Defining equation (physics)

In physics, defining equations are equations that define new quantities in terms of base quantities.[1] This article uses the current SI system of units, not natural or characteristic units.

Contents

Treatment of vectors

There are many forms of vector notation. In this section the following is specifically used throughout the article, closely matching standard use; upright boldface is for a vector quantity and upright boldface with a hat (circumflex) is for a unit vector.

The standard ordered vector basis for spherical polar coordinates;  \scriptstyle{\boldsymbol{\hat{r}}, \boldsymbol{\hat{\theta}}, \boldsymbol{\hat{\phi}}} \,\! are used with the general forms of unit vectors below.

Common general themes of unit vectors occur throughout physics:

Unit vector Nomenclature Diagram
Tangent vector to a curve/flux line  \mathbf{\hat{t}}\,\! "200px" "200px"

A normal vector  \mathbf{\hat{n}} \,\! to the plane containing and defined by the radial position vector  r \mathbf{\hat{r}} \,\! and angular tangential direction of rotation  \theta \boldsymbol{\hat{\theta}} \,\! is necessary so that the vector equations of angular motion hold.

Normal to a surface tangent plane/plane containing radial position component and angular tangential component  \mathbf{\hat{n}}\,\!

In terms of polar coordinates;  \mathbf{\hat{n}} = \mathbf{\hat{r}} \times \boldsymbol{\hat{\theta}} \,\!

Binormal vector to tangent and normal  \mathbf{\hat{b}} = \mathbf{\hat{t}} \times \mathbf{\hat{n}} \,\![2]
Parallel to some axis/line  \mathbf{\hat{e}}_{\parallel} \,\! "200px"

One unit vector  \mathbf{\hat{e}}_{\parallel}\,\! aligned parallel to a principle direction (red line), and a perpendicular unit vector  \mathbf{\hat{e}}_{\bot}\,\! is in any radial direction relative to the principle line.

Perpendicular to some axis/line in some radial direction  \mathbf{\hat{e}}_{\bot} \,\!
Possible angular deviation relative to some axis/line  \mathbf{\hat{e}}_{\angle} \,\! "200px"

Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principle direction.

Classical mechanics

Mass and inertia

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Linear, surface, volumetric mass density λ or μ (especially in acoustics, see below) for Linear, σ for surface, ρ for volume.  \lambda = \mathrm{d} m/\mathrm{d} x \,\! (or  \mu = \mathrm{d} m/\mathrm{d} x \,\!)

 \sigma = \mathrm{d}^2 m/\mathrm{d} x_2 \mathrm{d} x_1 = \mathrm{d}^2 m/\mathrm{d} S \,\!  \rho = \mathrm{d}^3 m/\mathrm{d} x_3 \mathrm{d} x_2 \mathrm{d} x_1 = \mathrm{d} m/\mathrm{d} V \,\!

kg mn, n = 1, 2, 3 [M][L]n
Moment of mass[3] m (No common symbol) Point mass:

 \mathbf{m} = \mathbf{r}m \,\!

Descrete masses about an axis  x_i \,\!:
 \mathbf{m} = \sum_{i=1}^N \mathbf{r}_\mathrm{i} m_i \,\!

Continuum of mass about an axis  x_i \,\!:
 \mathbf{m} = \int \rho \left ( \mathbf{r} \right ) x_i \mathrm{d} \mathbf{r} \,\!

kg m [M][L]
Centre of mass rcom

(Symbols vary)

ith moment of mass  \mathbf{m}_\mathrm{i} = \mathbf{r}_\mathrm{i} m_i \,\!

Discrete masses:
 \mathbf{r}_\mathrm{com} = \frac{1}{M}\sum_i \mathbf{r}_\mathrm{i} m_i = \frac{1}{M}\sum_i \mathbf{m}_\mathrm{i} \,\!

Mass continuum:
 \mathbf{r}_\mathrm{com} = \frac{1}{M}\int \mathrm{d}\mathbf{m} = \frac{1}{M}\int \mathbf{r} \mathrm{d}m = \frac{1}{M}\int \mathbf{r} \rho \mathrm{d}V \,\!

m [L]
2-Body reduced mass m12, μ Pair of masses = m1 and m2  \mu = \left (m_1m_2 \right )/\left ( m_1 + m_2 \right) \,\! kg [M][L]2
Moment of inertia (MOI) I Discrete Masses:

 I = \sum_i \mathbf{m}_\mathrm{i} \cdot \mathbf{r}_\mathrm{i} = \sum_i \left | \mathbf{r}_\mathrm{i} \right | ^2 m \,\!

Mass continuum:
 I = \int \left | \mathbf{r} \right | ^2 \mathrm{d} m = \int \mathbf{r} \cdot \mathrm{d} \mathbf{m}  = \int \left | \mathbf{r} \right | ^2 \rho \mathrm{d}V \,\!

kg m2 s−1 [M][L]2

Derived kinematic quantities

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Velocity v  \mathbf{v} = \mathrm{d} \mathbf{r}/\mathrm{d} t \,\! m s−1 [L][T]−1
Acceleration a  \mathbf{a} = \mathrm{d} \mathbf{v}/\mathrm{d} t = \mathrm{d}^2 \mathbf{r}/\mathrm{d} t^2  \,\! m s−2 [L][T]−2
Jerk j  \mathbf{j} = \mathrm{d} \mathbf{a}/\mathrm{d} t = \mathrm{d}^3 \mathbf{r}/\mathrm{d} t^3 \,\! m s−3 [L][T]−3
Angular velocity ω  \boldsymbol{\omega} = \mathbf{\hat{n}} \left ( \mathrm{d} \theta /\mathrm{d} t \right ) \,\! rad s−1 [T]−1
Angular Acceleration α  \boldsymbol{\alpha} = \mathrm{d} \boldsymbol{\omega}/\mathrm{d} t = \mathbf{\hat{n}} \left ( \mathrm{d}^2 \theta / \mathrm{d} t^2 \right ) \,\! rad s−2 [L][T]−2

Derived dynamic quantities

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Momentum p  \mathbf{p}=m\mathbf{v} \,\! kg m s−1 [M][L][T]−1
Force F  \mathbf{F} = \mathrm{d} \mathbf{p}/\mathrm{d} t \,\! N = kg m s−2 [M][L][T]−2
Impulse Δp, I  \mathbf{I} = \Delta \mathbf{p} = \int_{t_1}^{t_2} \mathbf{F}\mathrm{d} t \,\! kg m s−1 [M][L][T]−1
Angular momentum about a position point r0, L, J, S  \mathbf{L} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{p} \,\!

Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point.

kg m2 s−1 [M][L]2[T]−1
Moment of a force about a position point r0,

Torque

τ, M  \boldsymbol{\tau} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{F} = \mathrm{d} \mathbf{L}/\mathrm{d} t \,\! N m = kg m2 s−2 [M][L]2[T]−2
Angular impulse ΔL (no common symbol)  \Delta \mathbf{L} = \int_{t_1}^{t_2} \boldsymbol{\tau}\mathrm{d} t \,\! kg m2 s−1 [M][L]2[T]−1

General energy definitions

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Mechanical work due

to a Resultant Force

W  W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{r} \,\! J = N m = kg m2 s−2 [M][L]2[T]−2
Work done ON mechanical

system, Work done BY

WON, WBY  \Delta W_\mathrm{ON} = - \Delta W_\mathrm{BY} \,\! J = N m = kg m2 s−2 [M][L]2[T]−2
Potential energy φ, Φ, U, V, Ep  \Delta W = - \Delta V \,\! J = N m = kg m2 s−2 [M][L]2[T]−2
Mechanical power P  P = \mathrm{d}E/\mathrm{d}t \,\! W = J s−1 [M][L]2[T]−3
Lagrangian L  L = T-V \,\! J [M][L]2[T]−2
Hamiltonian H  H = T + V \,\! J [M][L]2[T]−2
Action S,  \scriptstyle{\mathcal{S}} \,\!  \mathcal{S} = \int_{t_1}^{t_2} L {\mathrm{d}t} \,\! J s [M][L]2[T]−1

Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

- Wherever the force is zero, its potential energy is defined to be zero as well. - Whenever the force does work, potential energy is lost.

Transport mechanics

Here  \mathbf{\hat{t}} \,\! is a unit vector in the direction of the flow/current/flux.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Flow velocity vector field u  \mathbf{u}=\mathbf{u}\left ( \mathbf{r},t \right ) \,\! m s−1 [L][T]−1
Volume velocity, volume flux φV (no standard symbol) \phi_V = \int_S \mathbf{u} \cdot \mathrm{d}\mathbf{A}\,\! m3 s−1 [L]3 [T]−1
Mass current per unit volume s (no standard symbol) s = \mathrm{d}\rho / \mathrm{d}t \,\! kg m−3 s−1 [M] [L]−3 [T]−1
Mass current, mass flow rate Im  I_\mathrm{m} = \mathrm{d} m/\mathrm{d} t \,\! kg s−1 [M][T]−1
Mass current density jm  \mathbf{j}_\mathrm{m} \cdot \mathbf{\hat{n}} = \mathrm{d} I_\mathrm{m} /\mathrm{d} A = \mathrm{d}^2 m/\mathrm{d} A \mathrm{d} t \,\! kg m−2 s−1 [M][L]−2[T]−1
Momentum current Ip  I_\mathrm{p} = \mathrm{d} \left | \mathbf{p} \right |/\mathrm{d} t \,\! kg m s−2 [M][L][T]−2
Momentum current density jp  \mathbf{j}_\mathrm{p} \cdot \mathbf{\hat{n}} = \mathrm{d} I_\mathrm{p}/\mathrm{d} A  = \mathrm{d}^2 \left | \mathbf{p} \right | /\mathrm{d} A \mathrm{d} t \,\! kg m s−2 [M][L][T]−2

Properties of matter

Stress and strain

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
General stress,

Pressure

P, σ  \sigma = F/A \,\!

F may be any force applied to area A

Pa = N m−2 [M] [T] [L]−1
General strain ε  \epsilon = \Delta D / D \,\!

D = dimension (length, area, volume)  \Delta D \,\! = change in dimension

dimensionless dimensionless
General elastic modulus Emod  E_\mathrm{mod} = \sigma / \epsilon \,\! Pa = N m−2 [M] [T] [L]−1
Young's modulus E, Y  Y = \sigma / \left ( \Delta L/ L \right ) \,\! Pa = N m−2 [M] [T] [L]−1
Shear modulus G  G = \Delta x/L\,\! Pa = N m−2 [M] [T] [L]−1
Bulk modulus B  B = P/\left ( \Delta V / V \right ) \,\! Pa = N m−2 [M] [T] [L]−1
Compressibility C  C = 1/B \,\! Pa−1 = m2 N−1 [L] [M]−1 [T]−1

Thermodynamics

Thermodynamic quantities

General fundamental quantities

Quantity (common name/s) (Common) symbol/s SI units Dimension
Number of molecules N dimensionless dimensionless
Number of moles n mol [N]
Temperature T K [Θ]
Heat energy Q, q J [M][L]2[T]−2
Latent heat QL J [M][L]2[T]−2
Entropy S J K−1 [M][L]2[T]−2 [Θ]−1
Negentropy J J K−1 [M][L]2[T]−2 [Θ]−1

Thermal properties of matter

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
General heat/thermal capacity C  C = \partial Q/\partial T\,\! J K −1 [M][L]2[T]−2 [Θ]−1
Heat capacity (isobaric) Cp  C_{p} = \partial Q/\partial T\,\! J K −1 [M][L]2[T]−2 [Θ]−1
Specific heat capacity (isobaric) Cmp  C_{mp} = \partial^2 Q/\partial m \partial T \,\! J kg−1 K−1 [L]2[T]−2 [Θ]−1
Molar specific heat

Capacity (isobaric)

Cnp C_{np} = \partial^2 Q/\partial n \partial T \,\! J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1
Heat capacity (isochoric/volumetric) CV  C_{V} = \partial Q/\partial T \,\! J K −1 [M][L]2[T]−2 [Θ]−1
Specific heat capacity (isochoric) CmV  C_{mV} = \partial^2 Q/\partial m \partial T \,\! J kg−1 K−1 [L]2[T]−2 [Θ]−1
Molar specific heat capacity (isochoric) CnV  C_{nV} = \partial^2 Q/\partial n \partial T \,\! J K −1 mol−1 [M][L]2[T]−2 [Θ]−1 [N]−1
Specific latent heat L L = \partial Q/ \partial m \,\! J kg−1 [L]2[T]−2
Ratio of isobaric to isochoric heat capacity,

Heat capacity ratio, adiabatic index

γ \gamma = C_p/C_V = c_p/c_V = C_{mp}/C_{mV} \,\! dimensionless dimensionless

Thermal transfer

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Temperature gradient No standard symbol  \nabla T \,\! K m−1 [Θ][L]−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer P P = \mathrm{d} q/\mathrm{d} t \,\! W = J s−1 [M] [L]2 [T]−2
Thermal intensity I I = \mathrm{d} P/\mathrm{d} A= \mathrm{d}^2 q/\mathrm{d} A \mathrm{d} t \,\! W m−2 [M] [T]−3
Thermal/heat flux density (vector analogue of thermal intensity above) q \mathbf{q} \cdot \mathbf{\hat{t}} = I = \mathrm{d}^2 q/\mathrm{d} A \mathrm{d} t \,\! W m−2 [M] [T]−3

Waves

General fundamental quantities

A wave can be longnitudinal where the oscillations are parallel (or antiparallel) to the propagation direction, or transverse where the oscillations are perpendicular to the propagation direction. These oscillations are characterized by a periodically time-varying displacement in the parallel or perpendicular direction, and so the instantaneous velocity and acceleration are also periodic and time varying in these directions. But the wave profile (the apparent motion of the wave due to the successive oscillations of particles or fields about their equilibrium positions) propagates at the phase and group velocities parallel or antiparallel to the propagation direction, which is common to longitudinal and transverse waves. Below oscillatory displacement, velocity and acceleration refer to the kinematics in the oscillating directions of the wave - transverse or longitudinal (mathematical description is identical), the group and phase velocities are separate.

Quantity (common name/s) (Common) symbol/s SI units Dimension
Number of wave cycles N dimensionless dimensionless
(Oscillatory) displacement Symbol of any quantity which varies periodically, such as h, x, y (mechanical waves), x, s, η (longitudinal waves) I, V, E, B, H, D (electromagnetism), u, U (luminal waves), ψ, Ψ, Φ (quantum mechanics). Most general purposes use y, ψ, Ψ. For generality here, A is used and can be replaced by any other symbol, since others have specific, common uses.

\mathbf{A} = A \mathbf{\hat{e}}_{\parallel} \,\! for longitudinal waves,
\mathbf{A} = A \mathbf{\hat{e}}_{\bot} \,\! for transverse waves.

m [L]
(Oscillatory) displacement amplitude Any quantity symbol typically subscripted with 0, m or max, or the capitalized letter (if displacement was in lower case). Here for generality A0 is used and can be replaced. m [L]
(Oscillatory) velocity amplitude V, v0, vm. Here v0 is used. m s−1 [L][T]−1
(Oscillatory) acceleration amplitude A, a0, am. Here a0 is used. m s−2 [L][T]−2
Spatial position
Position of a point in space, not necessarily a point on the wave profile or any line of propagation
d, r m [M]
Wave profile displacement
Along propagation direction, distance travelled (path length) by one wave from the source point r0 to any point in space d (for longitudinal or transverse waves)
L, d, r

 \mathbf{r} \equiv r \mathbf{\hat{e}}_{\parallel} \equiv \mathbf{d} - \mathbf{r}_0 \,\!

m [L]
Phase angle δ, ε, φ rad dimensionless

General derived quantities

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Wavelength λ General definition (allows for FM):

\lambda = \mathrm{d} r/\mathrm{d} N \,\!

For non-FM waves this reduces to:
\lambda = \Delta r/\Delta N \,\!

m [L]
Wavenumber, k-vector, Wave vector k, σ Two definitions are in use:

\mathbf{k} = \left ( 2\pi/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!
\mathbf{k} = \left ( 1/\lambda \right ) \mathbf{\hat{e}}_{\angle} \,\!

m−1 [L]−1
Frequency f, ν General definition (allows for FM):

f = \mathrm{d} N/\mathrm{d} t \,\!

For non-FM waves this reduces to:
f = \Delta N/\Delta t \,\!

In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation:
f = 1/T \,\!

Hz = s−1 [T]−1
Angular frequency/ pulsatance ω \omega = 2\pi f = 2\pi / T \,\! Hz = s−1 [T]−1
Oscillatory velocity v, vt, v Longitudinal waves:

 \mathbf{v} = \mathbf{\hat{e}}_{\parallel} \left ( \partial A/\partial t \right ) \,\!

Transverse waves:
 \mathbf{v} = \mathbf{\hat{e}}_{\bot} \left ( \partial A/\partial t \right ) \,\!

m s−1 [L][T]−1
Oscillatory acceleration a, at Longitudinal waves:

 \mathbf{a} = \mathbf{\hat{e}}_{\parallel} \left ( \partial^2 A/\partial t^2 \right ) \,\!

Transverse waves:
 \mathbf{a} = \mathbf{\hat{e}}_{\bot} \left ( \partial^2 A/\partial t^2 \right ) \,\!

m s−2 [L][T]−2
Path length difference between two waves L, ΔL, Δx, Δr  \mathbf{r} =  \mathbf{r}_2 - \mathbf{r}_1 \,\! m [L]
Phase velocity vp General definition:

 \mathbf{v}_\mathrm{p} = \mathbf{\hat{e}}_{\parallel} \left ( \Delta r /\Delta t \right ) \,\!

In practice reduces to the useful form:
 \mathbf{v}_\mathrm{p} = \lambda f \mathbf{\hat{e}}_{\parallel} = \left ( \omega/k \right ) \mathbf{\hat{e}}_{\parallel} \,\!

m s−1 [L][T]−1
(Longitudinal) group velocity vg  \mathbf{v}_\mathrm{g} = \mathbf{\hat{e}}_{\parallel} \left ( \partial \omega /\partial k \right ) \,\! m s−1 [L][T]−1
Time delay, time lag/lead Δt  \Delta t = t_2 - t_1 \,\! s [T]
Phase difference δ, Δε, Δϕ  \Delta \phi = \phi_2 - \phi_1 \,\! rad dimensionless
Phase No standard symbol  \mathbf{k} \cdot \mathbf{r} \mp \omega t + \phi= 2\pi N \,\!

Physically;
upper sign: wave propagation in +r direction
lower sign: wave propagation in −r direction

Phase angle can lag if: ϕ > 0
or lead if: ϕ < 0.

rad dimensionless

Relation between space, time, angle analogues used to describe the phase:

 \frac{\Delta r}{\lambda} = \frac{\Delta t}{T} = \frac{\phi}{2\pi} \,\!

Modulation indicies

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
AM index:
h, hAM  h_{AM} = A/A_m \,\!

A = carrier amplitude
Am = peak amplitude of a component in the modulating signal

dimensionless dimensionless
FM index:
hFM  h_{FM} = \Delta f/f_m \,\!

Δf = max. deviation of the instantaneous frequency from the carrier frequency
fm = peak frequency of a component in the modulating signal

dimensionless dimensionless
PM index:
hPM  h_{PM} = \Delta \phi \,\!

Δϕ = peak phase deviation

dimensionless dimensionless

Acoustics

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Acoustic impedance Z Z = \rho v\,\!

v = speed of sound, ρ = volume density of medium

kg m−2 s−1 [M] [L]−2 [T]−1
Specific acoustic impedance z z = ZS\,\!

S = surface area

kg s−1 [M] [T]−1
Sound Level β \beta = \left ( \mathrm{dB} \right ) 10 \log \left | \frac{I}{I_0} \right | \,\! dimensionless dimensionless

Gravitation

A common misconseption occurs between centre of mass and centre of gravity. They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical descrition of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts. They are only equal if and only if the external gravitational field is uniform.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Centre of gravity rcog (symbols vary) ith moment of mass  \mathbf{m}_i = \mathbf{r}_i m_i \,\!

Centre of gravity for a descrete masses:
 \begin{align} \mathbf{r}_\mathrm{cog} & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right |}\sum_i \mathbf{m}_i \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right | \\
 & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\sum_i \mathbf{r}_i m_i \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right | \end{align}\,\!

Centre of a gravity for a continuum of mass:
 \begin{align} \mathbf{r}_\mathrm{cog} & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\int \left | \mathbf{g} \left ( \mathbf{r} \right ) \right |\mathrm{d}\mathbf{m} \\
 & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\int \mathbf{r} \left | \mathbf{g} \left ( \mathbf{r} \right ) \right | \mathrm{d}^n m \\
 & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\int \mathbf{r} \rho_n \left | \mathbf{g} \left ( \mathbf{r} \right ) \right | \mathrm{d}^n x \end{align} \,\!

m [L]
Standard gravitational parameter of a mass μ  \mu = Gm \,\! N m2 kg−1 [L]3 [T]−2
Gravitational field, field strength, potential gradient, acceleration g \mathbf{g} = \mathbf{F}/m \,\! N kg−1 = m s−2 [L][T]−2
Gravitational flux ΦG \Phi_G = \int_S \mathbf{g} \cdot \mathrm{d}\mathbf{A} \,\! m3 s−2 [L]3[T]−2
Absolute gravitational potential Φ, φ, U, V  U = - \frac{W_{\infty r}}{m} = - \frac{1}{m} \int_\infty^{r} \mathbf{F} \cdot \mathrm{d}\mathbf{r} = - \int_\infty^{r} \mathbf{g} \cdot \mathrm{d}\mathbf{r} \,\! J kg−1 [L]2[T]−2
Gravitational potential difference ΔΦ, Δφ, ΔU, ΔV  \Delta U = - \frac{W}{m} = - \frac{1}{m} \int_{r_1}^{r_2} \mathbf{F} \cdot \mathrm{d}\mathbf{r} = - \int_{r_1}^{r_2} \mathbf{g} \cdot \mathrm{d}\mathbf{r} \,\! J kg−1 [L]2[T]−2
Gravitational potential energy Ep  E_p = - W_{\infty r} \,\! J [M][L]2[T]−2
Gravitational torsion field Ω  \boldsymbol{\Omega} = 2 \boldsymbol{\xi} \,\! Hz = s−1 [T]−1
Gravitational torsion flux ΦΩ \Phi_\Omega = \int_S \boldsymbol{\Omega} \cdot \mathrm{d}\mathbf{A} \,\! N m s kg−1 = m2 s−1 [M]2 [T]−1
Gravitomagnetic field H, Bg, B, ξ  \mathbf{F} = m \left ( \mathbf{v} \times 2 \boldsymbol{\xi} \right ) \,\! Hz = s−1 [T]−1
Gravitomagnetic flux Φξ \Phi_\xi = \int_S \boldsymbol{\xi} \cdot \mathrm{d}\mathbf{A} \,\! N m s kg−1 = m2 s−1 [M]2 [T]−1
Gravitomagnetic vector potential [4] h  \mathbf{\xi} = \nabla \times \mathbf{h} \,\! m s−1 [M] [T]−1

Electromagnetism

Here subscripts e and m are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre). Dimensional analysis shows that magnetic charges relate by qm(Wb) = μ0 qm(Am).

Initial quantities

Quantity (common name/s) (Common) symbol/s SI units Dimension
Electric charge qe, q, Q C = As [I][T]
Monopole strength, magnetic charge qm, g, p Wb or Am [L]2[M][T]−2 [I]−1 (Wb)

[I][L] (Am)

Electric quantities

Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.

Electric transport

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Linear, surface, volumetric charge density λe for Linear, σe for surface, ρe for volume.  \lambda_e = \mathrm{d} q_e/\mathrm{d} x \,\!

 \sigma_e = \mathrm{d}^2 q_e/\mathrm{d} x_2 \mathrm{d} x_1 = \mathrm{d}^2 m/\mathrm{d} S \,\!  \rho_e = \mathrm{d}^3 q_e/\mathrm{d} x_3 \mathrm{d} x_2 \mathrm{d} x_1 = \mathrm{d} m/\mathrm{d} V \,\!

C mn, n = 1, 2, 3 [I][T][L]n
Capacitance C C = \mathrm{d}q/\mathrm{d}V\,\!

V = voltage, not volume.

F = C V−1 [I][T]3[L]−2[M]−1
Electric current I  I = \mathrm{d}q/\mathrm{d}t \,\! A [I]
Electric current density J \mathbf{J} \cdot \mathbf{\hat{n}} = \mathrm{d} I/\mathrm{d} A \,\! A m−2 [I][L]−2
Displacement current density Jd  \mathbf{J}_\mathrm{d} = \epsilon_0 \left ( \partial \mathbf{E} / \partial t \right ) = \partial \mathbf{D} / \partial t \,\! Am−2 [I][L]m−2
Convection current density Jc  \mathbf{J}_\mathrm{c} = \rho \mathbf{v} \,\! A m−2 [I] [L]m−2

Electric fields

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Electric field, field strength, flux density, potential gradient E \mathbf{E} =\mathbf{F}/q\,\! N C−1 = V m−1 [M][L][T]−3[I]−1
Electric flux ΦE \Phi_E = \int_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}\,\! N m2 C−1 [M][L]3[T]−3[I]−1
Absolute permittivity; ε  \epsilon = \epsilon_r \epsilon_0\,\! F m−1 [I]2 [T]4 [M]−1 [L]−3
Electric displacement field D \mathbf{D} =\mathbf{E}/\epsilon\,\! C m−2 [I][T][L]−2
Electric displacement flux ΦD \Phi_D = \int_S \mathbf{D} \cdot \mathrm{d} \mathbf{A}\,\! C [I][T]
Electric dipole moment p \mathbf{p} = 2q\mathbf{a}\,\!

a = charge separation directed from -ve to +ve charge

C m [I][T][L]
Electric Polarization, polarization density P \mathbf{P} = \mathrm{d} \langle \mathbf{p} \rangle /\mathrm{d} V \,\! C m−2 [I][T][L]−2
Absolute electric potential, EM scalar potential relative to point  r_0 \,\!

Theoretical:  r_0 = \infty \,\!
Practical:  r_0 = R_\mathrm{earth} \,\! (Earth's radius)

φ ,V  V = -\frac{W_{\infty r }}{q} = -\frac{1}{q}\int_\infty^r \mathbf{F} \cdot \mathrm{d} \mathbf{r} = -\int_{r_1}^{r_2} \mathbf{E} \cdot \mathrm{d} \mathbf{r}\,\! V = J C−1 [M] [L] [T]−3 [I]−1
Voltage, Electric potential difference ΔφV \Delta V = -\frac{\Delta W}{q} = -\frac{1}{q}\int_{r_1}^{r_2} \mathbf{F} \cdot \mathrm{d} \mathbf{r} = -\int_{r_1}^{r_2} \mathbf{E} \cdot \mathrm{d} \mathbf{r} \,\! V = J C−1 [M] [L] [T]−3 [I]−1
Electric potential energy U U = - W\,\! J [M][L]2[T]2

Magnetic quantities

Magnetic transport

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Linear, surface, volumetric pole density λm for Linear, σm for surface, ρm for volume.  \lambda_m = \mathrm{d} q_m/\mathrm{d} x \,\!

 \sigma_m = \mathrm{d}^2 q_m/\mathrm{d} x_2 \mathrm{d} x_1 = \mathrm{d}^2 m/\mathrm{d} S \,\!  \rho_m = \mathrm{d}^3 q_m/\mathrm{d} x_3 \mathrm{d} x_2 \mathrm{d} x_1 = \mathrm{d} m/\mathrm{d} V \,\!

Wb mn

A m−(n + 1),
n = 1, 2, 3

[L]2[M][T]−2 [I]−1 (Wb)

[I][L] (Am)

Monopole current Im  I_m = \mathrm{d}q_m/\mathrm{d}t \,\! Wb s−1

A m s−1

[L]2[M][T]−3 [I]−1 (Wb)

[I][L][T]−1 (Am)

Monopole current density Jm \mathbf{J}_\mathrm{m} \cdot \mathbf{\hat{n}} = \mathrm{d} I/\mathrm{d} A \,\! Wb s−1 m−2

A m−1 s−1

[M][T]−3 [I]−1 (Wb)

[I][L]−1[T]−1 (Am)

Magnetic fields

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Magnetic field, field strength, flux density, induction field B \mathbf{F} =q_e \left ( \mathbf{v}\times\mathbf{B} \right ) \,\! T = N A−1 m−1 = Wb m2 [M][T]−2[I]−1
Magnetic potential, EM vector potential A  \mathbf{B} = \nabla \times \mathbf{A} T m = N A−1 = Wb m3 [M][L][T]−2[I]−1
Magnetic flux ΦB \Phi_B = \int_S \mathbf{B} \cdot \mathrm{d} \mathbf{A}\,\! Wb = T m−2 [L]2[M][T]−2[I]−1
Magnetic field intensity, (AKA field strength) H \mathbf{H} = \mathbf{B}/\mu \,\! A m−1 [I] [L]−1
Magnetic moment, magnetic dipole moment m, μ, Π

Two definitions are possible:

using pole strengths,
\mathbf{m} = q_m \mathbf{a}\,\!

using currents:
\mathbf{m} = NIA\mathbf{\hat{n}}\,\!

a = pole separation N is the number of turns of conductor

A m2 [I][L]2
Magnetization M \mathbf{M} = \mathrm{d} \langle \mathbf{m} \rangle /\mathrm{d} V \,\! A m2 [I] [L]−1
Intensity of magnetization, magnetic polarization I, J \mathbf{I} = \mu \mathbf{M} \,\! T = N A−1 m−1 = Wb m2 [M][T]−2[I]−1
Self Inductance L Two equivalent definitions are possible:

L=N\left ( \mathrm{d} \Phi/\mathrm{d} I \right )\,\! L\left ( \mathrm{d} I/\mathrm{d} t \right )=-NV\,\!

H = Wb A−1 [L]2 [M] [T]−2 [I]−2
Mutual inductance M Again two equivalent definitions are possible:

M_1=N\left ( \mathrm{d} \Phi_2/\mathrm{d} I_1 \right )\,\! M\left ( \mathrm{d} I_2/\mathrm{d} t \right )=-NV_1\,\!

1,2 subscripts refer to two conductors/inductors mutually inducing voltage/ linking magnetic flux though each other. They can be interchanged for the required conductor/inductor;

M_2=N\left ( \mathrm{d} \Phi_1/\mathrm{d} I_2 \right )\,\!
M\left ( \mathrm{d} I_1/\mathrm{d} t \right )=-NV_2\,\!

H = Wb A−1 [L]2 [M] [T]−2 [I]−2
Gyromagnetic ratio (for charged particles in a magnetic field) γ \omega = \gamma B \,\! Hz T−1 [M]−1[T][I]

Electric circuits

DC circuits, general definitions

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Terminal Voltage for

Power Supply

Vter V = J C−1 [M] [L]2 [T]−3 [I]−1
Load Voltage for Circuit Vload V = J C−1 [M] [L]2 [T]−3 [I]−1
Internal resistance of power supply Rint  R_\mathrm{int} = V_\mathrm{ter}/I \,\! Ω = V A−1 = J s C−2 [M][L]2 [T]−3 [I]−2
Load resistance of circuit Rext  R_\mathrm{ext} = V_\mathrm{load}/I \,\! Ω = V A−1 = J s C−2 [M][L]2 [T]−3 [I]−2
Electromotive force (emf), voltage across entire circuit including power supply, external components and conductors E \mathcal{E} = V_\mathrm{ter} + V_\mathrm{load} \,\! V = J C−1 [M] [L]2 [T]−3 [I]−1

AC circuits

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Resistive load voltage VR  V_R = I_R R \,\! V = J C−1 [M] [L]2 [T]−3 [I]−1
Capacitive load coltage VC  V_C = I_C X_C\,\! V = J C−1 [M] [L]2 [T]−3 [I]−1
Inductive load coltage VL V_L = I_L X_L\,\! V = J C−1 [M] [L]2 [T]−3 [I]−1
Capacitive reactance XC X_C = \frac{1}{\omega_\mathrm{d} C} \,\! Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2
Inductive reactance XL  X_L = \omega_d L \,\! Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2
AC electrical impedance Z V = I Z\,\!

Z = \sqrt{R^2 + \left ( X_L - X_C \right )^2 } \,\!

Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2
Phase constant δ, φ \tan\phi= \frac{X_L - X_C}{R}\,\! dimensionless dimensionless
AC peak current I0 I_0 = I_\mathrm{rms} \sqrt{2}\,\! A [I]
AC root mean square current Irms  I_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} \left [ I \left ( t \right ) \right ]^2 \mathrm{d} t}  \,\! A [I]
AC peak voltage V0  V_0 = V_\mathrm{rms} \sqrt{2} \,\! V = J C−1 [M] [L]2 [T]−3 [I]−1
AC root mean square voltage Vrms  V_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} \left [ V \left ( t \right ) \right ]^2 \mathrm{d} t}  \,\! V = J C−1 [M] [L]2 [T]−3 [I]−1
AC emf, root mean square \mathcal{E}_\mathrm{rms}, \sqrt{\langle \mathcal{E} \rangle} \,\! \mathcal{E}_\mathrm{rms}=\mathcal{E}_\mathrm{m}/\sqrt{2}\,\! V = J C−1 [M] [L]2 [T]−3 [I]−1
AC average power  \langle P \rangle \,\!  \langle P \rangle =\mathcal{E}I_\mathrm{rms}\cos\phi\,\! W = J s−1 [M] [L]2 [T]−3
Capacitive time constant τC \tau_C = RC\,\! s [T]
Inductive time constant τL \tau_L = L/R\,\! s [T]

Magnetic circuits

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Magnetomotive force, mmf F, \mathcal{F}, \mathcal{M} \mathcal{M} = NI

N = number of turns of conductor

A [I]

Photonics

Geometric optics (luminal rays)

General fundamental quantities

Quantity (common name/s) (Common) symbol/s SI units Dimension
Object distance x, s, d, u, x1, s1, d1, u1 m [L]
Image distance x', s', d', v, x2, s2, d2, v2 m [L]
Object height y, h, y1, h1 m [L]
Image height y', h', H, y2, h2, H2 m [L]
Angle subtended by object θ, θo, θ1 rad dimensionless
Angle subtended by image θ', θi, θ2 rad dimensionless
Curvature radius of lens/mirror r, R m [L]
Focal length f m [L]
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Lens power P P = 1/f \,\! m−1 = D (dioptre) [L]−1
Lateral magnification m m = - x_2/x_1 = y_2/y_1 \,\! dimensionless dimensionless
Angular magnification m m = \theta_2/\theta_1 \,\! dimensionless dimensionless

Physical optics (EM luminal waves)

There are different forms of the Poynting vector, the most common are in terms of the E and B or E and H fields.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Poynting vector S, N \mathbf{N} = \frac{1}{\mu_0}\mathbf{E}\times\mathbf{B} = \mathbf{E}\times\mathbf{H} \,\! W m−2 [M][T]−3
Poynting flux, EM field power flow ΦS, ΦN  \Phi_N = \int_S \mathbf{N} \cdot \mathrm{d}\mathbf{S} \,\! W [M][L]2[T]−3
RMS Electric field of Light Erms E_\mathrm{rms} = \sqrt{\langle E^2 \rangle} = E/\sqrt{2}\,\! N C−1 = V m−1 [M][L][T]−3[I]−1
Radiation momentum p, pEM, pr  p_{EM} = U/c\,\! J s m−1 [M][L][T]−1
Radiation pressure Pr, pr, PEM P_{EM} = I/c = p_{EM}/At \,\! W m−2 [M][T]−3

Radiometry

Visulization of flux through differential area and solid angle. As always  \mathbf{\hat{n}} \,\! is the unit normal to the incidant surface A,  \mathrm{d} \mathbf{A} = \mathbf{\hat{n}}\mathrm{d}A \,\!, and  \mathbf{\hat{e}}_{\angle} \,\! is a unit vector in the direction of incident flux on the area element, θ is the angle between them. The factor  \mathbf{\hat{n}} \cdot \mathbf{\hat{e}}_{\angle} \mathrm{d}A = \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} = \cos \theta \mathrm{d}A \,\! arises when the flux is not normal to the surface element, so the area normal to the flux is reduced.


For spectral quantities two definitions are in use to refer to the same quantity, in terms of frequency or wavelength.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Radiant energy Q, E, Qe, Ee J [M][L]2[T]−2
Radiant exposure He  H_e = \mathrm{d} Q/\left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \,\! J m−2 [M][T]−3
Radiant energy density ωe  \omega_e = \mathrm{d} Q/\mathrm{d}V \,\! J m−3 [M][L]−3
Radiant flux, radiant power Φ, Φe  \Phi = \mathrm{d} Q/\mathrm{d} t \,\! W [M][L]2[T]−3
Radiant intensity I, Ie  I = \mathrm{d} \Phi/\mathrm{d} \Omega \,\! W sr−1 [M][L]2[T]−3
Radiance, intensity L, Le  L = \mathrm{d}^2 \Phi/\left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \mathrm{d} \Omega \,\! W sr−1 m−2 [M][T]−3
Irradiance E, I, Ee, Ie  E = \mathrm{d} \Phi/\left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \,\! W m−2 [M][T]−3
Radiant exitance, radiant emittance M, Me  M = \mathrm{d} \Phi/\left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \,\! W m−2 [M][T]−3
Radiosity J, Jν, Je, J  J = E + M \,\! W m−2 [M][T]−3
Spectral radiant flux, spectral radiant power Φλ, Φν, Φ, Φ  \Phi_\lambda = \frac{\mathrm{d}^2 Q}{\mathrm{d} \lambda \mathrm{d} t} , \quad \Phi_\nu = \frac{\mathrm{d}^2 Q}{\mathrm{d} \nu \mathrm{d} t } \,\! W m−1 (Φλ)
W Hz−1 = J (Φν)
[M][L]−3[T]−3 (Φλ)
[M][L]−2[T]−2 (Φν)
Spectral radiant intensity Iλ, Iν, I, I  I_\lambda = \frac{\mathrm{d}^2 \Phi}{\mathrm{d} \lambda \mathrm{d} \Omega} , \quad I_\nu = \frac{\mathrm{d}^2 \Phi}{\mathrm{d} \nu \mathrm{d} \Omega} \,\! W sr−1 m−1 (Iλ)
W sr−1 Hz−1 (Iν)
[M][L]−3[T]−3 (Iλ)
[M][L]2[T]−2 (Iν)
Spectral radiance Lλ, Lν, L, L  L_\lambda = \frac{\mathrm{d}^3 \Phi}{\mathrm{d} \lambda \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \mathrm{d} \Omega} , \quad L_\nu = \frac{\mathrm{d}^3 \Phi}{\mathrm{d} \nu \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) \mathrm{d} \Omega} \,\! W sr−1 m−3 (Lλ)
W sr−1 m−2 Hz−1 (Lν)
[M][L]−1[T]−3 (Lλ)
[M][L]−2[T]−2 (Lν)
Spectral irradiance Eλ, Eν, E, E  E_\lambda = \frac{\mathrm{d}^2 \Phi}{\mathrm{d} \lambda \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right ) } , \quad E_\nu = \frac{\mathrm{d}^2 \Phi}{\mathrm{d} \nu \left ( \mathbf{\hat{e}}_{\angle} \cdot \mathrm{d}\mathbf{A} \right )} \,\! W m−3 (Eλ)
W m−2 Hz−1 (Eν)
[M][L]−1[T]−3 (Eλ)
[M][L]−2[T]−2 (Eν)

Atomic/nuclear physics

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Number of atoms N = Number of atoms remaining at time t

N0 = Initial number of atoms at time t = 0
ND = Number of atoms decayed at time t

 N_0 = N + N_D \,\! dimensionless dimensionless
Decay rate, activity of a radioisotope A  A = \mathrm{d} N /\mathrm{d} t \,\! Bq = Hz = s−1 [T]−1
Decay constant λ  \lambda = A/N \,\! Bq = Hz = s−1 [T]−1
Half-life of a radioisotope t1/2, T1/2 Time taken for half the number of atoms present to decay

 t \rightarrow t + T_{1/2} \,\!
 N \rightarrow N / 2 \,\!

s [T]
Number of half-lives n (no standard symbol)  n = t / T_{1/2} \,\! dimensionless dimensionless
Radioisotope time constant, mean lifetime of an atom before decay τ (no standard symbol)  \tau = 1 / \lambda \,\! s [T]
Absorbed dose, total ionizing dose (total energy of radiation transferred to unit mass) D can only be found experimentally N/A Sv = J kg−1 (Sievert) [L]2[T]−2
Equivalent dose H  H = DQ \,\!

Q = radiation quality factor (dimensionless)

Sv = J kg−1 (Sievert) [L]2[T]−2
Effective dose E  E = \sum_j H_jW_j \,\!

Wj = weighting factors corresponding to radiosensitivities of matter (dimensionless)

 \sum_j W_j = 1 \,\!

Sv = J kg−1 (Sievert) [L]2[T]−2

Quantum mechanics

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Wave function ψ, Ψ To solve from the Schrödinger equation varies with situation and number of particles
Threshold frequency f0, ν0 Can only be found by experiment/calculated from work function below. Hz = s−1 [T]−1
Work function φ, Φ \phi = hf_0\,\! J [M][L]2[T]−2
Wavefunction probability density ρ \rho = \left | \Psi \right |^2 = \Psi^* \Psi m−3 [L]−3
Wavefunction probability current j Non-relativistic, no external field:

\mathbf{j} = \frac{-i\hbar}{2m}\left(\Psi^* \nabla \Psi - \Psi \nabla \Psi^*\right)  = \frac\hbar m \mathrm{Im}(\Psi^*\nabla\Psi)=\mathrm{Re}(\Psi^* \frac{\hbar}{im} \nabla \Psi)

star * is complex conjugate

m−2 s−1 [T]−1 [L]−2

Astrophysics

In astrophysics, L is used for luminosity (energy per unit time, equivalent to power) and F is used for energy flux (energy per unit time per unit area, equivalent to intensity in terms of area, not solid angle). They are not new quantities, simply different names.

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Comoving transverse distance DM pc (parsecs) [L]
Luminosity distance DL D_L = \sqrt{\frac{L}{4\pi F}} \, pc (parsecs) [L]
Apparent magnitude in band j (UV, visible and IR parts of EM spectrum) (Bolometric) m m_j= -\frac{5}{2} \log_{10} \left | \frac {F_j}{F_j^0} \right | \, dimensionless dimensionless
Absolute magnitude

(Bolometric)

M  M = m - 5 \left [ \left ( \log_{10}{D_L} \right ) - 1 \right ]\!\, dimensionless dimensionless
Distance modulus μ  \mu = m - M \!\, dimensionless dimensionless
Colour indices (No standard symbols)  U-B = M_U - M_B\!\,

 B-V = M_B - M_V\!\,

dimensionless dimensionless
Bolometric correction Cbol (No standard symbol)  \begin{align} C_\mathrm{bol} & = m_\mathrm{bol} - V \\
& = M_\mathrm{bol} - M_V 
\end{align} \!\, dimensionless dimensionless

See also

List of physics formulae
Defining equation (physical chemistry)

Footnotes

  1. ^ Warlimont, pp 12–13
  2. ^ Vector Analysis (2nd Edition), Schaum's Outlines Series, M. R. Spiegel, S. Lipschutz, D. Spellman, Mc Graw Hill, ISBN 978 0 07 161545 7
  3. ^ http://www.ltcconline.net/greenl/courses/202/multipleIntegration/MassMoments.htm, Section: Moments and center of mass
  4. ^ Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, ISBN 0-691-03323-4

References

  • Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978 0 470 01460 8
  • Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, 2nd Edition, 1978, John Murray, ISBN 0 7195 3382 1
  • Encyclopaedia of Physics, R.G. Lerner, G.L. Trigg, 2nd Edition, VHC Publishers, Hans Warlimont, Springer, 2005, pp 12–13
  • Physics for Scientists and Engineers: With Modern Physics (6th Edition), P.A. Tipler, G. Mosca, W.H. Freeman and Co, 2008, 9-781429-202657

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