List of elementary physics formulae

In physics, there are equations in every field to relate physical quantities to each other and perform calculations. Entire handbooks of equations can only summarize most of the full subject, else are highly specialized within a certain field. The scope of the article is as follows:

1. General equations which are constructed from definitions or physical laws, or to the same status universal principles, but themselves not formulae of these types.
2. General equations which are or allow any of the following:

large applicability on a specific but important topic,

large applicability on a general topic,

reduce to a number of idealized special cases.

3. Formulae which frequently appear in physics literature, in a small addition some perhaps less common formulae.
4. Level of study is typically that of advanced school/ introductory degree level Physics, and beyond.
5. For generality, vector calculus and multivariable calculus is the main formalism; inline with complex numbers and some linear algebra (such as matrices and coordinate systems).

Only SI units and their corresponding dimensions are used; no natural/characteristic units or non-dimensional equations are included.

Nomenclature

The short-hand notation for the square of a vector, as the dot product of a vector with itself is used: $\mathbf {A} ^{2}\equiv \mathbf {A} \cdot \mathbf {A} \cos 0\equiv \left|\mathbf {A} \right|^{2}\equiv A^{2}.\,\!$ No confusion should arise by mistaking it for the cross product since the cross product of a vector with itself is always the null vector.

All symbols are matched as to standard closely as possible, but due to a variety of notations for any given variable, every table below has locally defined variables.

Classical mechanics

Galilean frame transforms

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame travelling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Motion of entities	Inertial frames	Accelerating frames
Translation V = Constant relative velocity between two inertial frames F and F'. A = (Variable) relative acceleration between two accelerating frames F and F'.	Relative position $\mathbf {r} '=\mathbf {r} +\mathbf {V} t\,\!$ Relative velocity $\mathbf {v} '=\mathbf {v} +\mathbf {V} \,\!$ Equivalent accelerations $\mathbf {a} '=\mathbf {a}$	Relative accelerations $\mathbf {a} '=\mathbf {a} +\mathbf {A}$ Apparent/ficticous forces $\mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }$
Rotation Ω = Constant relative angular velocity between two frames F and F'. Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.	Relative angular position $\theta '=\theta +\Omega t\,\!$ Relative velocity ${\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}\,\!$ Equivalent accelerations ${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}$	Relative accelerations ${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}$ Apparent/ficticous torques ${\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }$
	Transformation of any vector T to a rotating frame ${\frac {{\rm {d}}\mathbf {T} '}{{\rm {d}}t}}={\frac {{\rm {d}}\mathbf {T} }{{\rm {d}}t}}-{\boldsymbol {\Omega }}\times \mathbf {T}$

General results

General results from classical mechanics

Physical situation	Nomenclature	Equations
General work-energy theorem (translation and rotation)	r = position vector of a point on the body, W = work done by an external agent F = force exterted by an external agent at r, τ = torque exerted by an external agent on the body at r due to F C = curve which the agent exerts the force/torque along	$W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\left|{\boldsymbol {\tau }}\right|}{\mathrm {d} \theta }\right)\,\!$
Velocity and acceleration of a rotating rigid body	ω = angular velocity of rotation α = angular acceleration of rotation v = acceleration at point r about some axis a = acceleration at point r about some axis	$\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!$ $\mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v} \,\!$
Translational and angular momentum of a rotating rigid body	m = moment of mass at r about some axis p = translational momentum at r about some axis L = angular momentum about some axis I = moment of inertia tensor about some axis	$\mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} \,\!$ $\mathbf {L} =\mathbf {I} \cdot {\boldsymbol {\omega }}\,\!$
Coriolis acceleration and force	c subscripts refer to coriolis	$\mathbf {a} _{c}=\mathbf {F} _{c}/m=-2{\boldsymbol {\omega \times v}}$
Euler's equations		$\mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}\,\!$

Mechanical oscillators

SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.

Equations of motion

Physical situation	Nomenclature	Translational equations	Angular equations
SHM	x = Transverse displacement θ = Angular displacement A = Transverse amplitude Θ = Angular amplitude	${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega ^{2}x\,\!$ Solution: $x=A\sin \left(\omega t+\phi \right)\,\!$	${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\omega ^{2}\theta \,\!$ Solution: $\theta =\Theta \sin \left(\omega t+\phi \right)\,\!$
Unforced DHM	b = damping constant κ = torsion constant	${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega ^{2}x=0\,\!$ Solution (see below for ω'): $x=Ae^{-bt/2m}\cos \left(\omega '\right)\,\!$ Resonant frequency: $\omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {b}{4m}}\right)^{2}}}\,\!$ Damping rate: $\gamma =b/m\,\!$ Expected lifetime of excitation: $\tau =1/\gamma \,\!$	${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\omega ^{2}\theta =0\,\!$ Solution: $\theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)\,\!$ Resonant frequency: $\omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {\kappa }{4m}}\right)^{2}}}\,\!$ Damping rate: $\gamma =\kappa /m\,\!$ Expected lifetime of excitation: $\tau =1/\gamma \,\!$

Angular frequencies

Physical situation	Nomenclature	Equations
Linear undamped unforced SHO	k = spring constant m = mass of oscillating bob	$\omega ={\sqrt {\frac {k}{m}}}\,\!$
Linear unforced DHO	k = spring constant b = Damping coefficient	$\omega '={\sqrt {{\frac {k}{m}}-\left({\frac {b}{2m}}\right)^{2}}}\,\!$
Low amplitude angular SHO	I = Moment of inertia about oscillating axis κ = torsion constant	$\omega ={\sqrt {\frac {I}{\kappa }}}\,\!$
Low amplitude simple pendulum	L = Length of pendulum g = Gravitational acceleration Θ = Angular amplitude	Approximate value $\omega ={\sqrt {\frac {g}{L}}}\,\!$ Exact value can be shown to be: $\omega ={\sqrt {\frac {g}{L}}}\left[1+\sum _{k=1}^{\infty }{\frac {\prod _{n=1}^{k}\left(2n-1\right)}{\prod _{n=1}^{m}\left(2n\right)}}\sin ^{2n}\Theta \right]\,\!$

Energy in mechanical oscillations

Physical situation	Nomenclature	Equations
SHM energy	T = kinetic energy U = potenial energy E = total energy	Potential energy $U={\frac {m}{2}}\left(x\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\cos ^{2}(\omega t+\phi )\,\!$ Maximum value at x = A: $U_{\mathrm {max} }{\frac {m}{2}}\left(\omega A\right)^{2}\,\!$ Kinetic energy $T={\frac {\omega ^{2}m}{2}}\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\sin ^{2}\left(\omega t+\phi \right)\,\!$ Total energy $E=T+U\,\!$
DHM energy		$E={\frac {m\left(\omega A\right)^{2}}{2}}e^{-bt/m}\,\!$

Fluid mechanics

Equations in fluid mechanics

Physical situation	Nomenclature	Equations
Fluid statics, pressure gradient	r = Position ρ = ρ(r) = Fluid density at gravitational equipotential containing r g = g(r) = Gravitational field strength at point r ∇P = Pressure gradient	$\nabla P=\rho \mathbf {g} \,\!$
Buoyancy equations	ρf = Mass density of the fluid Vimm = Immersed volume of body in fluid Fb = Buoyant force Fg = Gravitational force Wapp = Apparent weight of immersed body W = Actual weight of immersed body	Buoyant force $\mathbf {F} _{\mathrm {b} }=-\rho _{f}V_{\mathrm {imm} }\mathbf {g} =-\mathbf {F} _{\mathrm {g} }\,\!$ Apparent weight $\mathbf {W} _{\mathrm {app} }=\mathbf {W} -\mathbf {F} _{\mathrm {b} }\,\!$
Bernoulli's equation	pconstant is the total pressure at a point on a streamline	$p+\rho v^{2}/2+\rho gy=p_{\mathrm {constant} }\,\!$
Euler equations	ρ = fluid mass density u is the fluid velocity vector E = total volume energy density U = internal energy per unit mass of fluid p = pressure $\otimes$ denotes the tensor product	${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0\,\!$ ${\frac {\partial \rho {\mathbf {u} }}{\partial t}}+\nabla \cdot \left(\mathbf {u} \otimes \left(\rho \mathbf {u} \right)\right)+\nabla p=0\,\!$ ${\frac {\partial E}{\partial t}}+\nabla \cdot \left({\mathbf {u}}\left(E+p\right)\right)=0\,\!$ $E=\rho \left(U+{\frac {1}{2}}\mathbf {v} ^{2}\right)\,\!$
Convective acceleration		$\mathbf {a} =\left(\mathbf {v} \cdot \nabla \right)\mathbf {v}$
Navier-stokes equations	TD = Deviatoric stress tensor $\mathbf {f}$ = volume density of the body forces acting on the fluid $\nabla$ here is the del operator.	$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\nabla \cdot \mathbf {T} _{\mathrm {D} }+\mathbf {f}$

Thermodynamics

All temperatures are in thermodynamic units, not celsius or fahrenheit.

Kinetic theory

Ideal gas equations

Physical situation	Nomenclature	Equations
Ideal gas law	p = pressure V = volume of container T = temperature n = number of moles N = number of molecules k = Boltzmann’s constant	$pV=nRT=kTN\,\!$ ${\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {n_{1}T_{1}}{n_{2}T_{2}}}={\frac {N_{1}T_{1}}{N_{2}T_{2}}}\,\!$
Pressure of an ideal gas	m = mass of one molecule Mm = molar mass	$p={\frac {Nm\langle v^{2}\rangle }{3V}}={\frac {nM_{m}\langle v^{2}\rangle }{3V}}={\frac {1}{3}}\rho \langle v^{2}\rangle \,\!$

Phase transitions

Physical situation	Equations
Adiabatic transition	$\Delta Q=0,\quad \Delta U=W\,\!$
Isothermal transition	$\Delta U=0,\quad \Delta W=-\Delta Q\,\!$ For an ideal gas $W=kTN\ln(V_{2}/V_{1})\,\!$
Isobaric transition	p1 = p2, p = constant $\Delta W=p\Delta V,\quad \Delta U=\Delta Q+p\delta V\,\!$
Isochoric transition	V1 = V2, V = constant $\Delta W=0,\quad \Delta Q=\Delta U\,\!$
Adiabatic expansion	$p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }\,\!$ $T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}\,\!$
Free expansion	$\Delta U=0\,\!$
Work done by an expanding gas	Process $\Delta W=\int _{V_{1}}^{V_{1}}p\mathrm {d} V\,\!$ Net Work Done in Cyclic Processes $\Delta W=\oint _{\mathrm {cycle} }p\mathrm {d} V\,\!$

Statistical physics

Below are useful results from the Maxwell-Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting ideal gases.

Physical situation	Nomenclature	Equations
Maxwell–Boltzmann distribution	v = velocity of atom/molecule, m = mass of each molecule (all molecules are identical in kinetic theory), γ(p) = Lorentz factor as function of momentum (see below) Ratio of thermal to rest mass-energy of each molecule:$\theta =k_{B}T/mc^{2}\,\!$ </div"> K2 is the Modified Bessel function of the second kind.	Non-relativistic speeds $P\left(v\right)=4\pi \left({\frac {m}{2\pi k_{B}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{B}T}\,\!$ Relativistic speeds (Maxwell-Juttner distribution) $f(p)={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}e^{-\gamma (p)/\theta }$
Entropy Logarithm of the density of states	Pi = probability of system in microstate i Ω = total number of microstates	$S=-k_{B}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega \,\!$ where: $P_{i}=1/\Omega \,\!$
Entropy change		$\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}\,\!$ $\Delta S=k_{B}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}\,\!$
Entropic force		$\mathbf {F} _{\mathrm {S} }=-T\nabla S\,\!$
Equipartition theorem	df = degree of freedom	Average kinetic energy per degree of freedom $\langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT\,\!$ Internal energy $U=d_{f}\langle E_{\mathrm {k} }\rangle ={\frac {d_{f}}{2}}kT\,\!$

Corollaries of the non-relativistic Maxwell-Boltzmann distribution are below.

Physical situation	Nomenclature	Equations
Mean speed		$\langle v\rangle ={\sqrt {\frac {8k_{B}T}{\pi m}}}\,\!$
Root mean square speed		$v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {k_{B}T}{3m}}}\,\!$
Modal speed		$v_{\mathrm {mode} }={\sqrt {\frac {k_{B}T}{2m}}}\,\!$
Mean free path	σ = Effective cross-section n = Volume density of number of target particles Template:Ell = Mean free path	$\ell =1/{\sqrt {2}}n\sigma \,\!$

The four most common Maxwell's relations are:

Physical situation	Nomenclature	Equations
Thermodynamic potentials as functions of their natural variables	$U(S,V)\,$ = Internal energy $H(S,P)\,$ = Enthalpy $A(T,V)\,$ = Helmholtz free energy $G(T,P)\,$ = Gibbs free energy	$\left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}={\frac {\partial ^{2}U}{\partial S\partial V}}$ $\left({\frac {\partial T}{\partial P}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{P}={\frac {\partial ^{2}H}{\partial S\partial P}}$ $+\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\partial ^{2}A}{\partial T\partial V}}$ $-\left({\frac {\partial S}{\partial P}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial ^{2}G}{\partial T\partial P}}$

Thermal transfer

Physical situation	Nomenclature	Equations
Net intensity emission/absorption	Texternal = external temperature (outside of system) Tsystem = internal temperature (inside system) ε = emmisivity	$I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)\,\!$
Internal energy of a substance	CV = isovolumetric heat capacity of substance ΔT = temperature change of substance	$\Delta U=NC_{V}\Delta T\,\!$
Work done by an expanding ideal gas		$W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V=\int _{T_{1}}^{T_{2}}Nk\mathrm {d} T\,\!$
Meyer's equation	Cp = isobaric heat capacity CV = isovolumetric heat capacity n = number of moles	$C_{p}-C_{V}=nR\,\!$
Effective thermal conductivities	λi = thermal conductivity of substance i λnet = equivalent thermal conductivity	Series $\lambda _{\mathrm {net} }=\sum _{j}\lambda _{j}\,\!$ Parallel ${\frac {1}{\lambda }}_{\mathrm {net} }=\sum _{j}\left({\frac {1}{\lambda }}_{j}\right)\,\!$

Waves

In what follows n, m are any integers (Z = set of integers); $n,m\in \mathbf {Z} \,\!$.

Standing waves

Physical situation	Nomenclature	Equations
Harmonic frequencies	fn = nth mode of vibration, nth harmonic, (n-1)th overtone	$f_{n}={\frac {v}{\lambda _{n}}}={\frac {nv}{2L}}=nf_{1}\,\!$

Propagating waves

Sound waves

Physical situation	Nomenclature	Equations
Average wave power	P0 = Sound power due to source	$\langle P\rangle =\mu v\omega ^{2}x_{m}^{2}/2\,\!$
Sound intensity	Ω = Solid angle	$I=P_{0}/(\Omega r^{2})\,\!$ $I=P/A=\rho v\omega ^{2}s_{m}^{2}/2\,\!$
Acoustic beat frequency	f1, f2 = frequencies of two waves (nearly equal amplitudes)	$f_{\mathrm {beat} }=\left|f_{2}-f_{1}\right|\,\!$
Doppler effect for mechanical waves	V = speed of sound wave in medium f0 = Source frequency fr = Receiver frequency v0 = Source velocity vr = Receiver velocity	$f_{r}=f_{0}{\frac {V\pm v_{r}}{v\mp v_{0}}}\,\!$ upper signs indicate relative approach,lower signs indicate relative recession.
Mach cone angle (Supersonic shockwave, sonic boom)	v = speed of body v = local speed of sound θ = angle between direction of travel and conic evelope of superimposed wavefronts	$\sin \theta ={\frac {v}{v_{s}}}\,\!$
Acoustic pressure and displacement amplitudes	p0 = pressure amplitude s0 = displacement amplitude v = speed of sound ρ = local density of medium	$p_{0}=\left(v\rho \omega \right)s_{0}\,\!$
Wave functions for sound		Acoustic beats $s=\left[2s_{0}\cos \left(\omega 't\right)\right]\cos \left(\omega t\right)\,\!$ Sound displacement function $s=s_{0}\cos(kr-\omega t)\,\!$ Sound pressure-variation $p=p_{0}\sin(kr-\omega t)\,\!$

Gravitational waves

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Gravitational radiation for two orbiting bodies in the low-speed limit.

Physical situation	Nomenclature	Equations
Radiated power	P = Radiated power from system, t = time, r = separation between centres-of-mass m1, m2 = masses of the orbiting bodies	$P={\frac {\mathrm {d} E}{\mathrm {d} t}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}$
Orbital radius decay		${\frac {\mathrm {d} r}{\mathrm {d} t}}=-{\frac {64}{5}}{\frac {G^{3}}{c^{5}}}{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\$
Orbital lifetime	r0 = initial distance between the orbiting bodies	$t={\frac {5}{256}}{\frac {c^{5}}{G^{3}}}{\frac {r_{0}^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}\$

Superposition, interference, and diffraction

Physical situation	Nomenclature	Equations
Principle of superposition	N = number of waves	$y_{\mathrm {net} }=\sum _{i=1}^{N}y_{i}\,\!$
Resonance	ωd = driving angular frequency (external agent) ωnat = natural angular frequency (oscillator)	$\omega _{d}=\omega _{\mathrm {nat} }\,\!$
Phase and interference	Δr = path length difference φ = phase difference between any two successive wave cycles	${\frac {\Delta r}{\lambda }}={\frac {\Delta t}{T}}={\frac {\phi }{2\pi }}\,\!$ Constructive interference $n={\frac {\lambda }{\Delta x}}\,\!$ Destructive interference $n+{\frac {1}{2}}={\frac {\lambda }{\Delta x}}\,\!$

Wave propagation

A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media. In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency.

The phase velocity is the rate at which the phase of the wave propagates in space.

The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.

Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function called the Dispersion Relation. The use of the explicit form ω(k) is standard, since the phase velocity ω/k and the group velocity dω/dk usually have convenient representations by this function.

Physical situation	Nomenclature	Equations
Idealized non-dispersive media	p = (any type of) Stress or Pressure, ρ = Volume Mass Density, F = Tension Force, μ = Linear Mass Density of medium	$v={\sqrt {\frac {p}{\rho }}}={\sqrt {\frac {F}{\mu }}}\,\!$
Dispersion relation		Implicit form $D\left(\omega ,k\right)=0$ Explicit form $\omega \left(k\right)=0$
Amplitude modulation, AM		$A=A\left(t\right)$
Frequency modulation, FM		$f=f\left(t\right)$

General wave functions

Wave equations

Physical situation	Nomenclature	Wave equation	General solution/s
Non-dispersive Wave Equation in 3d	A = amplitude as function of position and time	$\nabla ^{2}A={\frac {1}{v_{\parallel }^{2}}}{\frac {\partial ^{2}A}{\partial t^{2}}}\,\!$	$A\left(\mathbf {r} ,t\right)=A\left(x-v_{\parallel }t\right)\,\!$
Exponentially damped waveform	A0 = Initial amplitude at time t = 0 b = damping parameter		$A=A_{0}e^{-bt}\sin \left(kx-\omega t+\phi \right)\,\!$
Korteweg–de Vries equation	α = constant	${\frac {\partial y}{\partial t}}+\alpha y{\frac {\partial y}{\partial x}}+{\frac {\partial ^{3}y}{\partial x^{3}}}=0\,\!$	$A(x,t)={\frac {3v_{\parallel }}{\alpha }}\mathrm {sech} ^{2}\left[{\frac {\sqrt {v_{\parallel }}}{2}}\left(x-v_{\parallel }t\right)\right]\,\!$

Sinusoidal solutions to the 3d wave equation

N different sinusoidal waves

Complex amplitude of wave n
$A_{n}=\left|A_{n}\right|e^{i\left(\mathbf {k} _{\mathrm {n} }\cdot \mathbf {r} -\omega _{n}t+\phi _{n}\right)}\,\!$

Resultant complex amplitude of all N waves
$A=\sum _{n=1}^{N}A_{n}\,\!$

Modulus of amplitude
$A={\sqrt {AA^{*}}}={\sqrt {\sum _{n=1}^{N}\sum _{m=1}^{N}\left|A_{n}\right|\left|A_{m}\right|\cos \left[\left(\mathbf {k} _{n}-\mathbf {k} _{m}\right)\cdot \mathbf {r} +\left(\omega _{n}-\omega _{m}\right)t+\left(\phi _{n}-\phi _{m}\right)\right]}}\,\!$

The transverse displacements are simply the real parts of the complex amplitudes.

1 dimensional corollaries for two sinusoidal waves

The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.

Wavefunction	Nomenclature	Superposition	Resultant
Standing wave		${\begin{aligned}y_{1}+y_{2}&=A\sin \left(kx-\omega t\right)\\&+A\sin \left(kx+\omega t\right)\end{aligned}}\,\!$	$y=A\sin \left(kx\right)\cos \left(\omega t\right)\,\!$
Beats	$\langle \omega \rangle ={\frac {\omega _{1}+\omega _{2}}{2}}\,\!$ $\langle k\rangle ={\frac {k_{1}+k_{2}}{2}}\,\!$ $\Delta \omega =\omega _{1}-\omega _{2}\,\!$ $\Delta k=k_{1}-k_{2}\,\!$	${\begin{aligned}y_{1}+y_{2}&=A\sin \left(k_{1}x-\omega _{1}t\right)\\&+A\sin \left(k_{2}x+\omega _{2}t\right)\end{aligned}}\,\!$	$y=2A\sin \left(\langle k\rangle x-\langle \omega \rangle t\right)\cos \left({\frac {\Delta k}{2}}x-{\frac {\Delta \omega }{2}}t\right)\,\!$
Coherent interference		${\begin{aligned}y_{1}+y_{2}&=A\sin \left(kx-\omega t\right)\\&+A\sin \left(kx+\omega t+\phi \right)\end{aligned}}\,\!$	$y=2A\cos \left({\frac {\phi }{2}}\right)\sin \left(kx-\omega t+{\frac {\phi }{2}}\right)\,\!$

Gravitation

Gravitational fields

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General classical equations also occur in relativity for weak-field limits. It can be shown that a uniform spherically symmetric mass distribution generates an equivalent gravitational field to a point mass, so all formulae for point masses apply to bodies which can be modelled in this way.

Physical situation	Nomenclature	Equations
Gravitational potential gradient and field	U = gravitational potential C = curved path traversed by a mass in the field	$\mathbf {g} =-\nabla U$ $\Delta U=-\int _{C}\mathbf {g} \cdot d\mathbf {r} \,\!$
Point mass		$\mathbf {g} ={\frac {Gm}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!$
At a point in a local array of point masses		$\mathbf {g} =\sum _{i}\mathbf {g} _{i}=G\sum _{i}{\frac {m_{i}}{\left|\mathbf {r} _{i}-\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} _{i}\,\!$
Gravitational torque and potential energy due to non-uniform fields and mass moments	V = volume of space occupied by the charge distribution	${\boldsymbol {\tau }}=\int _{V_{n}}\left(\mathrm {d} \mathbf {m} \times \mathbf {g} +\mathbf {m} \times \mathrm {d} \mathbf {g} \right)\,\!$ $U=\int _{V_{n}}\left(\mathrm {d} \mathbf {m} \cdot \mathbf {g} +\mathbf {m} \cdot \mathrm {d} \mathbf {g} \right)\,\!$
Gravitational field for a rotating body	φ = zenith angle relative to rotation axis $\mathbf {\hat {a}} \,\!$ = unit vector perpendicular to rotation (zenith) axis, radial from it	$\mathbf {g} =-{\frac {GM}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} -\left|{\boldsymbol {\omega }}\right|^{2}\left|\mathbf {r} \right|\sin \phi \mathbf {\hat {a}} \,\!$

Weak-gravitational field relativistic equations

Physical situation	Nomenclature	Equations
Gravitomagnetic field for a rotating body	ξ = gravitomagnetic field	${\boldsymbol {\xi }}={\frac {G}{2c^{2}}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {\hat {r}} )\mathbf {\hat {r}} }{\left|\mathbf {r} \right|^{3}}}$

Gravitational potentials

General classical equations.

Physical situation	Nomenclature	Equations
Potential energy from gravity, integral from Newton's law		$U=-{\frac {Gm_{1}m_{2}}{\left|\mathbf {r} \right|}}\approx m\left|\mathbf {g} \right|y\,\!$
Escape speed	M = Mass of body (e.g. planet) to escape from r = radius of body	$v={\sqrt {\frac {2GM}{r}}}\,\!$
Orbital energy	m = mass of orbiting body (e.g. planet) M = mass of central body (e.g. star) ω = angular velocity of orbiting mass r = separation between centres of mass T = kinetic energy U = gravitational potential energy (sometimes called "gravitational binding energy" for this instance)	${\begin{aligned}E&=T+U\\&=-{\frac {GmM}{\left|\mathbf {r} \right|}}+{\frac {1}{2}}m\left|\mathbf {v} \right|^{2}\\&=m\left(-{\frac {GM}{\left|\mathbf {r} \right|}}+{\frac {\left|{\boldsymbol {\omega }}\times \mathbf {r} \right|^{2}}{2}}\right)\\&=-{\frac {GmM}{2\left|\mathbf {r} \right|}}\end{aligned}}\,\!$

Electromagnetism

Electric fields

General Classical Equations

Physical situation	Equations
Electric potential gradient and field	$\mathbf {E} =-\nabla V$ $\Delta V=-\int _{r_{1}}^{r_{1}}\mathbf {E} \cdot d\mathbf {r} \,\!$
Point charge	$\mathbf {E} ={\frac {q}{4\pi \epsilon _{0}\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!$
At a point in a local array of point charges	$\mathbf {E} =\sum \mathbf {E} _{i}={\frac {1}{4\pi \epsilon _{0}}}\sum _{i}{\frac {q_{i}}{\left|\mathbf {r} _{i}-\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} _{i}\,\!$
At a point due to a continuum of charge	$\mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}\int _{V}{\frac {\mathbf {r} \rho \mathrm {d} V}{\left|\mathbf {r} \right|^{3}}}\,\!$
Electrostatic torque and potential energy due to non-uniform fields and dipole moments	${\boldsymbol {\tau }}=\int _{V}\left(\mathrm {d} \mathbf {p} \times \mathbf {E} +\mathbf {p} \times \mathrm {d} \mathbf {E} \right)\,\!$ $U=\int _{V}\left(\mathrm {d} \mathbf {p} \cdot \mathbf {E} +\mathbf {p} \cdot \mathrm {d} \mathbf {E} \right)\,\!$

Magnetic fields and moments

{{#invoke:see also|seealso}}

General classical equations

Physical situation	Equations
Magnetic potential, EM vector potential	$\mathbf {B} =\nabla \times \mathbf {A}$
Due to a magnetic moment	$\mathbf {A} ={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times \mathbf {r} }{\left|\mathbf {r} \right|^{3}}}$ $\mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }={\frac {\mu _{0}}{4\pi }}\left({\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{\left|\mathbf {r} \right|^{5}}}-{\frac {\mathbf {m} }{\left|\mathbf {r} \right|^{3}}}\right)$
Magnetic moment due to a current distribution	$\mathbf {m} ={\frac {1}{2}}\int _{V}\mathbf {r} \times \mathbf {J} \mathrm {d} V$
Magnetostatic torque and potential energy due to non-uniform fields and dipole moments	${\boldsymbol {\tau }}=\int _{V}\left(\mathrm {d} \mathbf {m} \times \mathbf {B} +\mathbf {m} \times \mathrm {d} \mathbf {B} \right)\,\!$ $U=\int _{V}\left(\mathrm {d} \mathbf {m} \cdot \mathbf {B} +\mathbf {m} \cdot \mathrm {d} \mathbf {B} \right)\,\!$

Electromagnetic induction

Physical situation	Nomenclature	Equations
Transformation of voltage	N = number of turns of conductor η = energy efficiency	${\frac {V_{s}}{V_{p}}}={\frac {N_{s}}{N_{p}}}={\frac {I_{p}}{I_{s}}}=\eta \,\!$

Electric circuits and electronics

Below N = number of conductors or circuit components. Subcript net refers to the equivalent and resultant property value.

Physical situation	Nomenclature	Series	Parallel
Resistors and conductors	Ri = resistance of resistor or conductor i Gi = conductance of conductor or conductor i	$R_{\mathrm {net} }=\sum _{i=1}^{N}R_{i}\,\!$ ${\frac {1}{G_{\mathrm {net} }}}=\sum _{i=1}^{N}{\frac {1}{G_{i}}}\,\!$	${\frac {1}{R_{\mathrm {net} }}}=\sum _{i=1}^{N}{\frac {1}{R_{i}}}\,\!$ $G_{\mathrm {net} }=\sum _{i}\sum _{i=1}^{N}\,\!$
Charge, capacitors, currents	qi = capacitance of capacitor i qi = charge of charge carrier i	$q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!$ $C_{\mathrm {net} }=\sum _{i=1}^{N}C_{i}\,\!$ $I_{\mathrm {net} }=\sum _{i=1}^{N}I_{i}\,\!$	$q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!$ $C_{\mathrm {net} }=\sum _{i=1}^{N}C_{i}\,\!$ $I_{\mathrm {net} }=\sum _{i=1}^{N}I_{i}\,\!$
Inductors	Li = self inductance of inductor i Lij = self inductance element ij of L matrix Mij = mutual inductance between inductors i and j	${\frac {1}{L_{\mathrm {net} }}}=\sum _{i}{\frac {1}{L_{i}}}\,\!$	$V_{i}=\sum _{j=1}^{N}L_{ij}{\frac {\mathrm {d} I_{j}}{\mathrm {d} t}}\,\!$

Series circuit equations

Circuit	DC Circuit equations	AC Circuit equations
RC circuits	Circuit equation $R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!$ Capacitor charge $q=C{\mathcal {E}}\left(1-e^{-t/RC}\right)\,\!$ Capacitor discharge $q=C{\mathcal {E}}e^{-t/RC}\,\!$
RL circuits	Circuit equation $L{\frac {\mathrm {d} I}{\mathrm {d} t}}+RI={\mathcal {E}}\,\!$ Inductor current rise $I={\frac {\mathcal {E}}{R}}\left(1-e^{-Rt/L}\right)\,\!$ Inductor current fall $I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-Rt/L}\,\!$
LC circuits	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\,\!$	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$ Circuit resonant frequency $\omega _{\mathrm {res} }=1/{\sqrt {LC}}\,\!$ Circuit charge $q=q_{0}\cos(\omega t+\phi )\,\!$ Circuit current $I=-\omega q_{0}\sin(\omega t+\phi )\,\!$ Circuit electrical potential energy $U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!$ Circuit magnetic potential energy $U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!$
RLC Circuits	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!$	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$ Circuit charge $q=q_{0}eT^{-Rt/2L}\cos(\omega 't+\phi )\,\!$

Light

Luminal electromagnetic waves

Physical situation	Nomenclature	Equations
Energy density in an EM wave	$\langle u\rangle \,\!$ = mean energy density	For a dielectric: $\langle u\rangle ={\frac {1}{2}}\left(\epsilon \mathbf {E} ^{2}+\mu \mathbf {B} ^{2}\right)\,\!$
Kinetic and potential momenta (non-standard terms in use)		Potential momentum: $\mathbf {p} _{\mathrm {p} }=q\mathbf {A} \,\!$ Kinetic momentum: $\mathbf {p} _{\mathrm {k} }=m\mathbf {v} \,\!$ Cononical momentum: $\mathbf {p} =m\mathbf {v} +q\mathbf {A} \,\!$
Irradiance, light intensity	$\langle \mathbf {S} \rangle \,\!$ = time averaged poynting vector I = irradiance I0 = intensity of source P0 = power of point source Ω = solid angle r = radial position from source	$I=\langle \mathbf {S} \rangle =E_{\mathrm {rms} }^{2}/c\mu _{0}\,\!$ At a spherical surface: $I={\frac {P_{0}}{\Omega \left|r\right|^{2}}}\,\!$
Doppler effect for light (relativistic)		$\lambda =\lambda _{0}{\sqrt {\frac {c-v}{c+v}}}\,\!$ $v=|\Delta \lambda |c/\lambda _{0}\,\!$
Cherenkov radiation, cone angle	n = refractive index v = speed of particle θ = cone angle	$\cos \theta ={\frac {v}{nc}}=v{\sqrt {\epsilon _{0}\mu _{0}}}\,\!$
Electric and magnetic amplitudes	E = electric field H = magnetic field strength	For a dielectric $\left|\mathbf {E} \right|={\sqrt {\frac {\epsilon }{\mu }}}\left|\mathbf {H} \right|\,\!$
EM wave components		Electric $\mathbf {E} =\mathbf {E} _{0}\sin(kx-\omega t)\,\!$ Magnetic $\mathbf {B} =\mathbf {B} _{0}\sin(kx-\omega t)\,\!$

Geometric optics

Physical situation	Nomenclature	Equations
Critical angle	n1 = refractive index of initial medium n2 = refractive index of final medium θc = critical angle	$\sin \theta _{c}={\frac {n_{2}}{n_{1}}}\,\!$
Thin lens equation	f = lens focal length x1 = object length x2 = image length r1 = incident curvature radius r2 = refracted curvature radius	${\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}={\frac {1}{f}}\,\!$ Lens focal length from refraction indices ${\frac {1}{f}}=\left({\frac {{n}_{\mathrm {lens} }}{n_{\mathrm {med} }-1}}\right)\left({\frac {1}{r_{1}}}-{\frac {1}{r_{2}}}\right)\,\!$
Image distance in a plane mirror		$x_{2}=-x_{1}\,\!$
Spherical mirror	r = curvature radius of mirror	Spherical mirror equation ${\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}={\frac {1}{f}}={\frac {2}{r}}\,\!$ Image distance in a spherical mirror ${\frac {n_{1}}{x_{1}}}+{\frac {n_{2}}{x_{2}}}={\frac {\left(n_{2}-n_{1}\right)}{r}}\,\!$

Subscripts 1 and 2 refer to initial and final optical media respectively.

These ratios are sometimes also used, following simply from other definitions of refractive index, wave phase velocity, and the luminal speed equation:

${\frac {n_{1}}{n_{2}}}={\frac {v_{2}}{v_{1}}}={\frac {\lambda _{2}}{\lambda _{1}}}={\sqrt {\frac {\epsilon _{1}\mu _{1}}{\epsilon _{2}\mu _{2}}}}\,\!$

where:

• ε = permittivity of medium,
• μ = permeability of medium,
• λ = wavelength of light in medium,
• v = speed of light in media.

Polarization

Physical situation	Nomenclature	Equations
Angle of total polarisation	θB = Reflective polarization angle, Brewster's angle	$\tan \theta _{B}=n_{2}/n_{1}\,\!$
intensity from polarized light, Malus' law	I0 = Initial intensity, I = Transmitted intensity, θ = Polarization angle between polarizer transmission axes and electric field vector	$I=I_{0}\cos ^{2}\theta \,\!$

Diffraction and interference

Property or effect	Nomenclature	Equation
Thin film in air	n1 = refractive index of initial medium (before film interference) n2 = refractive index of final medium (after film interference)	Minima: $N\lambda /n_{2}\,\!$ Maxima:$2L=(N+1/2)\lambda /n_{2}\,\!$
The grating equation	a = width of aperture, slit width α = incident angle to the normal of the grating plane	${\frac {\delta }{2\pi }}\lambda =a\left(\sin \theta +\sin \alpha \right)\,\!$
Rayleigh's criterion		$\theta _{R}=1.22\lambda /\,\!d$
Bragg's law (solid state diffraction)	d = lattice spacing δ = phase difference between two waves	${\frac {\delta }{2\pi }}\lambda =2d\sin \theta \,\!$ For constructive interference: $\delta /2\pi =n\,\!$ For destructive interference: $\delta /2\pi =n/2\,\!$ where $n\in \mathbf {N} \,\!$
Single slit diffraction intensity	I0 = source intensity Wave phase through apertures $\phi ={\frac {2\pi a}{\lambda }}\sin \theta \,\!$	$I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}\right]^{2}\,\!$
N-slit diffraction (N ≥ 2)	d = centre-to-centre separation of slits N = number of slits Phase between N waves emerging from each slit $\delta ={\frac {2\pi d}{\lambda }}\sin \theta \,\!$	$I=I_{0}\left[{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!$
N-slit diffraction (all N)		$I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!$
Circular aperture intensity	a = radius of the circular aperture J1 is a Bessel function	$I=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2}$
Amplitude for a general planar aperture	Cartesian and spherical polar coordinates are used, xy plane contains aperture A, amplitude at position r r' = source point in the aperture Einc, magnitude of incident electric field at aperture	Near-field (Fresnel) $A\left(\mathbf {r} \right)\propto \iint _{\mathrm {aperture} }E_{\mathrm {inc} }\left(\mathbf {r} '\right)~{\frac {e^{ik\left|\mathbf {r} -\mathbf {r} '\right|}}{4\pi \left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} x'\mathrm {d} y'$ Far-field (Fraunhofer) $A\left(\mathbf {r} \right)\propto {\frac {e^{ikr}}{4\pi r}}\iint _{\mathrm {aperture} }E_{\mathrm {inc} }\left(\mathbf {r} '\right)e^{-ik\left[\sin \theta \left(\cos \phi x'+\sin \phi y'\right)\right]}\mathrm {d} x'\mathrm {d} y'$
Huygen-Fresnel-Kirchhoff principle	r0 = position from source to aperture, incident on it r = position from aperture diffracted from it to a point α0 = incident angle with respect to the normal, from source to aperture α = diffracted angle, from aperture to a point S = imaginary surface bounded by aperture $\mathbf {\hat {n}} \,\!$ = unit normal vector to the aperture $\mathbf {r} _{0}\cdot \mathbf {\hat {n}} =\left|\mathbf {r} _{0}\right|\cos \alpha _{0}\,\!$ $\mathbf {r} \cdot \mathbf {\hat {n}} =\left|\mathbf {r} \right|\cos \alpha \,\!$ $\left|\mathbf {r} \right|\left|\mathbf {r} _{0}\right|\ll \lambda \,\!$	$A\mathbf {(} \mathbf {r} )={\frac {-i}{2\lambda }}\iint _{\mathrm {aperture} }{\frac {e^{i\mathbf {k} \cdot \left(\mathbf {r} +\mathbf {r} _{0}\right)}}{\left|\mathbf {r} \right|\left|\mathbf {r} _{0}\right|}}\left[\cos \alpha _{0}-\cos \alpha \right]\mathrm {d} S\,\!$
Kirchhoff's diffraction formula		$A\left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\iint _{\mathrm {aperture} }{\frac {e^{i\mathbf {k} \cdot \mathbf {r} _{0}}}{\left|\mathbf {r} _{0}\right|}}\left[i\left|\mathbf {k} \right|U_{0}\left(\mathbf {r} _{0}\right)\cos {\alpha }+{\frac {\partial A_{0}\left(\mathbf {r} _{0}\right)}{\partial n}}\right]\mathrm {d} S$

Quantum mechanics

Light

Property/Effect	Nomenclature	Equation
Photoelectric equation		$K_{\mathrm {max} }=hf-\Phi \,\!$
photon momentum		$p=hf/c=h/\lambda \,\!$
Cutoff wavelength		$\lambda _{\mathrm {min} }=hc/K_{0}\,\!$

Wave–particle duality

Property or effect	Nomenclature	Equation
Planck–Einstein equation	E = energy of particle ω = 2πf is the angular frequency and frequency of the particle ħ = 2π/h are the Plank constants c = speed of light	$E=hf=hc/\lambda \,\!$ $E=\hbar \omega =\hbar ck\,\!$
de Broglie wavelength	p = momentum of particle |k| = 2π/λ is the magnitude of the particle's wavevector k λ = wavelength of particle	$\lambda p=h\,\!$ $\mathbf {p} =\hbar \mathbf {k} \,\!$
Heisenberg's uncertainty principle		$\Delta x\Delta p\geq {\frac {\hbar }{2}}\,\!$ $\Delta E\Delta t\geq {\frac {\hbar }{2}}\,\!$

Probability Distributions

Property or effect	Nomenclature	Equation
Density of states		$N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}\,\!$
Fermi-Dirac distribution (fermions)	P(Ei) = probability of energy Ei g(Ei) = degeneracy of energy Ei (no of states with same energy) μ = chemical potential	$P(E_{i})=g(E_{i})/(e^{(E-\mu )/kT}+1)\,\!$
Bose-Einstein distribution (bosons)		$P(E_{i})=g(E_{i})/(e^{(E_{i}-\mu )/kT}-1)\,\!$

Angular momentum

{{#invoke:main|main}}

Property or effect	Nomenclature	Equation
Angular momentum quantum numbers	s = spin quantum number ms = spin magnetic quantum number ℓ = Azimuthal quantum number mℓ = azimuthal magnetic quantum number mj = total angular momentum magnetic quantum number j = total angular momentum quantum number	Spin projection: $m_{s}\in \{-s,-s+1\cdots s-1,s\}\,\!$ Orbital: $m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\,\!$ $m_{\ell }\in \{0\cdots n-1\}\,\!$ Total: ${\begin{aligned}&j=\ell +s\\&j\in \{|\ell -s|,|\ell -s|+1\cdots |\ell +s|-1,|\ell +s|\}\\\end{aligned}}\,\!$
Angular momentum magnitudes	angular momementa: S = Spin, L = orbital, J = total	Spin magnitude: $|\mathbf {S} |=\hbar {\sqrt {s(s+1)}}\,\!$ Orbital magnitude: $|\mathbf {L} |=\hbar {\sqrt {\ell (\ell +1)}}\,\!$ Total magnitude: $\mathbf {J} =\mathbf {L} +\mathbf {S} \,\!$ $|\mathbf {J} |=\hbar {\sqrt {j(j+1)}}\,\!$
Angular momentum components		Spin: $S_{z}=m_{s}\hbar \,\!$ Orbital: $L_{z}=m_{\ell }\hbar \,\!$

Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Property or effect	Nomenclature	Equation
orbital magnetic dipole moment	e = electron charge me = electron rest mass L = electron orbital angular momentum gTemplate:Ell = orbital Landé g-factor μB = Bohr magneton	${\boldsymbol {\mu }}_{\ell }=-e\mathbf {L} /2m_{e}=g_{\ell }{\frac {\mu _{B}}{\hbar }}\mathbf {L} \,\!$ z-component: $\mu _{\ell ,z}=-m_{\ell }\mu _{B}\,\!$
spin magnetic dipole moment	S = electron spin angular momentum gs = spin Landé g-factor	${\boldsymbol {\mu }}_{s}=-e\mathbf {S} /m_{e}=g_{s}{\frac {\mu _{B}}{\hbar }}\mathbf {S} \,\!$ z-component: $\mu _{s,z}=-eS_{z}/m_{e}=g_{s}eS_{z}/2m_{e}\,\!$
dipole moment potential	U = potential energy of dipole in field	$U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$

The Hydrogen atom

Property or effect	Nomenclature	Equation
Energy levels	En = energy eigenvalue n = principal quantum number e = electron charge me = electron rest mass ε0 = permittivity of free space h = Plank's constant	$E_{n}=-me^{4}/8\epsilon _{0}^{2}h^{2}n^{2}=13.61eV/n^{2}\,\!$
Spectrum	λ = wavelength of emitted photon, during electronic transition from Ei to Ej	${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$

Wavefunctions

The general form wavefunction for a system of particles, each with position r_i and z-component of spin s_{z i}. Sums are over the descrete variable s_z, integrals over continuous positions r.

For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necerssary). Following are general mathematical results, used in calculations.

Property or effect	Nomenclature	Equation
Wavefunction for N particles in 3d	r = (r1, r2... rN) sz = (sz 1, sz 2... sz N)	In function notation: $\Psi =\Psi \left(\mathbf {r} ,\mathbf {s_{z}} ,t\right)$ in bra-ket notation: $|\Psi \rangle =\sum _{s_{z1}}\sum _{s_{z2}}\cdots \sum _{s_{zN}}\int _{V_{1}}\int _{V_{2}}\cdots \int _{V_{N}}\mathrm {d} \mathbf {r} _{1}\mathrm {d} \mathbf {r} _{2}\cdots \mathrm {d} \mathbf {r} _{N}\Psi |\mathbf {r} ,\mathbf {s_{z}} \rangle$ for non-interacting particles: $\Psi =\prod _{n=1}^{N}\Psi \left(\mathbf {r} _{n},s_{zn},t\right)$
Position-momentum Fourier transform (1 particle in 3d)	Φ = momentum-space wavefunction Ψ = position-space wavefunction	${\begin{aligned}\Phi (\mathbf {p} ,s_{z},t)&={\frac {1}{\sqrt {2\pi \hbar }}}\int \limits _{\mathrm {all\,space} }e^{-i\mathbf {p} \cdot \mathbf {r} /\hbar }\Psi (\mathbf {r} ,s_{z},t)\mathrm {d} ^{3}\mathbf {r} \\&\upharpoonleft \downharpoonright \\\Psi (\mathbf {r} ,s_{z},t)&={\frac {1}{\sqrt {2\pi \hbar }}}\int \limits _{\mathrm {all\,space} }e^{+i\mathbf {p} \cdot \mathbf {r} /\hbar }\Phi (\mathbf {r} ,s_{z},t)\mathrm {d} ^{3}\mathbf {p} _{n}\\\end{aligned}}$
General probability distribution	Vj = volume (3d region) particle may occupy, P = Probability that particle 1 has position r1 in volume V1 with spin sz1 and particle 2 has position r2 in volume V2 with spin sz2, etc.	$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!$
General normalization condition		$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\int \limits _{\mathrm {all\,space} }\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!$

Relativity

Lorentz frame tranforms

If V = any 4-vector, then the Lorentz transformation has the general form:

$\mathbf {V} '={\boldsymbol {\Lambda }}\mathbf {V}$

where Λ is the transformation matrix, in general the components are complicated (see main article for details).

4-vectors and frame-invariant results

Invariance and unification of physical quantities both arise from four-vectors.. The inner product of a 4-vector with itself is equal to a scalar (by definition of the inner product), and since the 4-vectors are physical quantities their magnitudes correspond to physical quantities also.

Property/effect	3-vector	4-vector	Invariant result
Space-time events	3-Position: r = (x1, x2, x3) $\mathbf {r} \cdot \mathbf {r} \equiv r^{2}\equiv x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\,\!$		4-Position: X = (ct, x1, x2, x3) $\mathbf {X} \cdot \mathbf {X} =\left(c\tau \right)^{2}\,\!$ ${\begin{aligned}&\left(ct\right)^{2}-\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)\\&=\left(ct\right)^{2}-r^{2}\\&=\chi ^{2}=\left(c\tau \right)^{2}\end{aligned}}\,\!$ τ = Proper time χ = Proper length
Momentum-energy invariance	$\mathbf {p} =m\mathbf {u} \,\!$ 3-Momentum: p = (p1, p2, p3) $\mathbf {p} \cdot \mathbf {p} \equiv p^{2}\equiv p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\,\!$	4-Position: P = (E/c, p1, p2, p3) $\mathbf {P} =m\mathbf {U} \,\!$	$\mathbf {P} \cdot \mathbf {P} =\left(mc\right)^{2}\,\!$ ${\begin{aligned}&\left({\frac {E}{c}}\right)^{2}-\left(p_{1}^{2}+p_{2}^{2}+p_{3}^{2}\right)\\&=\left({\frac {E}{c}}\right)^{2}-p^{2}\\&=\left(mc\right)^{2}\end{aligned}}\,\!$ which leads to: $E^{2}=\left(pc\right)^{2}+\left(mc^{2}\right)^{2}\,\!$ E = total energy m = invariant mass
Velocity	3-Velocity: u = (u1, u2, u3) $\mathbf {u} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\,\!$	4-Velocity: U = (u0, u1, u2, u3) $\mathbf {U} ={\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} \tau }}=\gamma \left(c,\mathbf {u} \right)$	$\mathbf {U} \cdot \mathbf {U} =c^{2}\,\!$
Acceleration	3-Acceleration: a = (a1, a2, a3) $\mathbf {a} ={\frac {\mathrm {d} \mathbf {u} }{\mathrm {d} t}}\,\!$	4-Acceleration: A = (a0, a1, a2, a3) $\mathbf {A} ={\frac {\mathrm {d} \mathbf {U} }{\mathrm {d} \tau }}=\gamma \left(c{\frac {\mathrm {d} \gamma }{\mathrm {d} t}},{\frac {\mathrm {d} \gamma }{\mathrm {d} t}}\mathbf {u} +\gamma \mathbf {a} \right)$	$\mathbf {A} \cdot \mathbf {A} =0\,\!$
Force	3-Force: f = (f1, f2, f3) $\mathbf {f} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}\,\!$	4-Force: F = (f0, f1, f2, f3) $\mathbf {F} ={\frac {\mathrm {d} \mathbf {P} }{\mathrm {d} \tau }}=\gamma m\left(c{\frac {\mathrm {d} \gamma }{\mathrm {d} t}},{\frac {\mathrm {d} \gamma }{\mathrm {d} t}}\mathbf {u} +\gamma \mathbf {a} \right)$	$\mathbf {F} \cdot \mathbf {F} =0\,\!$

Corollaries of Lorentz transforms and 4-vector invariance

Subscripts 0 refer to the frame F₀, the frame observing frame F travel at speed v relative to F₀. In frame F₀ all subscript 0 quantities are proper lengths or times, or rest masses.

Physical situation	Nomenclature	Equations
Lorentz factor	$\Delta t=\gamma \Delta t_{0}\,\!$	$\gamma \left(v\right)={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\,\!$ $\gamma \left(p\right)={\sqrt {1+\left({\frac {p}{mc}}\right)^{2}}}\,\!$
Time dilation	t0 = elapsed time measured in inertial frame F0 = proper time for frame F0 t = elapsed time measured in inertial frame F moving at speed v relative to frame F0	$\Delta t=\gamma \Delta t_{0}\,\!$
Length contraction	x0 = measured distance in inertial frame F0 = proper length for frame F0 x = measured length in inertial frame F moving at speed v relative to frame F0	$\Delta x={\frac {\Delta x_{0}}{\gamma }}\,\!$
Relativistic mass	m0 = measured mass in inertial frame F0 = rest mass for frame F0 m = measured mass in inertial frame F moving at speed v relative to frame F0	$m=\gamma m_{0}\,\!$
3-Momentum		$\mathbf {p} =\gamma m\mathbf {v} \,\!$
Mass-energy equivalence		$E=mc^{2}\,\!$ Rest mass-energy $E_{0}=m_{0}c^{2}\,\!$
Momentum-energy for massless particles		$E=pc\,\!$
Kinetic energy		$T=\left(\gamma -1\right)mc^{2}\,\!$

Nuclear and particle physics

Physical situation	Nomenclature	Equations
Mass number	A = (Relative) atomic mass = Mass number = Number of protons and neutrons N = Number of neutrons Z = Atomic number = Number of protons = Number of electrons	$A=Z+N\,\!$
Mass in nuclei	M'nuc = Mass of nucleus, bound nucleons MΣ = Sum of masses for isolated nucleons mp = proton rest mass mn = neutron rest mass	$M_{\Sigma }=Zm_{p}+Nm_{n}\,\!$ $M_{\Sigma }>M_{N}\,\!$ $\Delta M=M_{\Sigma }-M_{\mathrm {nuc} }\,\!$ $\Delta E=\Delta Mc^{2}\,\!$
Nuclear radius	r0 ≈ 1.2 fm	$r=r_{0}A^{1/3}\,\!$ hence (approximately) nuclear volume ∝ A nuclear surface ∝ A2/3
Radioactive decay	N0 = Initial number of atoms N = Number of atoms at time t λ = Decay constant t = Time	Statistical decay of a radionuclide: ${\frac {\mathrm {d} N}{\mathrm {d} t}}=-\lambda N$ $N=N_{0}e^{-\lambda t}\,\!$
Nuclear binding energy, empirical curve	Dimensionless parameters to fit experiment: EB = binding energy, av = nuclear volume coefficient, as = nuclear surface coefficient, ac = electrostatic interaction coefficient, aa = symmetry/asymmetry extent coefficient for the numbers of neutrons/protons,	${\begin{aligned}E_{B}=&a_{v}A-a_{s}A^{2/3}-a_{c}Z(Z-1)A^{-1/3}\\&-a_{a}(N-Z)^{2}A^{-1}+12\delta (N,Z)A^{-1/2}\\\end{aligned}}$ where (due to pairing of nuclei) δ(N, Z) = +1 even N, even Z, δ(N, Z) = −1 odd N, odd Z, δ(N, Z) = 0 odd A
Radiation flux	I0 = Initial intensity/Flux of radiation I = Number of atoms at time t μ = Linear absorption coefficient x = Thickness of substance	$I=I_{0}e^{-\mu x}\,\!$
Fermi energy		$E_{F}=(3/16{\sqrt {2}}\pi )^{2/3}h^{2}n^{2/3}m\,\!$
Hubble's law		$v=Hr\,\!$

Fundamental forces

Name	Equations
Strong force	${\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,,\\\end{aligned}}\,\!$
Electroweak interaction	:${\mathcal {L}}_{EW}={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.\,\!$ ${\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }\,\!$ ${\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}\,\!$ ${\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}\,\!$ ${\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.\,\!$
Quantum electrodynamics	${\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,\,\!$

See also

• Variables commonly used in physics
• Continuity equation
• Constitutive equation
• Operator (physics)
• Defining equation (physical chemistry)
• Physical constant

References

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Sources

• The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
• Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0-7195-3382-1
• Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
• McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
• Physics with Modern Applications, L.H. Greenberg, Holt-Saunders International W.B. Saunders and Co, 1978, ISBN 0-7216-4247-0
• Principles of Physics, J.B. Marion, W.F. Hornyak, Holt-Saunders International Saunders College, 1984, ISBN 4-8337-0195-2
• Physics for Scientists and Engineers: With Modern Physics (6th Edition), P.A. Tipler, G. Mosca, W.H. Freeman and Co, 2008, 9-781429-202657
• 3000 Solved Problems in Physics, Schaum Series, A. Halpern, Mc Graw Hill, 1988, ISBN 978-0-07-025734-4
• Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8
• The Physics of Vibrations and Waves (3rd edition), H.J. Pain, John Wiley & Sons, 1983, ISBN 0-471-90182-2
• Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0

External links

• Physics formulae at xs4all.nl

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• Alternative version on wikiversity Tables of Physics Formulae Wikiversity
• http://mathworld.wolfram.com/SphericalCoordinates.html