• 107328  Infos

# List of elementary physics formulae

A list of elementary physics formulae commonly appearing in high-school and college introductory physics courses
Where possible this list has avoided using any specific system of units Axioms are in green definitions are in blue and theorems are black

## SI Prefixes

significand$times 1000^n$

n SI prefix name etymology
8 Y yotta- [1]octo
7 Z zetta- septem
6 E exa- hexa
5 P peta- penta
4 T tera- tetra
3 G giga- giant
2 M mega- great
1 k kilo- thousand (Greek)
-1 m milli- thousand (Latin)
-2 µ micro- small
-3 n nano- dwarf
-4 p pico- small
-5 f femto- (fifteen/3 = 5) (Danish)
-6 a atto- (eighteen/3 = 6) (Danish)
-7 z zepto- septem
-8 y yocto- [2]octo

## Translation and rotation

 time $t$ $t$ mass $m$ $m$ position $x$ $theta$ duration $Delta t$ $Delta t$ displacement $Delta x$ $Delta theta$ conservation of mass $Delta m$ $Delta m$ velocity $v = dx/dt$ $omega = dtheta/dt$ acceleration $a = dv/dt$ $alpha = domega/dt$ jerk $j = da/dt$ $j = dalpha/dt$ inertia $\left\{color\left\{blue\right\} m = int dm = Sigma m_i \right\}$ $\left\{color\left\{blue\right\} I = int r^2 dm = Sigma r^2m_i \right\}$ kinetic energy $E_k = mv^2/2$ $E_k = Iomega^2/2$ momentum $P = mv$ $L = Iomega$ conservation of mass $Delta m = 0$ $Delta m = 0$ conservation of energy $Delta E = 0$ $Delta E = 0$ conservation of momentum $Delta P = 0$ $Delta L = 0$ force $\left\{color\left\{blue\right\}f = dP/dt \right\} = ma = -dU/dx$ $\left\{color\left\{blue\right\}tau = dL/dt \right\} = Ialpha$ $= mathbf\left\{r\right\} times mathbf\left\{f\right\}$ impulse $\left\{color\left\{blue\right\}J=int f dt\right\}$ $\left\{color\left\{blue\right\}J=int tau dt\right\}$ work $\left\{color\left\{blue\right\}W = int f dx \right\} = mathbf\left\{f\right\} cdot mathbf\left\{d\right\}$ $\left\{color\left\{blue\right\}W = int tau dtheta \right\}$ power $\left\{color\left\{blue\right\} P = dW/dt \right\} = fv$ $\left\{color\left\{blue\right\} P = dW/dt \right\} = tauomega$ Newton's Third Law $f_\left\{ab\right\} = - f_\left\{ba\right\}$ $tau_\left\{ab\right\} = -tau_\left\{ba\right\}$

## Equations for constant acceleration

 displacement $Delta v = at$ $Delta omega = alpha t$ time $Delta\left(v^2\right) = 2aDelta x$ $Delta\left(omega^2\right) = 2alphaDelta theta$ acceleration $Delta x = tDelta v/2$ $Delta theta = tDelta omega/2$ initial velocity $Delta x = -at^2/2 + v_2t$ $Delta theta = -alpha t^2/2 + omega_2t$ final velocity $Delta x = +at^2/2 + v_1t$ $Delta theta = +alpha t^2/2 + omega_1t$

## Common forces

 spring force lies parallel to spring $f_k = -kd$ weight points toward the centre of gravity $f_g = -f_n = mg$ tension lies within the cord $f_t = f$ static friction maxiumum lies tangent to the surface $f_\left\{s\right\}=mu_sf_n$ kinetic friction lies tangent to the surface $f_\left\{k\right\}=mu_kf_n$ drag force tangent to the path $f_\left\{d\right\} =mu_drho a v^2/2$

## Mapping rotation to translation

 centripetal force points to the axis of rotation $f_c = mv^2/r$ angular to linear displacement $x = theta r$ angular to linear speed $v = theta omega$ angular to linear acceleration tangential component $a_t = alpha r$ angular to linear acceleration radial component $a_r = omega^2r$ angular to linear acceleration centripetal acceleration $alpha=v^2/r$

## Other

 work done by a spring positive when relaxes $W_s = -kDelta\left(x^2\right)/2$ inertial frames $x_\left\{PA\right\} = x_\left\{PB\right\} + x_\left\{AB\right\}$ $v_\left\{PA\right\} = v_\left\{PB\right\} + v_\left\{AB\right\}$ $a_\left\{PA\right\} = a_\left\{PB\right\} + 0$ greatest distance $1/8$ rotations trajectory $y=xtantheta-gx^2/\left(2V_0costheta\right)^2$ flight distance $v_0^2sin\left\{2theta\right\}/g$ flight time? $T=2pi r/v$ terminal velocity $v_t=sqrt\left\{2fg/\left(mu_drho A\right)\right\}$ elastic potential energy $u\left(x\right)=kx^2/2$ mechanical energy $\left\{color\left\{blue\right\} E_\left\{mec\right\}=K + U\right\}$ work by type of energy $W = Delta E_\left\{mec\right\} + Delta E_\left\{th\right\} + Delta E_\left\{int\right\}$ friction creates heat $Delta E_\left\{th\right\} = f_kd$ center of mass COM $mathbf\left\{r\right\}_\left\{com\right\}=M^\left\{-1\right\}Sigma m_i mathbf\left\{r\right\}_i$ $x_\left\{com\right\}=M^\left\{-1\right\}int x dm, cdots$ for constant density: $x_\left\{com\right\}=V^\left\{-1\right\}int x dV, cdots$ COM is in all planes of symmetry  elastic collision $Delta E_k = 0$ inelastic collision $Delta E_k =$maximum ? $mathbf\left\{P\right\}_\left\{1i\right\}+mathbf\left\{P\right\}_\left\{2i\right\}=mathbf\left\{P\right\}_\left\{1f\right\}+mathbf\left\{P\right\}_\left\{2f\right\}$ system COM remains inert $mathbf\left\{v\right\}_\left\{com\right\}=\left\{\left(mathbf\left\{P\right\}_\left\{1i\right\}+mathbf\left\{P\right\}_\left\{2i\right\}\right)over\left(M_1+M_2\right)\right\} = const$ elastic collision 1D, M2 stationary $v_\left\{1f\right\}=\left\{\left(m_1 - m_2\right)over\left(m_1 + m_2\right)\right\}v_\left\{1i\right\}$ $v_\left\{2f\right\}=\left\{\left(2m_1\right)over\left(m_1 + m_2\right)\right\}v_\left\{1i\right\}$ first rocket equationthrust $t = Rv_\left\{rel\right\}=Ma$ second rocket equation $-v_i + v_f = v_\left\{rel\right\}ln\left(M_i/M_f\right)$ parallel axis theorem $I = I_\left\{com\right\} + Mh^2$ smooth rolling $v_\left\{com\right\}=Romega$ $x_\left\{arc\right\}=Rtheta$ $x_\left\{com\right\}=Ralpha$ down a ramp axis x $a_\left\{comx\right\}=-frac\left\{gsintheta\right\}\left\{I_\left\{com\right\}/MR^2\right\}$

## Thermodynamics

 irreversible  adiabatic $Delta Q = 0$ molecules $N$ temperature $T$ degrees of freedom $f$ linear expansion $dL/dt = alpha L$ volume expansion $dV/dt = 3 alpha V$ heat $Delta E_Q$ due to $Delta T$ heat capacity $C_\left\{th\right\} = Delta T/Q$ specific heat $c_\left\{th\right\} = mDelta T/Q$ heat of vaporization $L_v = Q/m$ heat of fusion $L_f = Q/m$ work due to gas expansion $W = int_\left\{i\right\}^\left\{f\right\}pdV$ adiabatic $Delta E_\left\{int\right\} = W$ constant volume $Delta E_\left\{int\right\} = Q$ free expansion $Delta E_\left\{int\right\} = 0$ closed cycle $Q + W = 0$ thermal conductivity $kappa$ thermal resistance $R=L/ kappa$ conduction rate $P_\left\{cond\right\} = A\left(T_H - T_C\right)/\left(L/kappa\right) = Q/t$ steady state conduction through a composite slab $P_\left\{cond\right\} = A\left(T_H - T_C\right)/Sigma\left(L/kappa\right)$ Stefan-Boltzmann constant $sigma = 567*10^\left\{-8\right\}?$ thermal radiation $P_\left\{rad\right\} = sigma epsilon A T ^4_\left\{sys\right\}$ thermal absorption $P_\left\{rad\right\} = sigma epsilon A T ^4_\left\{env\right\}$ Boltzmann Constant $k = 138 times 10^\left\{-23\right\}J/K$ ideal gas law $PV = kTN$ work constant temperature $W=kTNln\left(V_f/V_i\right)$ work constant volume $W=0$ work constant pressure $W=pDelta V$ translational energy $E_\left\{kavg\right\} = kTf/2$ internal energy $E_\left\{int\right\} = NkTf/2$ mean speed $v_\left\{avg\right\}= sqrt\left\{\left(kT/m\right)\left(8/pi\right)\right\}$ mode speed $v_\left\{prb\right\} = sqrt\left\{\left(kT/m\right)2\right\}$ root mean square speed $v_\left\{rms\right\} = sqrt\left\{\left(kT/m\right)3\right\}$ mean free path $lambda = 1/\left( sqrt\left\{2\right\} pi d^2 N / V\right)$? Maxwell's Distribution] $P\left(v\right)=4pi\left(m/\left(2pi kT\right)\right)^\left\{3/2\right\}V^2e^\left\{-\left(mv^2/\left(2kT\right)\right)\right\}$ molecular specific heat at a constant volume $C_V = Q/\left(NDelta T\right)$ ? $Delta E_\left\{int\right\} = NC_V Delta T$ molecular specific heat at a constant pressure $C_p = Q/\left(NDelta T\right)$ ? $W = p Delta V = Nk Delta T$ ? $k = C_p - C_V$ adiabatic expansion $pV^\left\{gamma\right\} = constant$ adiabatic expansion $TV^\left\{gamma - 1\right\} = constant$ four special processes on page 529  multiplicity of configurations $W = N!/n_1!n_2!$ microstate in one half of the box $n_1 n_2$ Boltzmann's equation] $S = klnW$ entropy $\left\{color\left\{blue\right\}S = - ksum_i P_i ln P_i !\right\}$ change in entropy $Delta S = int_i^f\left(1/T\right)dQ approx Q/T_\left\{avg\right\}$ change in entropy $Delta S = kNln\left(V_f/V_i\right) + NC_Vln\left(T_f/T_i\right)$ force due to entropy $f = -Tds/dx$ engine efficiency W|/|Q_H| Carnot engine efficiency Q_H|-|Q_L|)/|Q_H| = (T_H-T_L)/T_H refrigerator performance Q_L|/|W| Carnot refrigerator performance Q_L|/(|Q_H|-|Q_L|) = T_L/(T_H-T_L) Zeroth Law of Thermodynamics temperature is an equivalence relation First Law of Thermodynamics $Delta E_\left\{int\right\} = Q + W$ Second Law of Thermodynamics $Delta S ge 0$ Third Law of Thermodynamics $S = S_\left\{structural\right\} + CT$

## Waves

 phasor  node  antinode  period $T$ amplitude $x_m$ decibel $dB$ torsion constant $kappa = -tau / theta$ amplitude $x_m$ frequency $f = 1/T = omega /\left(2pi\right)$ angular frequency $omega = 2pi f = 2pi / T$ phase angle $phi$ phase $\left(omega t + phi\right)$ damping constant $b$ damping force $f_d = -bv$ phase $ky -omega t$ angular number] $k$ phase constant $phi$ linear density $mu$ harmonic number $n$ harmonic series $f = v/lambda = nv/\left(2L\right)$ wavelength $lambda = k/\left(2pi\right)$ bulk modulus $B = Delta p /\left(Delta V / V\right)$ path length difference $Delta L$ resonance $omega_d = omega$ phase difference $phi = 2 pi Delta L / lambda$ fully constructive interference $Delta L/lambda = n$ fully destructive interference $Delta L/lambda = n+05$ intensity $I = P/A = rho v omega^2 s^2_m/2$ source power $P_s$ sound intensity by distance $I = P_s/\left(4pi r^2\right)$ standard reference intensity $I_0$ sound level $Beta = \left(10 dB\right)log\left(I/I_0\right)$ pipe two open ends $f=v/lambda = nv/\left(2L\right)$ pipe one open end $f = v/lambda = nv/\left(4L\right)$ for n odd beats $s\left(t\right) =\text{'} tcos omega t$ beat frequency $f_\left\{beat\right\} = f_1 - f_2$ Doppler effect $f\text{'} = f\left(v+-v_D\right)/\left(v+-v_S\right)$ Mach cone angle $sin theta = v/v_s$ average wave power $P_\left\{avg\right\}=mu v omega^2 x_m^2/2$ pressure amplitude $Delta p_m = \left(vrho omega \right)x_m$ wave equation $frac\left\{partial y\right\}\left\{partial x^2\right\} = frac\left\{1\right\}\left\{v^2\right\} frac\left\{partial ^2 y\right\}\left\{partial t^2\right\}$ wave superposition $x\text{'}\left(yt\right) = x_1\left(yt\right) + x_2\left(yt\right)$ wave speed $v = omega/k = lambda/T = lambda f$ wave speed on a stretched string $v=sqrt\left\{f_t/mu\right\}$ angular frequency of a linear simple harmonic oscillator $omega = sqrt\left\{k/m\right\}$ angular frequency of an angular simple harmonic oscillator $omega = sqrt\left\{I/kappa\right\}$ angular frequency of a low amplitude simple pendulum $omega = sqrt\left\{L/g\right\}$ angular frequency of a low amplitude physical pendulum $omega = sqrt\left\{I/mgh\right\}$ speed of sound $v = sqrt\left\{B/ rho \right\}$ damped angular frequency $omega \text{'} = sqrt\left\{\left(k/m\right)-\left(b^2/4m^2\right)\right\}$ displacement $x\left(t\right)=cos\left(omega t + phi\right)x_m$ damped displacement $x\left(t\right)=cos\left(omega \text{'}t+phi\right)x_m\left(e^\left\{-bt/2m\right\}\right)$ velocity $v\left(t\right)=sin\left(omega t + phi\right)x_m \left( - omega\right)$ acceleration $a\left(t\right)=cos\left(omega t + phi\right)x_m\left( - omega^2 \right)$ transverse sinusoidal wave $x\left(yt\right) = sin\left(ky-omega t\right)x_m$ wave travelling backwards $x\left(yt\right) = sin\left(ky+omega t\right)x_m$ resultant wave $x\text{'}\left(yt\right) = sin\left(ky-omega t + phi/2\right)x_m\left(2cosphi/2\right)$ standing wave $x\text{'}\left(yt\right) = cos\left(omega t\right)\left(2ysin ky\right)$ sound displacement function $x\left(yt\right) = cos\left(ky-omega t\right)x_m$ sound pressure-variation function $Delta p\left(yt\right) = sin\left(ky-omega t\right)Delta p_m$ potential harmonic energy $E_U\left(t\right) = kx^2/2 = kx_m^2cos^2\left(omega t + phi\right)/2$ kinetic harmonic energy $E_K\left(t\right) = kx^2/2 = kx_m^2sin^2\left(omega t + phi\right)/2$ total harmonic energy $E\left(t\right) = kx_m^2/2 = E_U + E_K$ damped mechanical energy $E_\left\{mec\right\}\left(t\right) = ke^\left\{-bt/m\right\}x^2_m/2$

## Gravitation

 gravity $f_G = Gm_1m_2/r^2$ potential energy $E_G = -Gm_1m_2/r$ potential energy $E_g = ma_\left\{g\right\}y$ standard gravity $\left\{color\left\{blue\right\} a_g = Gm_\left\{Earth\right\}/r_\left\{Earth\right\}^2 \right\} approx 981m/s^2$ $Wg=mgdcos theta$ ? $Delta k = W_\left\{lift\right\} + W_g$ potential work $Delta u = -W$ understand path independence $W_\left\{ab1\right\}=W_\left\{ab2\right\}=cdots$ ? $W_\left\{pull\right\} = - W_s$

## Electricity

 electric constant $epsilon_0 F/m$ electrostatic constant $k = 1/\left(4piepsilon_0\right) m/F$ Coulomb's law q_1 /r^2 elementary charge $e$ in Coloumbs electric charge $q = ne$ for integer n conservation of charge $Delta e = 0$ test charge $q_0$ electric field $mathbf\left\{E\right\} =mathbf\left\{F\right\}/q_0$ electric field of a point charge $mathbf\left\{E\right\}q =mathbf\left\{F\right\}$ electric field lines end at a negative charge r hat? $hat\left\{r\right\}$ electric field of a point charge $mathbf\left\{E\right\} = mathbf\left\{F\right\}/q_0 = \left(1/\left(epsilon_04pi\right)\right)q/r^2hat\left\{r\right\}$ electric moment] $mathbf\left\{p\right\}$ ? $z$ electric field of a moment] $E = \left(1/\left(epsilon_0 2pi\right)\right)\left(p/z^3\right)$ electric field of a charged ring $E = qz/\left(epsilon_04pi\left(z^2 + R^2\right)^\left(3/2\right)$ surface charge density $sigma = Q/A$ electric field of a charged disk $E = \left(sigma/epsilon_0 2\right)\left(1 - z/sqrt\left\{z^2 + R^2\right\}\right)$ torque on a dipole $mathbf\left\{tau\right\}=mathbf\left\{p\right\}timesmathbf\left\{E\right\}$ potential of a dipole] $U = -mathbf\left\{p\right\}cdotmathbf\left\{E\right\}$ work done on a dipole $W_a=-W=Delta U$ Gaussian surface ? flux ? ? $mathbf\left\{A\right\}$ electric flux $Phi = ointmathbf\left\{E\right\}cdot d mathbf\left\{A\right\}$ Gauss' law $q_\left\{enc\right\} = epsilon_0Phi$ Gauss' law $q_\left\{enc\right\} = epsilon_0ointmathbf\left\{E\right\}cdot d mathbf\left\{A\right\}$ volume charge density $rho = Q/V$ linear charge density $lambda = Q/L$ acceleration due to charge $a = qE/m$ conducting surface $E = sigma / epsilon_0$ electric field of a line of charge $E = lambda / epsilon_0 2pi r$ non-conducting sheet of charge $E = sigma /epsilon_0 2$ electric field outside spherical shell r>=R $E = q/epsilon_0 4 pi r^2$ electric field inside spherical shell r $E = 0$ electric field of uniform charge r<=R $E = qr/epsilon_0 4 pi R^3$ electric potential energy $U$ Work done by electric potential energy $Delta U=-W$ finite energy of the system] $U - W_infty$ electric potential $V = U/q$ electric potential difference $Delta V = Delta U/q = -W/q$ potential defined $V = -W_\left\{infty\right\}/q$ potential from the electric field $Delta V = -int_i^f mathbf\left\{E\right\}cdot dmathbf\left\{s\right\}$ potential of a point charge $V = q/epsilon_0 4 pi r$ potential of a charge group] $V = Sigma V_i = \left(1/epsilon_0 4 pi\right) Sigma q_i/r_i$ potential of a dipole $V = pcostheta/epsilon_0 4 pi r^2$ potential of continuous charge $V = int dV = \left(1/epsilon_0 4 pi\right)int dq/r$ field from potential $mathbf\left\{E\right\} = nabla V$ potential of a pair of point charges $U = W = q_2V = q_1q_2/epsilon_04pi r$ capacitance $C = q/V$ parallel plate capacitor $C = epsilon_0A/d$ cylindrical capacitor $C = epsilon_0 2 pi L/ln\left(b/a\right)$ spherical capacitor $C = epsilon_0 4 pi ba/\left(b-a\right)$ isolated spherical capacitor $C = epsilon_0 4 pi R$ parallel capacitors $C_\left\{eq\right\} = Sigma C_i$ series capacitors $1/C_\left\{eq\right\} = Sigma 1/C_i$ potential stored in a capacitor] $U=q^2/2C = CV^2/2$ energy density $u = epsilon_0 E^2/2$ dielectric constant $kappa ge 1$ dimensions of capacitance $C = epsilon_0mathcal\left\{L\right\}$ dielectric $epsilon_0 to epsilon_0kappa$ Gauss' law with dialectric $q = epsilon_0 oint kappamathbf\left\{E\right\}cdot d mathbf\left\{A\right\}$ electric displacement $mathbf\left\{D\right\} = epsilon_0 kappa mathbf\left\{E\right\}$ current $i = dq/dt$ charge density $mathbf\left\{J\right\} = i/A$ ? $i = int JdA$ drift speed $mathbf\left\{v\right\}_d$ ? $mathbf\left\{J\right\} = nemathbf\left\{v\right\}_d/m^3$ resistance $R = V/i$ resistivity in ohm-meters $rho = mathbf\left\{E\right\}/mathbf\left\{J\right\}$ conductivity $sigma = mathbf\left\{J\right\}/mathbf\left\{E\right\} = 1/rho$ ? $R/rho = L/A$ variation of with temperature] $rho - rho_0 = rho_0alpha\left(T-T_0\right)$ temperature of resistivity] $alpha$ Ohm's law $V=iR$ electrical power $P=iV$ resistive dissipation $P = i^2R = V^2/R$ emf $mathcal\left\{E\right\} = dW/dq = iR$ rules for calculating emf loop resistance emf internal resistance $i = mathcal\left\{E\right\}/\left(R+r\right)$ resistors in series $R_\left\{eq\right\}=Sigma R_i$ resistors in parallel $1/R_\left\{eq\right\} = 1/ Sigma R_i$ potential difference across a real battery $p = mathcal\left\{E\right\} - iR$ power of an emf device $P_\left\{emf\right\} = imathcal\left\{E\right\}$ Kirchoff's junction rule $i_\left\{in\right\} = i_\left\{out\right\}$ RC charging a capacitor $q = Cmathcal\left\{E\right\}\left(1-e^\left\{-t/RC\right\}$ RC charging a capacitor $i = \left(mathcal\left\{E\right\}/R\right)e^\left\{-t/RC\right\}$ RC charging a capcitor $V_C = mathcal\left\{E\right\}\left(1-e^\left\{-t/RC\right\}$ capacitive constant] $tau = RC$ magnetic field $mathbf\left\{F\right\}_B = qmathbf\left\{v\right\}timesmathbf\left\{B\right\}$ Hall effect $n = Bi/Vle$ circulating charged particle q|vB=mv^2/r cyclotron resonance condition $f = f_\left\{osc\right\}$ force on a current $mathbf\left\{F\right\}_B=imathbf\left\{L\right\}timesmathbf\left\{B\right\}$ magnetic moment $mu=NiA$ magnetic torque] $mathbf\left\{tau\right\}=mathbf\left\{mu\right\}timesmathbf\left\{B\right\}$ magnetic potential energy $U\left(theta\right)=-mathbf\left\{mu\right\}cdotmathbf\left\{B\right\}$ magnetism constant $mu_0$ in Tm/A Biot-Savart law $dmathbf\left\{B\right\} = \left(mu_0/4pi\right)\left(idmathbf\left\{s\right\}timeshat\left\{r\right\}/r^2\right)$ magnetic field due to a long straight wire $B = mu_0i/2pi R$ magnetic field due to a semi-infinite straight wire $B=mu_0i/4pi R$ magnetic field at the center of a circular arc $B=mu_0iphi/4pi R$ Ampere's law $oint mathbf\left\{B\right\}cdot dmathbf\left\{s\right\} = mu_0i_\left\{enc\right\}$ ideal solenoid $B=mu_0in$ toroid $B=mu_0iN/2pi r$ current carrying coil $mathbf\left\{B\right\}=mu_0mathbf\left\{mu\right\}/2pi z^3$ magnetic flux through A $Phi _B = int mathbf\left\{B\right\}cdot dmathbf\left\{A\right\}$ ? $Phi_B = BA$ Faraday's law $mathcal\left\{E\right\}=dPhi_B/dt$ Lenz's law $?$ Faraday's law $ointmathbf\left\{E\right\}cdot dmathbf\left\{s\right\} = -dPhi_B/dt$ inductance $L=NPhi_B/i$ solenoid $L/l=mu_0n^2A$ self-induced emf $mathcal\left\{E\right\}_L = -Ldi/dt$ RL circuit $Ldi/dt+Ri=mathcal\left\{E\right\}$ RL circuit rise of current $i = mathcal\left\{E\right\}/R\left(1-e^\left\{-t/tau_L\right\}\right)$ RL circuit constant] $tau_L=L/R$ RL circuit decay of current $i=mathcal\left\{E\right\}e^\left\{-t/tau_L\right\}/R=i_0e^\left\{-t/tau_L\right\}$ magnetic energy $U_B=Li^2/2$ magnetic density] $u_B=B^2/2mu_0$ mutual induction $mathcal\left\{E\right\}_1=-Mdi_2/dtmathcal\left\{E\right\}_2=-Mdi_1/dt$ LC circuit $omega = 1/sqrt\left\{LC\right\}$ LC oscillations $Ld^2q/dt^2+q/C = 0$ LC charge $q = Qcos\left(omega t + phi\right)$ LC current $i=-omega Q sin\left(omega t + phi\right)$ LC electrical energy $U_E=q^2/2C=Q^2cos^2\left(omega t + phi\right)/2C$ $U_B=Q^2sin^2\left(omega t + phi\right)/2C$ RLC circuit ODE $Ld^2/q/dt^2 + Rdq/dt +q/C = 0$ RLC circuit ODE solution $q = QeT^\left\{-Rt/2L\right\}cos\left(omega\text{'}t+phi\right)$ resistive load $V_R=I_RR$ capacitive reactance $X_C = 1/omega_d C$ capacitive load $V_C = I_C X_C$ inductive reactance $X_L = omega_d L$ inductive load $V_L = I_L X_L$ phase constant $tanphi=X_L - X_C /R$ electromagnetic resonance $omega_d = omega = 1/sqrt\left\{LC\right\}$ rms current $I_\left\{rms\right\}=I/sqrt\left\{2\right\}$ rms voltage $V_\left\{rms\right\}=V/sqrt\left\{2\right\}$ rms emf $mathcal\left\{E\right\}_\left\{rms\right\}=mathcal\left\{E\right\}_m/sqrt\left\{2\right\}$ average power $P_\left\{avg\right\}=mathcal\left\{E\right\}I_\left\{rms\right\}cosphi$ transformation of voltage $V_s N_p = V_p N_s$ transformation of currents $I_s N_s = I_p N_p$ transformer reistance $R_eq = \left(Np/Ns\right)^2R$ Gauss' law for magnetic fields $Phi_B = oint mathbf\left\{B\right\} cdot d mathbf\left\{A\right\} = 0$ Maxwell's law of induction $oint mathbf\left\{B\right\} /cdot d mathbf\left\{s\right\} = mu_0epsilon_0dPhi_E/dt$ Ampere-Maxwell law $oint mathbf\left\{B\right\} cdot d mathbf\left\{s\right\} = mu_0epsilon_0dPhi_E/dt + mu_0i_\left\{enc\right\}$ displacement current $i_d = epsilon_0 dPsi_E/dt$ Ampere-Maxwell law $oint mathbf\left\{B\right\} cdot dmathbf\left\{s\right\} = mu_0i_\left\{denc\right\} + mu_0i_\left\{enc\right\}$ induced field inside a circular capacitor] $B = \left(mu_0i_d/2pi R^2\right)r$ induced field outside a circular capacitor] $B = mu_0i_d/2pi rr$ spin magnetic moment] $mathbf\left\{mu_s\right\} = -emathbf\left\{S\right\}/m$ Bohr magneton $mu_B = eh/4pi m$ ? $U = -mathbf\left\{mu\right\}_scdotmathbf\left\{B\right\}_\left\{ext\right\} = -mu_\left\{sz\right\}B_\left\{ext\right\}$ orbital magnetic moment] $mathbf\left\{mu\right\}_\left\{orb\right\}=-emathbf\left\{L\right\}_orb/2m$ ? $U = -mathbf\left\{mu\right\}_\left\{orb\right\}cdotmathbf\left\{B\right\}_\left\{ext\right\} = -mu_\left\{orbz\right\}B_\left\{ext\right\}$

## Light

 electric component] $E = E_m sin\left(kx-omega t\right)$ magnetic component] $B = B_m sin\left(kx-omega t\right)$ celerity $c = 1/sqrt\left\{mu_0epsilon_0\right\}$ magnitude ratio $E/B = c$ Poynting vector $mathbf\left\{S\right\} = mu_0^\left\{-1\right\}mathbf\left\{E\right\}timesmathbf\left\{B\right\}$ instantaneous flow rate] $S = E^2/cmu_0$ intensity $I = E^2_\left\{rms\right\}/cmu_0$ intensity at the sphere $I = P_s/4pi r^2$ total absorption $Delta p = Delta U/c$ total reflection back along path $Delta p = 2 Delta U/c$ total absorption $p_r = Ic$ total reflection back along path $p_r = 2I/c$ one-half rule $I = I_0/2$ cosine-squared rule $I = I_0cos^2theta$ law of reflection $theta_1\text{'}=theta_1$ law of refraction (snell's) $n_1sintheta_1 = n_2sintheta_2$ critical angle $theta_c = sin^\left\{-1\right\}n_2/n_1$ Brewster angle $theta_B = tan^\left\{-1\right\}n_2/n_1$ plane mirror $i = -p$ spherical mirror $f =r/2$ spherical mirror $1/p + 1/i = 1/f$ lateral magnification m|=h'/h lateral magnification $m=-i/p$ ? $n_1/p + n_2/i = \left(n_2 - n_1\right)/r$ thin lens $1/f = 1/p +1/i$ thin lens in air $1/f = \left(n-1\right)\left(1/r_1 - 1/r_2\right)$ index of refraction $n=c/v$ path length difference $Delta L = d sintheta$ maxima - bright fringes $d sintheta = mlambda$ for m natural minima - dark fringes $d sintheta = \left(m + 1/2\right)lambda$ for m natural double-slit interference intensity $I = 4I_0cos^2\left(theta/2\right)$ ? $theta = 2pi d sintheta / lambda$ bright film in air maxima $2L = \left(m + 1/2\right)lambda/n_2$ bright film in air minima $\left(m\right)lambda/n_2$ single-slit minima $a sin theta = mlambda$ single-slit intensity $I\left(theta\right)=I_m\left(sinalpha/alpha\right)^2$ ? $alpha = phi/2 = pi a sintheta/lambda$ circular aperture first minimum $sintheta = 122lambda/d$ Rayleigh's criterion $theta_R = 122lambda/d$ double slit intensity $I\left(theta\right) = I_m\left(cos^2Beta\right)\left(sinalpha/alpha\right)^2$ diffraction grating maxima lines $dsintheta = mlambda$ for natural m diffraction grating half-width $Deltatheta_\left\{hw\right\} = lambda/Ndcostheta$ diffraction grating dispersion $D=m/d costheta$ diffraction grating resolving power $R=Nm$ Bragg's law $2dsintheta = mlambda$

## Relativity

 The Relativity Postulate  The of Light Postulate]  Lorentz factor $gamma = 1/sqrt\left\{1-\left(v/c\right)^2\right\}$ speed parameter $Beta = v/c$ time dilation $Delta t = gamma Delta t_0$ length contraction $L = L_0/gamma$ Lorentz transformation $x\text{'}=gamma\left(x-vt\right)$ $y\text{'} = y$ $z\text{'} = z$ $t\text{'} = gamma\left(t-xv/c^2\right)$ Doppler effect for light $f=f_0sqrt\left\{1-Beta/1+Beta\right\}$ wavelength Doppler shift Deltalambda|c/lambda_0 momentum $mathbf\left\{p\right\}=gamma mmathbf\left\{v\right\}$ rest energy $E_0 = mc^2$ total energy $E = E_0 + K = mc^2 + K = gamma mc^2$ $Q = -Delta mc^2$ kinetic energy $K = E - mc^2 = gamma mc^2 - mc^2 = mc^2\left(gamma -1\right)$ momentum and energy triangle] $E = sqrt\left\{\left(pc\right)^2 + \left(mc^2\right)^2\right\}$

## Particles

 quantum constant $h$ in energy/frequency photon energy $E = hf$ photoelectric equation $hf = K_\left\{max\right\} + Phi$ photon momentum $p = hf/c = h/lambda$ de Broglie wavelength $lambda = h/p$ Schrodinger's equation one dimensional motion $d^2psi/dx^2 + 8pi^2m\left[3\right]psi/h^2 = 0$ Schrodinger's equation free particle $d^2psi/dx^2 + k^2psi = 0$ Heisenberg's principle] $Delta x cdot Delta p_x ge hbar$ infinite well] $E_n = h^2n^2/8mL^2$ energy changes $Delta E = hf$ wave function of a trapped electron $psi_n\left(x\right) = A sin\left(npi x/L\right)$ for positive int n probability of detection $p\left(x\right) = psi^2_n\left(x\right)dx$ normalization $int psi^2_n\left(x\right)dx = 1$ hydrogen orbital energy $E_n = 1361eV/n^2$ for positive int n hydrogen changes] $1/lambda = R\left(1/n^2_\left\{low\right\} - 1/n^2_\left\{high\right\}$ hydrogen radial density] $P\left(r\right) = 4r^2/a^3e^\left\{2r/a\right\}$ spin magnetic quantum number $m_s in \left\{-1/2+1/2\right\}$ orbital magnetic moment] $mathbf\left\{mu\right\}_\left\{orb\right\} = -emathbf\left\{L\right\}/2m$ orbital magnetic moment components] $mathbf\left\{mu\right\}_\left\{orbz\right\} = -m_mathcal\left\{L\right\}mu_B$ Bohr magneton $mu_B = ehbar/2m$ angular components] $L_z = mmathcal\left\{L\right\}hbar$ spin momentum magnitude] $S = hbarsqrt\left\{s\left(s+1\right)\right\}$ cutoff wavelength $lambda_\left\{min\right\} = hc/K_0$ density of states $N\left(E\right) = 8sqrt\left\{2\right\}pi m^\left\{3/2\right\}E^\left\{1/2\right\}/h^3$ occupancy probability $P\left(E\right) = 1/\left(e^\left\{\left(E-E_F\right)/kT\right\}+1\right)$ Fermi energy $E_F = \left(3/16sqrt\left\{2\right\}pi\right)^\left\{2/3\right\}h^2n^\left\{2/3\right\}m$ mass number $A = Z+N$ nuclear radii $r=r_0A^\left\{1/3\right\}$ mass excess $Delta = M - A$ radioactive decay $N = N_0e^\left\{-lambda t\right\}$ conservation of number]  conservation of number]  conservation of strangeness  the eightfold way  weak force  strong force  Hubble constant $H = 710km/s$ Hubble's law $v=Hr$