# 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 accelerations
$\mathbf {a} '=\mathbf {a} +\mathbf {A}$ 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 accelerations
${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}$ Transformation of any vector T to a rotating frame

### 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
$\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!$ 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
$\mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} \,\!$ 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.

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

## 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
$pV=nRT=kTN\,\!$ Pressure of an ideal gas
• m = mass of one molecule
• Mm = molar mass
$p={\frac {Nm\langle v^{2}\rangle }{3V}}={\frac {nM_{m}\langle v^{2}\rangle }{3V}}={\frac {1}{3}}\rho \langle v^{2}\rangle \,\!$ ### 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

</div">

K2 is the Modified Bessel function of the second kind.

Non-relativistic speeds
Entropy Logarithm of the density of states
• Pi = probability of system in microstate i
• Ω = total number of microstates
$S=-k_{B}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega \,\!$ Entropy change $\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}\,\!$ Entropic force $\mathbf {F} _{\mathrm {S} }=-T\nabla S\,\!$ Equipartition theorem
• df = degree of freedom
Average kinetic energy per degree of freedom

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

The four most common Maxwell's relations are:

### 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

### Thermal efficiencies

Physical situation Nomenclature Equations
Thermodynamic engines
• η = efficiency
• W = work done by engine
• QH = heat energy in higher temperature reservoir
• QC = heat energy in lower temperature reservoir
• TH = temperature of higher temp. reservoir
• TC = temperature of lower temp. reservoir
Thermodynamic engine:
Refrigeration
• K = coefficient of refrigeration performance
Refrigeration performance

## Waves

### 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

$I=P_{0}/(\Omega r^{2})\,\!$ 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
• v0 = Source 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
Gravitational waves

{{#invoke:main|main}}

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

Physical situation Nomenclature Equations
• 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}}}\$ • 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
${\frac {\Delta r}{\lambda }}={\frac {\Delta t}{T}}={\frac {\phi }{2\pi }}\,\!$ ### 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.

### General wave functions

#### Sinusoidal solutions to the 3d wave equation

N different sinusoidal waves

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.

## Gravitation

### Gravitational fields

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
• U = gravitational potential
• C = curved path traversed by a mass in the field
$\mathbf {g} =-\nabla U$ 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
${\boldsymbol {\tau }}=\int _{V_{n}}\left(\mathrm {d} \mathbf {m} \times \mathbf {g} +\mathbf {m} \times \mathrm {d} \mathbf {g} \right)\,\!$ Gravitational field for a rotating body $\mathbf {g} =-{\frac {GM}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} -\left|{\boldsymbol {\omega }}\right|^{2}\left|\mathbf {r} \right|\sin \phi \mathbf {\hat {a}} \,\!$ 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

### Magnetic fields and moments

General classical equations

### 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.

## Light

### 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
${\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}={\frac {1}{f}}\,\!$ Image distance in a plane mirror $x_{2}=-x_{1}\,\!$ Spherical mirror
• r = curvature radius of mirror
Spherical mirror equation

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:

where:

### Polarization

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

### Wave–particle duality

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}}

Magnetic moments

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

The Hydrogen atom
Property or effect Nomenclature Equation
Energy levels
$E_{n}=-me^{4}/8\epsilon _{0}^{2}h^{2}n^{2}=13.61eV/n^{2}\,\!$ 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} Wavefunctions

The general form wavefunction for a system of particles, each with position ri and z-component of spin sz i. Sums are over the descrete variable sz, 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.

## 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.

### 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 $\gamma \left(v\right)={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\,\!$ 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 $E=mc^{2}\,\!$ 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

r0 ≈ 1.2 fm

$r=r_{0}A^{1/3}\,\!$ hence (approximately)
• nuclear volume ∝ A
• nuclear surface ∝ A2/3
• N0 = Initial number of atoms
• N = Number of atoms at time t
• λ = Decay constant
• t = Time
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
$I=I_{0}e^{-\mu x}\,\!$ Fermi energy $E_{F}=(3/16{\sqrt {2}}\pi )^{2/3}h^{2}n^{2/3}m\,\!$ Hubble's law $v=Hr\,\!$ 