acceleration via angular speed and radius

    \[a_{n} = \omega^{2} r\]

  • (doc) Centripetal acceleration via linear speed and radius

    \[a_{n} = \frac{v^{2}}{r}\]

  • (doc) Classical addition of velocities

    \[v_{OA} = v_{OB} + v_{BA}\]

  • (doc) Displacement in simple harmonic motion

    \[q = q_\text{max} \cos{\left(\omega t + \varphi \right)}\]

  • (doc) Position via constant acceleration and time

    \[x = x_{0} + v_{0} t + \frac{a t^{2}}{2}\]

  • (doc) Position via constant speed and time

    \[x = x_{0} + v t\]

  • (doc) Speed via angular speed and radius

    \[v = \omega r\]

  • (doc) Speed via constant acceleration and time

    \[v = v_{0} + a t\]

  • (doc) Tangential acceleration via angular acceleration and radius

    \[a_{\tau} = \alpha r\]

• (doc) laws.kinematics.damped_oscillations

  • (doc) Damped angular frequency

    \[\omega_\text{d} = \omega \sqrt{1 - \zeta^{2}}\]

  • (doc) Damping ratio from decay constant and undamped frequency

    \[\zeta = \frac{\lambda}{\omega}\]

  • (doc) Displacement in critical damping

    \[d = \exp{\left(- \omega t \right)} \left(x_{0} + \left(v_{0} + x_{0} \omega\right) t\right)\]

  • (doc) Displacement in underdamping

    \[d = a \exp{\left(- \lambda t \right)} \cos{\left(\omega_\text{d} t + \varphi \right)}\]

• (doc) laws.kinematics.rotational_inertia

  • (doc) Rotational inertia about axis and through center of mass

    \[I = I_\text{com} + m d^{2}\]

  • (doc) Rotational inertia in terms of Cartesian integral

    \[I = \int\limits_{z_{0}}^{z_{1}}\int\limits_{y_{0}}^{y_{1}}\int\limits_{x_{0}}^{x_{1}} \rho{\left(x,y,z \right)} r^{2}{\left(x,y,z \right)}\, dx\, dy\, dz\]

  • (doc) Rotational inertia in terms of a cylindrical integral

    \[I = \int\limits_{h_{0}}^{h_{1}}\int\limits_{\varphi_{0}}^{\varphi_{1}}\int\limits_{r_{0}}^{r_{1}} \rho{\left(r,\varphi,h \right)} r^{3}\, dr\, d\varphi\, dh\]

  • (doc) Rotational inertia is additive

    \[I = \sum_k {I}_{k}\]

  • (doc) Rotational inertia of a particle

    \[I = m r^{2}\]

• (doc) laws.kinematics.rotational_inertia.geometries

  • (doc) Slab about perpendicular axis through center

    \[I = \frac{m \left(l_{1}^{2} + l_{2}^{2}\right)}{12}\]

  • (doc) Solid disk about central axis

    \[I = \frac{m r^{2}}{2}\]

  • (doc) Thin rod about axis through center perpendicular to length

    \[I = \frac{m l^{2}}{12}\]

• (doc) laws.kinematics.vector

  • (doc) Absolute velocity of arbitrary motion

    \[{\vec v}_\text{abs} = {\vec v}_\text{rel} + {\vec v}_\text{tr}\]

  • (doc) Acceleration due to non-uniform rotation

    \[{\vec a}_\text{rot} = \left[ \frac{d}{d t} {\vec \omega} \left( t \right), {\vec r} \right]\]

  • (doc) Acceleration is normal plus tangential acceleration

    \[{\vec a} = {\vec a}_\text{n} + {\vec a}_\text{t}\]

  • (doc) Acceleration of transfer between relative frames

    \[{\vec a}_\text{tr} = {\vec a}_{0} + {\vec a}_\text{cp} + {\vec a}_\text{rot}\]

  • (doc) Centrifugal acceleration via centripetal acceleration

    \[{\vec a}_\text{cf} = - {\vec a}_\text{cp}\]

  • (doc) Centripetal acceleration via cross product

    \[{\vec a}_\text{cp} = \left[ {\vec \omega}, \left[ {\vec \omega}, {\vec r} \right] \right]\]

  • (doc) Centripetal acceleration via vector rejection

    \[{\vec a}_\text{cp} = {\vec \omega} \left( {\vec r}, {\vec \omega} \right) - {\vec r} \left( {\vec \omega}, {\vec \omega} \right)\]

  • (doc) Coriolis acceleration

    \[{\vec a}_\text{Cor} = 2 \left[ {\vec v}_\text{rel}, {\vec \omega} \right]\]

  • (doc) Linear displacement is angular displacement cross radius

    \[{\vec s} = \left[ {\vec \theta}, {\vec r} \right]\]

  • (doc) Velocity of transfer between reference frames

    \[{\vec v}_\text{tr} = {\vec v}_{0} + \left[ {\vec \omega}, {\vec r} \right]\]

  • (doc) Velocity relative to reference frame

    \[{\vec v} \left( t \right) = \frac{d}{d t} {\vec r} \left( t \right)\]

• (doc) laws.nuclear

  • (doc) Diffusion area from diffusion coefficient and absorption cross section

    \[L^{2} = \frac{D}{\Sigma_\text{a}}\]

  • (doc) Diffusion equation from neutron flux

    \[- D \nabla^{2} \Phi{\left(x \right)} + \Sigma_\text{a} \Phi{\left(x \right)} = \frac{\nu}{k_\text{eff}} \Sigma_\text{f} \Phi{\left(x \right)}\]

  • (doc) Effective multiplication factor from infinite multiplication factor and probabilities

    \[k_\text{eff} = k_{\infty} P_\text{FNL} P_\text{TNL}\]

  • (doc) Fast fission factor from resonance escape probability

    \[\varepsilon = 1 + \frac{1 - p}{p} \frac{\nu_\text{f} P_\text{FAF} u_\text{f}}{\nu_\text{t} P_\text{TNL} P_\text{TAF} f}\]

  • (doc) Fast non-leakage probability from Fermi age and geometric buckling

    \[P_\text{FNL} = \exp{\left(- B_\text{g}^2 \tau \right)}\]

  • (doc) Infinite multiplication factor formula

    \[k_{\infty} = \eta \varepsilon p f\]

  • (doc) Infinite multiplication factor from macroscopic cross sections

    \[k_{\infty} = \frac{\nu \Sigma_\text{f}}{\Sigma_\text{a}}\]

  • (doc) Solution to the exponential decay equation

    \[X = X_{0} \cdot 2^{- \frac{t}{t_{1/2}}}\]

  • (doc) Macroscopic cross section from mean free path

    \[\Sigma = \frac{1}{\lambda}\]

  • (doc) Macroscopic cross section from microscopic cross section and number density

    \[\Sigma = \sigma n\]

  • (doc) Macroscopic transport cross section from macroscopic scattering cross section

    \[\Sigma_\text{tr} = \sigma_\text{s} \left(1 - \mu\right)\]

  • (doc) Migration area from diffusion length and Fermi age

    \[M^{2} = L^{2} + \tau\]

  • (doc) Average cosine of scattering angle from mass number

    \[\mu = \frac{2}{3 A}\]

  • (doc) Diffusion coefficient from macroscopic scattering cross section

    \[D = \frac{1}{3 \Sigma_\text{tr}}\]

  • (doc) Reproduction factor from macroscopic cross sections in fuel

    \[\eta = \frac{\nu \Sigma_\text{f}^\text{f}}{\Sigma_\text{a}^\text{f}}\]

  • (doc) Resonance escape probability from resonance absorption integral

    \[p = \exp{\left(- \frac{n J_\text{eff}}{\xi \Sigma_\text{s}} \right)}\]

  • (doc) Thermal non-leakage probability from diffusion area and geometric buckling

    \[P_\text{TNL} = \frac{1}{1 + L_\text{th}^2 B_\text{g}^2}\]

  • (doc) Thermal utilization factor from macroscopic absorption cross sections

    \[f = \frac{\Sigma_\text{a}^\text{f}}{\Sigma_\text{a}}\]

• (doc) laws.nuclear.buckling

  • (doc) Geometric buckling for uniform cylinder

    \[B_\text{g}^2 = \left(\frac{2.405}{r}\right)^{2} + \left(\frac{\pi}{h}\right)^{2}\]

  • (doc) Geometric buckling for uniform parallelepiped

    \[B_\text{g}^2 = \left(\frac{\pi}{l_{1}}\right)^{2} + \left(\frac{\pi}{l_{2}}\right)^{2} + \left(\frac{\pi}{l_{3}}\right)^{2}\]

  • (doc) Geometric buckling for uniform slab

    \[B_\text{g}^2 = \left(\frac{\pi}{h}\right)^{2}\]

  • (doc) Geometric buckling for uniform sphere

    \[B_\text{g}^2 = \left(\frac{\pi}{r}\right)^{2}\]

  • (doc) Geometric buckling from multiplication factors and diffusion area

    \[B_\text{g}^2 = \frac{\frac{k_{\infty}}{k_\text{eff}} - 1}{L^{2}}\]

  • (doc) Geometric buckling from macroscopic cross sections and diffusion coefficient

    \[B_\text{g}^2 = \frac{\frac{\nu}{k_\text{eff}} \Sigma_\text{f} - \Sigma_\text{a}}{D}\]

  • (doc) Geometric buckling from neutron flux

    \[B_\text{g}^2 = - \frac{\nabla^{2} \Phi{\left(x \right)}}{\Phi{\left(x \right)}}\]

  • (doc) Material buckling from material cross sections and diffusion coefficient

    \[B_\text{m}^2 = \frac{\nu \Sigma_\text{f} - \Sigma_\text{a}}{D}\]

  • (doc) Neutron flux for uniform cylinder

    \[\Phi = \Phi_{0} J_{0}\left(\frac{2.405}{r_{0}} r\right) \cos{\left(\frac{\pi}{h_{0}} h \right)}\]

  • (doc) Neutron flux for uniform parallelepiped

    \[\Phi = \Phi_{0} \cos{\left(\frac{\pi}{l_{2}} x_{1} \right)} \cos{\left(\frac{\pi}{l_{1}} x_{2} \right)} \cos{\left(\frac{\pi}{l_{3}} x_{3} \right)}\]

  • (doc) Neutron flux for uniform slab

    \[\Phi = \Phi_{0} \cos{\left(\frac{\pi}{h} z \right)}\]

  • (doc) Neutron flux for uniform sphere

    \[\Phi = \Phi_{0} \frac{\sin{\left(\frac{\pi}{r_{0}} r \right)}}{r}\]

• (doc) laws.optics

  • (doc) Angle of light deflection in prism

    \[b = a \left(n - 1\right)\]

  • (doc) Angular magnification of telescope

    \[M_\text{A} = \frac{F}{f}\]

  • (doc) Bragg diffraction from angle of diffraction and wavelength

    \[d = \frac{N \lambda}{2 \sin{\left(\varphi \right)}}\]

  • (doc) Film thickness for minimum interference

    \[h = \frac{k \lambda}{2 n \cos{\left(\varphi \right)}}\]

  • (doc) Focal length of a concave spherical mirror

    \[f = \frac{r}{2}\]

  • (doc) Interference due to two slits

    \[\Lambda = \frac{x d}{l}\]

  • (doc) Interference maximum

    \[\Lambda = N \lambda\]

  • (doc) Interference minimum

    \[\Lambda = \frac{\left(2 N + 1\right) \lambda}{2}\]

  • (doc) Irradiance of light after polarizer

    \[E_\text{e} = E_{\text{e}0} k \cos^{2}{\left(\varphi \right)}\]

  • (doc) Lens focus from object and image

    \[\frac{1}{f} = \frac{1}{d_\text{o}} + \frac{1}{d_\text{i}}\]

  • (doc) Light pressure

    \[p = \frac{I \left(1 + R\right)}{c}\]

  • (doc) Linear magnification from distance to object and distance to image

    \[M = \frac{d_\text{i}}{d_\text{o}}\]

  • (doc) Linear magnification from object height and image height

    \[M = \frac{h_\text{i}}{h_\text{o}}\]

  • (doc) Optical distance difference from optical distances

    \[\Delta \Lambda = \Lambda_{2} - \Lambda_{1}\]

  • (doc) Optical path length from geometrical path length and refractive index

    \[\Lambda = n s\]

  • (doc) Optical power from focus distance

    \[D = \frac{1}{f}\]

  • (doc) Optical power from thin lens radii and refractive indices

    \[D = \left(n - n_{0}\right) \left(\frac{1}{r_{1}} - \frac{1}{r_{2}}\right)\]

  • (doc) Optical power of spherical lens from refractive indices and distances

    \[- \frac{n_{0}}{d_\text{o}} + \frac{n}{d_\text{i}} = \frac{n - n_{0}}{r}\]

  • (doc) Radiation intensity from energy, area, and time

    \[I = \frac{E}{A t}\]

  • (doc) Radius of dark Newton’s ring formula

    \[r = \sqrt{\frac{N R \lambda}{n}}\]

  • (doc) Refraction angle from enviroments

    \[n_{1} \sin{\left(\varphi_{1} \right)} = n_{2} \sin{\left(\varphi_{2} \right)}\]

  • (doc) Relative aperture of telescope

    \[A = \frac{D}{f}\]

  • (doc) Resolution of telescope

    \[\theta = A \frac{\lambda}{D}\]

• (doc) laws.quantities

  • (doc) Fractional change is change over initial value

    \[e_{X} = \frac{\Delta X}{X}\]

  • (doc) Quantity is areal density times volume

    \[X = \sigma_{X} A\]

  • (doc) Extensive quantity is linear density times length

    \[X = \lambda_{X} l\]

  • (doc) Quantity is molar quantity times amount of substance

    \[X = x_{m} n\]

  • (doc) Quantity is specific quantity times mass

    \[X = x m\]

  • (doc) Quantity is volumetric density times volume

    \[X = \rho_{X} V\]

• (doc) laws.quantum_mechanics

  • (doc) Probability density of quantum state

    \[\rho = \left|{\psi}\right|^{2}\]

• (doc) laws.quantum_mechanics.harmonic_oscillator

  • (doc) Energy levels of harmonic oscillator

    \[E_{n} = \left(N + \frac{1}{2}\right) \hbar \omega\]

  • (doc) Quantum harmonic oscillator equation

    \[- \frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi{\left(x \right)} + \frac{m \omega^{2}}{2} x^{2} \psi{\left(x \right)} = E \psi{\left(x \right)}\]

  • (doc) Wave eigenfunctions of quantum harmonic oscillator

    \[\psi = \frac{\sqrt[4]{\frac{m \omega}{\pi \hbar}}}{\sqrt{2^{N} N!}} \exp{\left(- \frac{m \omega}{2 \hbar} x^{2} \right)} H_{N}\left(\sqrt{\frac{m \omega}{\hbar}} x\right)\]

• (doc) laws.quantum_mechanics.schrodinger

  • (doc) Free particle plane wave solution

    \[\psi = \exp{\left(\frac{i}{\hbar} \left(p x - E t\right) \right)}\]

  • (doc) Time dependent Schrödinger equation in one dimension

    \[- \frac{\hbar^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} \psi{\left(x,t \right)} + U{\left(x \right)} \psi{\left(x,t \right)} = i \hbar \frac{\partial}{\partial t} \psi{\left(x,t \right)}\]

  • (doc) Time dependent solution via time independent solution

    \[\Psi{\left(x,t \right)} = \psi{\left(x \right)} \exp{\left(- \frac{i}{\hbar} E t \right)}\]

  • (doc) Time independent solution in one dimension

    \[- \frac{\hbar^{2}}{2 m} \frac{d^{2}}{d x^{2}} \psi{\left(x \right)} + U{\left(x \right)} \psi{\left(x \right)} = E \psi{\left(x \right)}\]

• (doc) laws.relativistic

  • (doc) Coordinate conversion at constant velocity

    \[x_{2} = \frac{x_{1} - v t_{1}}{\sqrt{1 - \left(\frac{v}{c}\right)^{2}}}\]

  • (doc) Lorentz transformation of time

    \[t' = \frac{t - \frac{v x}{c^{2}}}{\sqrt{1 - \left(\frac{v}{c}\right)^{2}}}\]

  • (doc) Proper time for timelike intervals

    \[\Delta \tau = \frac{\Delta s}{c}\]

  • (doc) Relativistic kinetic energy

    \[K = \left(\gamma - 1\right) m_{0} c^{2}\]

  • (doc) Relativistic length via rest length and speed

    \[l = l_{0} \sqrt{1 - \frac{v^{2}}{c^{2}}}\]

  • (doc) Relativistic mass via rest mass and speed

    \[m = \frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]

  • (doc) Relativistic momentum via rest mass and speed

    \[p = \frac{m_{0} v}{\sqrt{1 - \left(\frac{v}{c}\right)^{2}}}\]

  • (doc) Relativistic sum of velocities

    \[v_{OL} = \frac{v_{OP} + v_{PL}}{1 + \frac{v_{OP} v_{PL}}{c^{2}}}\]

  • (doc) Relativistic time dilation

    \[t = \frac{\tau}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}\]

  • (doc) Spacetime interval via time and distance

    \[s^{2} = \left(c t\right)^{2} - d^{2}\]

  • (doc) Spacetime interval is Lorentz invariant

    \[s_{2} = s_{1}\]

  • (doc) Total energy via momentum and rest mass

    \[E^{2} = \left(p c\right)^{2} + \left(m_{0} c^{2}\right)^{2}\]

  • (doc) Total energy via relativistic mass

    \[E = m c^{2}\]

• (doc) laws.relativistic.vector

  • (doc) Acceleration from force and velocity

    \[{\vec a} = \frac{{\vec F} - \frac{\left( {\vec F}, {\vec v} \right)}{c^{2}} {\vec v}}{m_{0} \gamma}\]

  • (doc) Force from acceleration and velocity

    \[{\vec F} = \gamma^{3} m_{0} {\vec a}_\text{t} + \gamma m_{0} {\vec a}_\text{n}\]

  • (doc) Relativistic mass moment

    \[{\vec N} = m_{0} \gamma^{2} \left({\vec r} - {\vec v} t\right)\]

  • (doc) Relativistic velocity normal to movement

    \[{\vec u}_\text{n} = \frac{\left({\vec u'} - \frac{\left( {\vec u'}, {\vec v} \right)}{\left( {\vec v}, {\vec v} \right)} {\vec v}\right) \sqrt{1 - \frac{\left( {\vec v}, {\vec v} \right)}{c^{2}}}}{1 + \frac{\left( {\vec u'}, {\vec v} \right)}{c^{2}}}\]

  • (doc) Relativistic velocity tangential to movement

    \[{\vec u}_\text{t} = \frac{{\vec v} \left(\frac{\left( {\vec u'}, {\vec v} \right)}{\left( {\vec v}, {\vec v} \right)} + 1\right)}{1 + \frac{\left( {\vec u'}, {\vec v} \right)}{c^{2}}}\]

• (doc) laws.thermodynamics

  • (doc) Average kinetic energy of ideal gas from temperature

    \[\langle K \rangle = \frac{3 k_\text{B}}{2} T\]

  • (doc) Average speed in Maxwell—Boltzmann statistics

    \[\langle v \rangle = \sqrt{\frac{8 k_\text{B} T}{\pi m}}\]

  • (doc) Average square speed in Maxwell—Boltzmann statistics

    \[\langle v^{2} \rangle = \frac{3 k_\text{B} T}{m}\]

  • (doc) Canonical partition function of a classical discrete system

    \[Z = \sum_i {f}_{i}\]

  • (doc) Change in entropy of ideal gas from volume and temperature

    \[S = \frac{m}{M} \left(c_{V, \text{m}} \log \left( \frac{T_{1}}{T_{0}} \right) + R \log \left( \frac{V_{1}}{V_{0}} \right)\right)\]

  • (doc) Chemical potential is Gibbs energy per particle

    \[\mu = \frac{G}{N}\]

  • (doc) Chemical potential is particle count derivative of enthalpy

    \[\mu = \frac{\partial}{\partial N} H{\left(S,p,N \right)}\]

  • (doc) Chemical potential is particle count derivative of free energy

    \[\mu = \frac{\partial}{\partial N} F{\left(T,V,N \right)}\]

  • (doc) Chemical potential is particle count derivative of Gibbs energy

    \[\mu = \frac{\partial}{\partial N} G{\left(T,p,N \right)}\]

  • (doc) Chemical potential is particle count derivative of internal energy

    \[\mu = \frac{\partial}{\partial N} U{\left(S,V,N \right)}\]

  • (doc) Chemical potential of ideal gas

    \[\mu = k_\text{B} T \log \left( n \lambda^{3} \right)\]

  • (doc) Classical isochoric molar heat capacity of solids

    \[c_{p, \text{m}} = 3 R\]

  • (doc) Compressibility factor via intermolecular force potential

    \[Z = 1 + \frac{2 \pi N}{V} \int\limits_{0}^{\infty} \left(1 - \exp{\left(- \frac{U{\left(r \right)}}{k_\text{B} T} \right)}\right) r^{2}\, dr\]

  • (doc) Diffusion coefficient of spherical Brownian particles from temperature and dynamic viscosity

    \[D = \frac{R T}{6 N_\text{A} \pi r \mu}\]

  • (doc) Diffusion flux from diffusion coefficient and concentration gradient

    \[J{\left(x \right)} = - D \frac{d}{d x} c{\left(x \right)}\]

  • (doc) Dynamic viscosity from temperature

    \[\mu = \mu_{0} \frac{T_{0} + S}{T + S} \left(\frac{T}{T_{0}}\right)^{\frac{3}{2}}\]

  • (doc) Efficiency of heat engine

    \[\eta = 1 - \frac{Q_{r}}{Q_{h}}\]

  • (doc) Enthalpy derivative via volume derivative

    \[\frac{\partial}{\partial p} H{\left(T,p \right)} = V{\left(T,p \right)} - T \frac{\partial}{\partial T} V{\left(T,p \right)}\]

  • (doc) Enthalpy differential

    \[dH = T dS + V dp + \mu dN\]

  • (doc) Enthalpy is internal energy plus pressure energy

    \[H = U + p V\]

  • (doc) Enthalpy via Gibbs energy

    \[H = G{\left(T,p \right)} - T \frac{\partial}{\partial T} G{\left(T,p \right)}\]

  • (doc) Entropy change in reversible process

    \[d S = \frac{\delta Q}{T}\]

  • (doc) Entropy derivative via volume derivative

    \[\frac{\partial}{\partial p} S{\left(T,p \right)} = - \frac{\partial}{\partial T} V{\left(T,p \right)}\]

  • (doc) Entropy from statistical weight

    \[S = k_\text{B} \log \left( \Omega \right)\]

  • (doc) Entropy is derivative of free energy

    \[S = - \frac{\partial}{\partial T} F{\left(T,V,N \right)}\]

  • (doc) Entropy is derivative of Gibbs energy

    \[S = - \frac{\partial}{\partial T} G{\left(T,p,N \right)}\]

  • (doc) Entropy of independent subsystems is sum of their entropies

    \[S = \sum_i {S}_{i}\]

  • (doc) Fractional volume change via small temperature change

    \[e_{V} = \alpha_{V} \Delta T\]

  • (doc) Free energy differential

    \[dF = - S dT - p dV + \mu dN\]

  • (doc) Gas mixture pressure from partial pressures

    \[p = \sum_i {p}_{i}\]

  • (doc) Gas pressure change from temperature

    \[\Delta p = p_{0} \left(\frac{T}{T_\text{std}} - 1\right)\]

  • (doc) Gibbs energy differential

    \[dG = - S dT + V dp + \mu dN\]

  • (doc) Gibbs energy via enthalpy

    \[G = H - T S\]

  • (doc) Grashof number

    \[\text{Gr} = \frac{g \alpha_{V} \left(T_\text{s} - T_\text{b}\right) l_\text{c}^{3}}{\nu^{2}}\]

  • (doc) Heat is heat capacity times temperature change

    \[Q = C \Delta T\]

  • (doc) Heat of combustion via mass

    \[Q = \varepsilon_{q} m\]

  • (doc) Heat of vaporization via mass

    \[Q = \varepsilon_{L} m\]

  • (doc) Helmholtz free energy via internal energy

    \[F = U - T S\]

  • (doc) Infinitesimal work in quasistatic process

    \[\delta W = p d V\]

  • (doc) Intensive parameters relation

    \[S d T - V d p + N d \mu = 0\]

  • (doc) Internal energy change of ideal gas via temperature

    \[dU = C_{V} dT\]

  • (doc) Internal energy change via heat and work

    \[d U = \delta Q - \delta W\]

  • (doc) Internal energy differential

    \[d U = T d S - p d V + \mu d N\]

  • (doc) Internal energy of ideal gas via temperature

    \[U = \frac{3 m R T}{2 M}\]

  • (doc) Internal energy via Helmholtz free energy

    \[U = F{\left(T,V \right)} - T \frac{\partial}{\partial T} F{\left(T,V \right)}\]

  • (doc) Isentropic speed of sound via pressure derivative

    \[v_\text{s} = \sqrt{\frac{\partial}{\partial \rho} p{\left(\rho,S \right)}}\]

  • (doc) Isobaric molar heat capacity of ideal gas via adiabatic index

    \[c_{p, \text{m}} = \frac{R \gamma}{\gamma - 1}\]

  • (doc) Isobaric potential from heat capacity

    \[\Delta G_\text{m} = \Delta H_\text{m} - T \Delta S_\text{m} - \Delta c_\text{m} T \left(\log \left( \frac{T}{T_\text{lab}} \right) + \frac{T_\text{lab}}{T} - 1\right)\]

  • (doc) Isobaric potential of temperature dependent heat capacity

    \[\Delta G_\text{m} = \Delta H_\text{m} - T S_\text{m} - T \left(a \left(\log \left( \frac{T_\text{lab}}{T} \right) + \frac{T_\text{lab}}{T} - 1\right) + b \left(\frac{T}{2} + \frac{T_\text{lab}^{2}}{2 T} - T_\text{lab}\right) + c \left(\frac{1}{T^{2} \cdot 2} - \frac{1}{T_\text{lab} T} + \frac{1}{T_\text{lab}^{2} \cdot 2}\right)\right)\]

  • (doc) Isochoric and isobaric heat capacities of homogeneous substance

    \[C_{p} - C_{V} = \frac{V T \alpha_{V}^{2}}{\beta_{T}}\]

  • (doc) Isochoric and isobaric heat capacities of ideal gas

    \[C_{p} - C_{V} = n R\]

  • (doc) Isochoric molar heat capacity of ideal gas via adiabatic index

    \[c_{V, \text{m}} = \frac{R}{\gamma - 1}\]

  • (doc) Isochoric molar heat capacity of ideal gas via degrees of freedom

    \[c_{V, \text{m}} = \frac{f}{2} R\]

  • (doc) Laplace pressure of spherical shapes

    \[P_\text{L} = \frac{2 \gamma}{r}\]

  • (doc) Latent heat of fusion via mass

    \[Q = \varepsilon_{\lambda} m\]

  • (doc) Mean free path of random motion

    \[\lambda = \frac{1}{\sqrt{2} \pi D^{2} n}\]

  • (doc) Number of impacts on the wall from area and speed

    \[N = \frac{n A v t}{2}\]

  • (doc) Prandtl number via dynamic viscosity and thermal conductivity

    \[\text{Pr} = \frac{c_{p} \mu}{k}\]

  • (doc) Pressure and temperature in isochoric process

    \[\frac{p_{0}}{p_{1}} = \frac{T_{0}}{T_{1}}\]

  • (doc) Pressure and volume in isothermal process

    \[p_{0} V_{0} = p_{1} V_{1}\]

  • (doc) Pressure from number density and kinetic energy

    \[p = \frac{2 n}{3} \langle K \rangle\]

  • (doc) Pressure of ideal gas from height and temperature

    \[p = p_{0} \exp{\left(- \frac{g m \Delta h}{k_\text{B} T} \right)}\]

  • (doc) Probability of finding ideal gas molecules in volume

    \[P = \left(\frac{V}{V_{0}}\right)^{N}\]

  • (doc) Probability of ideal gas macrostate

    \[P_\text{macro} = \Omega \prod_i {P}_{i}^{{N}_{i}}\]

  • (doc) Quantum isochoric molar heat capacity of solids

    \[c_{V, \text{m}} = 3 R \frac{x^{2} \exp{\left(x \right)}}{\left(\exp{\left(x \right)} - 1\right)^{2}}\]

  • (doc) Radiance of black body from temperature

    \[M_\text{e} = \sigma T^{4}\]

  • (doc) Radiation power via temperature

    \[P = \sigma \varepsilon A T^{4}\]

  • (doc) Rate of energy conduction through slab

    \[P = \frac{k A \left|{\Delta T}\right|}{h}\]

  • (doc) Relative humidity is ratio of vapor pressure

    \[\varphi = \frac{p}{p_\text{s}}\]

  • (doc) Speed of sound in ideal gas

    \[v_\text{s} = \sqrt{\frac{\gamma R T}{M}}\]

  • (doc) Temperature derivative via volume derivative

    \[\frac{\partial}{\partial p} T{\left(p,H \right)} = \frac{T{\left(p,H \right)} \frac{\partial}{\partial T{\left(p,H \right)}} V{\left(T{\left(p,H \right)},p \right)} - V{\left(T{\left(p,H \right)},p \right)}}{C_{p}}\]

  • (doc) Temperature is derivative of internal energy

    \[T = \frac{\partial}{\partial S} U{\left(S,V,N \right)}\]

  • (doc) Total energy transfer is zero in adiabatically isolated system

    \[\sum_i {E}_{i} = 0\]

  • (doc) Total particle count is sum of occupancies

    \[N = \sum_i {N}_{i}\]

  • (doc) Volume and temperature in isobaric process

    \[\frac{V_{0}}{V_{1}} = \frac{T_{0}}{T_{1}}\]

  • (doc) Volumetric and linear expansion coefficients in isotropic materials

    \[\alpha_{V} = 3 \alpha_{l}\]

  • (doc) Volumetric expansion coefficient of ideal gas

    \[\alpha_{V} = \frac{1}{T}\]

  • (doc) Work is integral of pressure over volume

    \[W = \int\limits_{V_{0}}^{V_{1}} p{\left(V \right)}\, dV\]

  • (doc) Work of ideal gas in isobaric process

    \[W = p \left(V_{1} - V_{0}\right)\]

  • (doc) Work of ideal gas in isothermal process

    \[W = n R T \log \left( \frac{V_{1}}{V_{0}} \right)\]

• (doc) laws.thermodynamics.bose_einstein_statistics

  • (doc) Single particle state distribution

    \[n_{i} = \frac{1}{\exp{\left(\frac{E_{i} - \mu}{k_\text{B} T} \right)} - 1}\]

• (doc) laws.thermodynamics.dielectrics

  • (doc) Enthalpy change via entropy change and electric field change

    \[dH = T dS - D dE\]

  • (doc) Enthalpy of dielectrics

    \[H = U - E D\]

  • (doc) Free energy change via temperature change and electric displacement change

    \[dH = - S dT + E dD\]

  • (doc) Gibbs energy change via temperature change and electric displacement change

    \[dG = - S dT - D dE\]

  • (doc) Gibbs energy of dielectrics

    \[G = F - E D\]

  • (doc) Internal energy change via heat and electric displacement change

    \[dU = \delta Q + E dD\]

• (doc) laws.thermodynamics.equations_of_state

  • (doc) Dieterici equation

    \[p \left(V_{m} - b\right) = R T \exp{\left(- \frac{a}{R T V_{m}} \right)}\]

  • (doc) Ideal gas equation

    \[p V = n R T\]

• (doc) laws.thermodynamics.equations_of_state.van_der_waals

  • (doc) Critical molar volume

    \[v_{\text{c},\text{m}} = 3 b\]

  • (doc) Critical pressure

    \[p_\text{c} = \frac{a}{27 b^{2}}\]

  • (doc) Critical temperature

    \[T_\text{c} = \frac{8 a}{27 R b}\]

  • (doc) Dimensionless equation

    \[\left(p_{r} + \frac{3}{V_{r}^{2}}\right) \left(V_{r} - \frac{1}{3}\right) = \frac{8 T_{r}}{3}\]

  • (doc) Molar internal energy

    \[u_\text{m} = \int c_{V, \text{m}}{\left(T \right)}\, dT - \frac{a}{v_\text{m}}\]

  • (doc) Reduced pressure

    \[p_{r} = \frac{p}{p_\text{c}}\]

  • (doc) Reduced temperature

    \[T_{r} = \frac{T}{T_\text{c}}\]

  • (doc) Reduced volume

    \[V_{r} = \frac{V}{V_\text{c}}\]

  • (doc) Second virial coefficient

    \[C_{2} = b - \frac{a}{R T}\]

  • (doc) Van der Waals equation

    \[\left(p + \frac{a}{v_\text{m}^{2}}\right) \left(v_\text{m} - b\right) = R T\]

• (doc) laws.thermodynamics.euler_relations

  • (doc) Enthalpy formula

    \[H = T S + \mu N\]

  • (doc) Gibbs energy formula

    \[G = \mu N\]

  • (doc) Internal energy formula

    \[U = T S - p V + \mu N\]

• (doc) laws.thermodynamics.fermi_dirac_statistics

  • (doc) Single-particle state distribution

    \[N_{i} = \frac{1}{\exp{\left(\frac{E_{i} - \mu}{k_\text{B} T} \right)} + 1}\]

• (doc) laws.thermodynamics.heat_transfer

  • (doc) Equation in homogeneous medium in one dimension

    \[\frac{\partial}{\partial t} T{\left(x,t \right)} = \alpha \frac{\partial^{2}}{\partial x^{2}} T{\left(x,t \right)}\]

  • (doc) General heat equation in 3D

    \[\rho c_{p} \frac{\partial}{\partial t} T{\left({\vec r},t \right)} = \text{div} \, k{\left({\vec r} \right)} \text{grad} \, T{\left({\vec r},t \right)} + q{\left({\vec r} \right)}\]

  • (doc) General equation in one dimension

    \[\rho c_{p} \frac{\partial}{\partial t} T{\left(x,t \right)} = \frac{\partial}{\partial x} k{\left(x \right)} \frac{\partial}{\partial x} T{\left(x,t \right)} + q{\left(x,t \right)}\]

  • (doc) Solution with zero temperature boundaries

    \[T_{n} = B_{n} \sin{\left(\frac{N \pi x}{x_\text{max}} \right)} \exp{\left(- \alpha \left(\frac{N \pi}{x_\text{max}}\right)^{2} t \right)}\]

• (doc) laws.thermodynamics.maxwell_boltzmann_statistics

  • (doc) Energy distribution

    \[f(E) = \frac{2 \sqrt{\frac{E}{\pi}}}{T^{\frac{3}{2}} k_\text{B}^{\frac{3}{2}}} \exp{\left(- \frac{E}{k_\text{B} T} \right)}\]

  • (doc) Most probable speed

    \[v_\text{prob} = \sqrt{\frac{2 k_\text{B} T}{m}}\]

  • (doc) Single-particle discrete distribution

    \[N_{i} = \frac{N}{Z} \exp{\left(- \frac{E_{i}}{k_\text{B} T} \right)}\]

  • (doc) Speed distribution

    \[f(v) = \sqrt{\frac{2}{\pi}} \left(\frac{m}{k_\text{B} T}\right)^{\frac{3}{2}} v^{2} \exp{\left(- \frac{m v^{2}}{2 k_\text{B} T} \right)}\]

  • (doc) Statistical weight of macrostate

    \[\Omega = \sum_i {N}_{i}! \prod_i {N}_{i}!^{-1}\]

  • (doc) Velocity component distribution

    \[f(v_{k)} = \sqrt{\frac{m}{2 \pi k_\text{B} T}} \exp{\left(- \frac{m v_{k}^{2}}{2 k_\text{B} T} \right)}\]

• (doc) laws.thermodynamics.relativistic

  • (doc) Reduced temperature in Maxwell—Jüttner statistics

    \[\theta = \frac{k_\text{B} T}{m c^{2}}\]

• (doc) laws.waves

  • (doc) Average power of sinusoidal wave on stretched string

    \[P = \frac{\mu v \omega^{2} u_\text{max}^{2}}{2}\]

  • (doc) Displacement in interfering waves

    \[u = 2 u_\text{max} \cos \left( \frac{\varphi}{2} \right) \sin \left( k x - \omega t + \frac{\varphi}{2} \right)\]

  • (doc) Displacement in standing wave

    \[u = 2 u_\text{max} \sin(k x) \cos(\omega t)\]

  • (doc) Frequency shift from speed in arbitrary motion

    \[f_\text{o} = \frac{f_\text{s} \left(v - v_\text{o} \cos{\left(\theta_\text{o} \right)}\right)}{v - v_\text{s} \cos{\left(\theta_\text{s} \right)}}\]

  • (doc) Frequency shift from speed in collinear motion

    \[f_\text{o} = \frac{f_\text{s} \left(v - v_\text{o}\right)}{v + v_\text{s}}\]

  • (doc) Fully constructive interference condition

    \[\varphi = 2 \pi N\]

  • (doc) Fully destructive interference condition

    \[\varphi = \left(1 + 2 N\right) \pi\]

  • (doc) Group velocity from dispersion relation

    \[v_\text{g} = \frac{d}{d k} \omega{\left(k \right)}\]

  • (doc) Intensity of sound wave via displacement amplitude

    \[I = \frac{\rho v \omega^{2} s_\text{max}^{2}}{2}\]

  • (doc) Light frequency change is proportional to gravitational potential change

    \[\frac{df}{f} = - \frac{d \phi}{c^{2}}\]

  • (doc) Peak wavelength via temperature

    \[\lambda_\text{peak} = \frac{b}{T}\]

  • (doc) Phase of traveling wave

    \[\varphi = k x - \omega t\]

  • (doc) Phase shift between two points

    \[\varphi = \frac{2 \pi d}{\lambda}\]

  • (doc) Phase speed of wave on stretched string

    \[v = \sqrt{\frac{T}{\mu}}\]

  • (doc) Phase speed from angular frequency and wavenumber

    \[v = \frac{\omega}{k}\]

  • (doc) Photoelectron energy from photon energy

    \[K_\text{max} = E - W\]

  • (doc) Photon energy is proportional to angular frequency

    \[E = \hbar \omega\]

  • (doc) Photon energy is proportional to linear frequency

    \[E = h f\]

  • (doc) Photon momentum is proportional to angular wavenumber

    \[p = \hbar k\]

  • (doc) Photon momentum is proportional to energy

    \[p = \frac{E}{c}\]

  • (doc) Position of antinodes in standing wave

    \[x = \frac{\left(N + \frac{1}{2}\right) \lambda}{2}\]

  • (doc) Position of nodes in standing wave

    \[x = \frac{N \lambda}{2}\]

  • (doc) Pressure amplitude in sound wave

    \[(\Delta p)_\text{max} = v \rho \omega s_\text{max}\]

  • (doc) Refractive index via permittivity and permeability

    \[n = \sqrt{\varepsilon_\text{r} \mu_\text{r}}\]

  • (doc) Resonant frequencies of stretched string with fixed ends

    \[f = \frac{N v}{2 l}\]

  • (doc) Sine of Mach cone angle via Mach number

    \[\sin{\left(\varphi \right)} = \frac{1}{\text{M}}\]

  • (doc) Speed of light via vacuum permittivity and permeability

    \[c = \frac{1}{\sqrt{\varepsilon_0 \mu_0}}\]

  • (doc) Speed of sound via bulk modulus and density

    \[v = \sqrt{\frac{K}{\rho}}\]

  • (doc) General solution to wave equation in one dimension

    \[u = f{\left(\varphi \right)}\]

  • (doc) Wavelength from phase speed and period

    \[\lambda = v T\]

  • (doc) Wavelength of standing wave in string with fixed ends

    \[\frac{N \lambda}{2} = l\]

  • (doc) Wave speed from medium permittivity and permeability

    \[v = \frac{c}{\sqrt{\varepsilon_\text{r} \mu_\text{r}}}\]

  • (doc) Wave speed from medium

    \[v = \frac{c}{n}\]

• (doc) laws.waves.blackbody_radiation

  • (doc) Spectral energy density at all frequencies

    \[w_{f} = \frac{8 \pi h f^{3}}{c^{3}} \frac{1}{\exp{\left(\frac{h f}{k_\text{B} T} \right)} - 1}\]

  • (doc) Spectral energy density at high frequency limit

    \[w_{f} = \frac{8 \pi h f^{3}}{c^{3}} \exp{\left(- \frac{h f}{k_\text{B} T} \right)}\]

  • (doc) Spectral energy density at low frequency limit

    \[w_{f} = \frac{8 \pi f^{2} k_\text{B} T}{c^{3}}\]

• (doc) laws.waves.relativistic

  • (doc) Frequency shift from speed and angle

    \[f_\text{o} = \frac{f_\text{s} \sqrt{c^{2} - v^{2}}}{c - v \cos{\left(\varphi \right)}}\]

  • (doc) Longitudinal frequency shift from speeds

    \[f_\text{o} = \frac{f_\text{s} \left(1 - \frac{v_\text{o}}{v}\right)}{1 + \frac{v_\text{s}}{v}} \sqrt{\frac{1 - \left(\frac{v_\text{s}}{c}\right)^{2}}{1 - \left(\frac{v_\text{o}}{c}\right)^{2}}}\]

  • (doc) Longitudinal frequency shift from speed

    \[f_\text{o} = f_\text{s} \sqrt{\frac{c - v}{c + v}}\]