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: ${\displaystyle \mathbf {A} ^{2}\equiv \mathbf {A} \cdot \mathbf {A} \cos 0\equiv \left|\mathbf {A} \right|^{2}\equiv A^{2}.\,\!}$ 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
${\displaystyle \mathbf {r} '=\mathbf {r} +\mathbf {V} t\,\!}$
Relative accelerations
${\displaystyle \mathbf {a} '=\mathbf {a} +\mathbf {A} }$
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
${\displaystyle \theta '=\theta +\Omega t\,\!}$
Relative accelerations
${\displaystyle {\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}}$
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
${\displaystyle W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\left|{\boldsymbol {\tau }}\right|}{\mathrm {d} \theta }\right)\,\!}$
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
${\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!}$
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
${\displaystyle \mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m} \,\!}$
Coriolis acceleration and force c subscripts refer to coriolis ${\displaystyle \mathbf {a} _{c}=\mathbf {F} _{c}/m=-2{\boldsymbol {\omega \times v}}}$
Euler's equations ${\displaystyle \mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}\,\!}$

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
${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega ^{2}x\,\!}$ ${\displaystyle {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\omega ^{2}\theta \,\!}$
Unforced DHM
• b = damping constant
• κ = torsion constant
${\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega ^{2}x=0\,\!}$

Solution (see below for ω'):
${\displaystyle x=Ae^{-bt/2m}\cos \left(\omega '\right)\,\!}$

${\displaystyle \tau =1/\gamma \,\!}$

${\displaystyle {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\omega ^{2}\theta =0\,\!}$

${\displaystyle \tau =1/\gamma \,\!}$

Angular frequencies
Physical situation Nomenclature Equations
Linear undamped unforced SHO
• k = spring constant
• m = mass of oscillating bob
${\displaystyle \omega ={\sqrt {\frac {k}{m}}}\,\!}$
Linear unforced DHO
• k = spring constant
• b = Damping coefficient
${\displaystyle \omega '={\sqrt {{\frac {k}{m}}-\left({\frac {b}{2m}}\right)^{2}}}\,\!}$
Low amplitude angular SHO
• I = Moment of inertia about oscillating axis
• κ = torsion constant
${\displaystyle \omega ={\sqrt {\frac {I}{\kappa }}}\,\!}$
Low amplitude simple pendulum
• L = Length of pendulum
• g = Gravitational acceleration
• Θ = Angular amplitude
Approximate value
Energy in mechanical oscillations
Physical situation Nomenclature Equations
SHM energy
• T = kinetic energy
• U = potenial energy
• E = total energy
Potential energy
DHM energy ${\displaystyle E={\frac {m\left(\omega A\right)^{2}}{2}}e^{-bt/m}\,\!}$

Fluid mechanics

Equations in fluid mechanics
Physical situation Nomenclature Equations
Fluid statics,
• r = Position
• ρ = ρ(r) = Fluid density at gravitational equipotential containing r
• g = g(r) = Gravitational field strength at point r
${\displaystyle \nabla P=\rho \mathbf {g} \,\!}$
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
Bernoulli's equation pconstant is the total pressure at a point on a streamline ${\displaystyle p+\rho v^{2}/2+\rho gy=p_{\mathrm {constant} }\,\!}$
Euler equations
${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0\,\!}$
Convective acceleration ${\displaystyle \mathbf {a} =\left(\mathbf {v} \cdot \nabla \right)\mathbf {v} }$
Navier-stokes equations ${\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\nabla \cdot \mathbf {T} _{\mathrm {D} }+\mathbf {f} }$

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

Phase transitions

Physical situation Equations
Adiabatic transition ${\displaystyle \Delta Q=0,\quad \Delta U=W\,\!}$
Isothermal transition ${\displaystyle \Delta U=0,\quad \Delta W=-\Delta Q\,\!}$
Isobaric transition p1 = p2, p = constant
Isochoric transition V1 = V2, V = constant
Adiabatic expansion ${\displaystyle p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }\,\!}$
Free expansion ${\displaystyle \Delta U=0\,\!}$
Work done by an expanding gas Process

Net Work Done in Cyclic Processes
${\displaystyle \Delta W=\oint _{\mathrm {cycle} }p\mathrm {d} V\,\!}$

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

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K2 is the Modified Bessel function of the second kind.

Non-relativistic speeds

Relativistic speeds (Maxwell-Juttner distribution)
${\displaystyle f(p)={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}e^{-\gamma (p)/\theta }}$

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

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

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

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
${\displaystyle I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)\,\!}$
Internal energy of a substance
• CV = isovolumetric heat capacity of substance
• ΔT = temperature change of substance
${\displaystyle \Delta U=NC_{V}\Delta T\,\!}$
Work done by an expanding ideal gas ${\displaystyle W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V=\int _{T_{1}}^{T_{2}}Nk\mathrm {d} T\,\!}$
Meyer's equation
• Cp = isobaric heat capacity
• CV = isovolumetric heat capacity
• n = number of moles
${\displaystyle C_{p}-C_{V}=nR\,\!}$
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

Carnot refrigeration performance ${\displaystyle K_{C}=|Q_{L}|/(|Q_{H}|-|Q_{L}|)=T_{L}/(T_{H}-T_{L})\,\!}$

Waves

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

Standing waves

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

Propagating waves

Sound waves
Physical situation Nomenclature Equations
Average wave power P0 = Sound power due to source ${\displaystyle \langle P\rangle =\mu v\omega ^{2}x_{m}^{2}/2\,\!}$
Sound intensity

Ω = Solid angle

${\displaystyle I=P_{0}/(\Omega r^{2})\,\!}$
Acoustic beat frequency
• f1, f2 = frequencies of two waves (nearly equal amplitudes)
${\displaystyle f_{\mathrm {beat} }=\left|f_{2}-f_{1}\right|\,\!}$
Doppler effect for mechanical waves
• V = speed of sound wave in medium
• f0 = Source frequency
• v0 = Source velocity
${\displaystyle f_{r}=f_{0}{\frac {V\pm v_{r}}{v\mp v_{0}}}\,\!}$

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
${\displaystyle \sin \theta ={\frac {v}{v_{s}}}\,\!}$
Acoustic pressure and displacement amplitudes
• p0 = pressure amplitude
• s0 = displacement amplitude
• v = speed of sound
• ρ = local density of medium
${\displaystyle p_{0}=\left(v\rho \omega \right)s_{0}\,\!}$
Wave functions for sound Acoustic beats

Sound displacement function ${\displaystyle s=s_{0}\cos(kr-\omega t)\,\!}$

Sound pressure-variation ${\displaystyle p=p_{0}\sin(kr-\omega t)\,\!}$

Gravitational waves

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Gravitational radiation for two orbiting bodies in the low-speed limit. [1]

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

Superposition, interference, and diffraction

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

Constructive interference ${\displaystyle n={\frac {\lambda }{\Delta x}}\,\!}$

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
${\displaystyle v={\sqrt {\frac {p}{\rho }}}={\sqrt {\frac {F}{\mu }}}\,\!}$
Dispersion relation Implicit form
Amplitude modulation, AM ${\displaystyle A=A\left(t\right)}$
Frequency modulation, FM ${\displaystyle f=f\left(t\right)}$

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
${\displaystyle \nabla ^{2}A={\frac {1}{v_{\parallel }^{2}}}{\frac {\partial ^{2}A}{\partial t^{2}}}\,\!}$ ${\displaystyle A\left(\mathbf {r} ,t\right)=A\left(x-v_{\parallel }t\right)\,\!}$
Exponentially damped waveform
• A0 = Initial amplitude at time t = 0
• b = damping parameter
${\displaystyle A=A_{0}e^{-bt}\sin \left(kx-\omega t+\phi \right)\,\!}$
Korteweg–de Vries equation[2]
• α = constant
${\displaystyle {\frac {\partial y}{\partial t}}+\alpha y{\frac {\partial y}{\partial x}}+{\frac {\partial ^{3}y}{\partial x^{3}}}=0\,\!}$ ${\displaystyle A(x,t)={\frac {3v_{\parallel }}{\alpha }}\mathrm {sech} ^{2}\left[{\frac {\sqrt {v_{\parallel }}}{2}}\left(x-v_{\parallel }t\right)\right]\,\!}$

Sinusoidal solutions to the 3d wave equation

N different sinusoidal waves

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

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
${\displaystyle \mathbf {g} =-\nabla U}$
Point mass ${\displaystyle \mathbf {g} ={\frac {Gm}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!}$
At a point in a local array of point masses ${\displaystyle \mathbf {g} =\sum _{i}\mathbf {g} _{i}=G\sum _{i}{\frac {m_{i}}{\left|\mathbf {r} _{i}-\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} _{i}\,\!}$
Gravitational torque and potential energy due to non-uniform fields and mass moments
• V = volume of space occupied by the charge distribution
${\displaystyle {\boldsymbol {\tau }}=\int _{V_{n}}\left(\mathrm {d} \mathbf {m} \times \mathbf {g} +\mathbf {m} \times \mathrm {d} \mathbf {g} \right)\,\!}$
Gravitational field for a rotating body
${\displaystyle \mathbf {g} =-{\frac {GM}{\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} -\left|{\boldsymbol {\omega }}\right|^{2}\left|\mathbf {r} \right|\sin \phi \mathbf {\hat {a}} \,\!}$
Weak-gravitational field relativistic equations
Physical situation Nomenclature Equations
Gravitomagnetic field for a rotating body
• ξ = gravitomagnetic field
${\displaystyle {\boldsymbol {\xi }}={\frac {G}{2c^{2}}}{\frac {\mathbf {L} -3(\mathbf {L} \cdot \mathbf {\hat {r}} )\mathbf {\hat {r}} }{\left|\mathbf {r} \right|^{3}}}}$

Gravitational potentials

General classical equations.

Physical situation Nomenclature Equations
Potential energy from gravity, integral from Newton's law ${\displaystyle U=-{\frac {Gm_{1}m_{2}}{\left|\mathbf {r} \right|}}\approx m\left|\mathbf {g} \right|y\,\!}$
Escape speed
• M = Mass of body (e.g. planet) to escape from
• r = radius of body
${\displaystyle v={\sqrt {\frac {2GM}{r}}}\,\!}$
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)
{\displaystyle {\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 ${\displaystyle \mathbf {E} =-\nabla V}$
Point charge ${\displaystyle \mathbf {E} ={\frac {q}{4\pi \epsilon _{0}\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!}$
At a point in a local array of point charges ${\displaystyle \mathbf {E} =\sum \mathbf {E} _{i}={\frac {1}{4\pi \epsilon _{0}}}\sum _{i}{\frac {q_{i}}{\left|\mathbf {r} _{i}-\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} _{i}\,\!}$
At a point due to a continuum of charge ${\displaystyle \mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}\int _{V}{\frac {\mathbf {r} \rho \mathrm {d} V}{\left|\mathbf {r} \right|^{3}}}\,\!}$
Electrostatic torque and potential energy due to non-uniform fields and dipole moments ${\displaystyle {\boldsymbol {\tau }}=\int _{V}\left(\mathrm {d} \mathbf {p} \times \mathbf {E} +\mathbf {p} \times \mathrm {d} \mathbf {E} \right)\,\!}$

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
${\displaystyle {\frac {V_{s}}{V_{p}}}={\frac {N_{s}}{N_{p}}}={\frac {I_{p}}{I_{s}}}=\eta \,\!}$

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
${\displaystyle R_{\mathrm {net} }=\sum _{i=1}^{N}R_{i}\,\!}$ ${\displaystyle {\frac {1}{R_{\mathrm {net} }}}=\sum _{i=1}^{N}{\frac {1}{R_{i}}}\,\!}$
Charge, capacitors, currents
• qi = capacitance of capacitor i
• qi = charge of charge carrier i
${\displaystyle q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!}$ ${\displaystyle q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!}$
Inductors
• Li = self inductance of inductor i
• Lij = self inductance element ij of L matrix
• Mij = mutual inductance between inductors i and j
${\displaystyle {\frac {1}{L_{\mathrm {net} }}}=\sum _{i}{\frac {1}{L_{i}}}\,\!}$ ${\displaystyle V_{i}=\sum _{j=1}^{N}L_{ij}{\frac {\mathrm {d} I_{j}}{\mathrm {d} t}}\,\!}$
Circuit DC Circuit equations AC Circuit equations
Series circuit equations
RC circuits Circuit equation
RL circuits Circuit equation
LC circuits Circuit equation Circuit equation

Circuit resonant frequency ${\displaystyle \omega _{\mathrm {res} }=1/{\sqrt {LC}}\,\!}$

Circuit electrical potential energy ${\displaystyle U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!}$

Circuit magnetic potential energy ${\displaystyle U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!}$

RLC Circuits Circuit equation Circuit equation

Circuit charge

Light

Luminal electromagnetic waves

Physical situation Nomenclature Equations
Energy density in an EM wave For a dielectric:
Kinetic and potential momenta (non-standard terms in use) Potential momentum:
${\displaystyle I=\langle \mathbf {S} \rangle =E_{\mathrm {rms} }^{2}/c\mu _{0}\,\!}$
Doppler effect for light (relativistic) ${\displaystyle \lambda =\lambda _{0}{\sqrt {\frac {c-v}{c+v}}}\,\!}$
• n = refractive index
• v = speed of particle
• θ = cone angle
${\displaystyle \cos \theta ={\frac {v}{nc}}=v{\sqrt {\epsilon _{0}\mu _{0}}}\,\!}$
Electric and magnetic amplitudes
• E = electric field
• H = magnetic field strength
For a dielectric
EM wave components Electric

Magnetic

Geometric optics

Physical situation Nomenclature Equations
Critical angle
• n1 = refractive index of initial medium
• n2 = refractive index of final medium
• θc = critical angle
${\displaystyle \sin \theta _{c}={\frac {n_{2}}{n_{1}}}\,\!}$
Thin lens equation
• f = lens focal length
• x1 = object length
• x2 = image length
• r1 = incident curvature radius
• r2 = refracted curvature radius
${\displaystyle {\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}={\frac {1}{f}}\,\!}$
Image distance in a plane mirror ${\displaystyle x_{2}=-x_{1}\,\!}$
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
${\displaystyle \tan \theta _{B}=n_{2}/n_{1}\,\!}$
intensity from polarized light, Malus' law
• I0 = Initial intensity,
• I = Transmitted intensity,
• θ = Polarization angle between polarizer transmission axes and electric field vector
${\displaystyle I=I_{0}\cos ^{2}\theta \,\!}$

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)
The grating equation
• a = width of aperture, slit width
• α = incident angle to the normal of the grating plane
${\displaystyle {\frac {\delta }{2\pi }}\lambda =a\left(\sin \theta +\sin \alpha \right)\,\!}$
Rayleigh's criterion ${\displaystyle \theta _{R}=1.22\lambda /\,\!d}$
Bragg's law (solid state diffraction)
• d = lattice spacing
• δ = phase difference between two waves
${\displaystyle {\frac {\delta }{2\pi }}\lambda =2d\sin \theta \,\!}$
Single slit diffraction intensity
• I0 = source intensity
• Wave phase through apertures
${\displaystyle I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}\right]^{2}\,\!}$
N-slit diffraction (N ≥ 2)
• d = centre-to-centre separation of slits
• N = number of slits
• Phase between N waves emerging from each slit
${\displaystyle I=I_{0}\left[{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!}$
N-slit diffraction (all N) ${\displaystyle I=I_{0}\left[{\frac {\sin \left(\phi /2\right)}{\left(\phi /2\right)}}{\frac {\sin \left(N\delta /2\right)}{\sin \left(\delta /2\right)}}\right]^{2}\,\!}$
Circular aperture intensity
${\displaystyle I=I_{0}\left({\frac {2J_{1}(ka\sin \theta )}{ka\sin \theta }}\right)^{2}}$
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)
Huygen-Fresnel-Kirchhoff principle ${\displaystyle A\mathbf {(} \mathbf {r} )={\frac {-i}{2\lambda }}\iint _{\mathrm {aperture} }{\frac {e^{i\mathbf {k} \cdot \left(\mathbf {r} +\mathbf {r} _{0}\right)}}{\left|\mathbf {r} \right|\left|\mathbf {r} _{0}\right|}}\left[\cos \alpha _{0}-\cos \alpha \right]\mathrm {d} S\,\!}$
Kirchhoff's diffraction formula ${\displaystyle A\left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\iint _{\mathrm {aperture} }{\frac {e^{i\mathbf {k} \cdot \mathbf {r} _{0}}}{\left|\mathbf {r} _{0}\right|}}\left[i\left|\mathbf {k} \right|U_{0}\left(\mathbf {r} _{0}\right)\cos {\alpha }+{\frac {\partial A_{0}\left(\mathbf {r} _{0}\right)}{\partial n}}\right]\mathrm {d} S}$

Quantum mechanics

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

Wave–particle duality

Property or effect Nomenclature Equation
Planck–Einstein equation
${\displaystyle E=hf=hc/\lambda \,\!}$
de Broglie wavelength
• p = momentum of particle
• |k| = 2π/λ is the magnitude of the particle's wavevector k
• λ = wavelength of particle
${\displaystyle \lambda p=h\,\!}$
Heisenberg's uncertainty principle ${\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}\,\!}$
Probability Distributions
Property or effect Nomenclature Equation
Density of states ${\displaystyle N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}\,\!}$
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
${\displaystyle P(E_{i})=g(E_{i})/(e^{(E-\mu )/kT}+1)\,\!}$
Bose-Einstein distribution (bosons) ${\displaystyle P(E_{i})=g(E_{i})/(e^{(E_{i}-\mu )/kT}-1)\,\!}$

Angular momentum

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Property or effect Nomenclature Equation
Angular momentum quantum numbers
• s = spin quantum number
• ms = spin magnetic quantum number
• = Azimuthal quantum number
• m = azimuthal magnetic quantum number
• mj = total angular momentum magnetic quantum number
• j = total angular momentum quantum number
Spin projection:
Angular momentum magnitudes angular momementa:
• S = Spin,
• L = orbital,
• J = total
Spin magnitude:
Angular momentum components Spin:
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
${\displaystyle {\boldsymbol {\mu }}_{\ell }=-e\mathbf {L} /2m_{e}=g_{\ell }{\frac {\mu _{B}}{\hbar }}\mathbf {L} \,\!}$
spin magnetic dipole moment
${\displaystyle {\boldsymbol {\mu }}_{s}=-e\mathbf {S} /m_{e}=g_{s}{\frac {\mu _{B}}{\hbar }}\mathbf {S} \,\!}$
dipole moment potential
• U = potential energy of dipole in field
${\displaystyle U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!}$
The Hydrogen atom
Property or effect Nomenclature Equation
Energy levels
${\displaystyle E_{n}=-me^{4}/8\epsilon _{0}^{2}h^{2}n^{2}=13.61eV/n^{2}\,\!}$
Spectrum λ = wavelength of emitted photon, during electronic transition from Ei to Ej ${\displaystyle {\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.

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:

for non-interacting particles:

Position-momentum Fourier transform (1 particle in 3d)
• Φ = momentum-space wavefunction
• Ψ = position-space wavefunction
{\displaystyle {\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.
${\displaystyle P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!}$
General normalization condition ${\displaystyle P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\int \limits _{\mathrm {all\,space} }\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!}$

Relativity

Lorentz frame tranforms

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

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

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.[3]. 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 ${\displaystyle \gamma \left(v\right)={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\,\!}$
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
${\displaystyle \Delta t=\gamma \Delta t_{0}\,\!}$
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
${\displaystyle \Delta x={\frac {\Delta x_{0}}{\gamma }}\,\!}$
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
${\displaystyle m=\gamma m_{0}\,\!}$
3-Momentum ${\displaystyle \mathbf {p} =\gamma m\mathbf {v} \,\!}$
Mass-energy equivalence ${\displaystyle E=mc^{2}\,\!}$
Momentum-energy for massless particles ${\displaystyle E=pc\,\!}$
Kinetic energy ${\displaystyle T=\left(\gamma -1\right)mc^{2}\,\!}$

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
${\displaystyle A=Z+N\,\!}$
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

${\displaystyle r=r_{0}A^{1/3}\,\!}$ 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,
{\displaystyle {\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
• I0 = Initial intensity/Flux of radiation
• I = Number of atoms at time t
• μ = Linear absorption coefficient
• x = Thickness of substance
${\displaystyle I=I_{0}e^{-\mu x}\,\!}$
Fermi energy ${\displaystyle E_{F}=(3/16{\sqrt {2}}\pi )^{2/3}h^{2}n^{2/3}m\,\!}$
Hubble's law ${\displaystyle v=Hr\,\!}$

References

2. Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3
3. Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Manchester Physics Series, John Wiley & Sons, 2009, ISBN 978-0-470-01460-8
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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