Defining equation (physics)

Defining equations in physics are equations which define new quantities in terms of fundamental ones. Quantities follow the same heirachy as the chosen system of units; fundamental quantities have defined base units, from which other quantities May Be derived. This article uses the current SI system of units, not natural or characteristic units.

For the common nomenclature for fundamental base quantities used in this article see: Physical quantity.

Classical Mechanics

Mass and Inertia

Main Articles: Inertia, Mass

Mass can be considered to be inertial or gravitational.

Inertial mass is associated with a body's inertia to maintain uniform momentum, while acted on by zero resultant force. Similarly inertial mass corresponds to the moment of inertia of a body to maintain uniform angular momentum, while acted on by zero resultant torque AbOUT any axis through any spatial point.

Gravitational mass is associated with the cause of gravitational attraction.

Inertial and gravitational mass are found to be equal by Newton's 3rd Law, later by Einstein's Principle of Equivalence (for Acceleration).

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|M_inertiaa = M_gravityg |}

Masses occur as discrete or continuous distributions. "Discrete mass" and "continuum mass" are not different quantities; the only differance is the calculation either as summation (discrete) or integration (continuous). Centre of mass is not to be confused with centre of gravity (see Center of mass).

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Mass density of dimension n (V_n = n-space) n = 1 for linear mass density, n = 2 for surface mass density, n = 3 for volume mass density, etc	linear mass density λ , surface mass density σ , volume mass density ρ , no general symbol for any dimension	n-space mass density: $\rho_n = \frac{ \partial ^n m}{ \partial x_n \cdots \partial x_2 \partial x_1} = \frac{\partial m}{ \partial V_n} \,\!$ special cases are: $\lambda = \frac{ \partial m}{ \partial x} \,\!$ $\sigma = \frac{ \partial^2 m}{ \partial x_2 \partial x_1} = \frac{ \partial^2 m}{ \partial S} \,\!$ $\rho = \frac{ \partial^3 m}{ \partial x_3 \partial x_2 \partial x_1} = \frac{ \partial m}{ \partial V} \,\!$	kg m^-n	[M][L]^-n
Total discrete mass	m, M	M = ∑_im_i	kg m	[M][L]
Total continuum mass	m, M	n-space mass density M = ∫ρ_ndⁿx = ∫⋯∫∫ρ_ndx₁dx₂⋯dx_n Special cases are: M = ∫λdx M = ∫σdA = ∬σdx₁dx₂ M = ∫ρdV = ∭ρdx₁dx₂dx₃	kg	[M]
Moment of Mass	m (No common symbol)	Point mass: m = rm 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 : m = ∫ρ(r)x_idr	kg m	[M][L]
Centre of Mass	r_com (Symbols vary enormously)	i^th moment of mass m_i = r_im_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: $\mathbf{r}_\mathrm{com} = \frac{1}{M}\int \mathbf{r} \mathrm{d}^n m = \frac{1}{M}\int \mathbf{r} \rho_n \mathrm{d}^n x = \frac{1}{M}\int \mathbf{r} \mathrm{d}\mathbf{m} \,\!$	m	[L]
2-Body Reduced Mass	m₁₂, μ Pair of masses = m₁ and m₂	μ = (m₁m₂)/(m₁+m₂)	kg	[M][L]²
Moment of Inertia (MOI)	I	Discrete Masses: I = ∑_im_i ⋅ r_i = ∑_i\|r_i\|²m Mass continuum: I = ∫\|r\|²dm = ∫r ⋅ dm = ∫\|r\|²ρ_ndⁿx	kg m² s⁻¹	[M][L]²

Derived Kinematic Quantities

Main Articles: Kinematics, Cross product

For the inclusion of the scalar angle of rotational position θ, it is necessary to include a normal vector n̂ to the plane containing and defined by the position vector and tangential direction of rotation, so that the vector equations hold.

Using the basis vectors for polar coordinates, which are r̂, θ̂, ϕ̂ , the unit normal is n̂ = r̂ × θ̂ .

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Velocity	v	v = dr/dt	m s⁻¹	[L][T]⁻¹
Acceleration	a	$\mathbf{a} = \frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t } = \mathrm{d}^2 \mathbf{v}/\mathrm{d} t^2 \,\!$	m s⁻²	[L][T]⁻²
Jerk	j	j = da/dt = d³v/dt³	m s⁻³	[L][T]⁻³
Angular Velocity	ω	ω = n̂(dθ/dt)	rad s⁻¹	[T]⁻¹
Angular Acceleration	α	α = dω/dt = n̂(d²θ/dt²)	rad s⁻²	[L][T]⁻²

Derived Dynamic Quantities

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Momentum	p	p = mv	kg m s⁻¹	[M][L][T]⁻¹
Force	F	$\mathbf{F} = \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t } \,\!$	N = kg m s⁻²	[M][L][T]⁻²
Impulse	Δp, I	I = Δp = ∫_t₁^t₂Fdt	kg m s⁻¹	[M][L][T]⁻¹
Angular Momentum about a position point R	L, J, S	L = (r−R) × p	kg m² s⁻¹	[M][L]²[T]⁻¹
Total, Spin and Orbital Angular Momentum	L, J, S	L_total = L_spin + L_orbital	kg m² s⁻¹	[M][L]²[T]⁻¹
Moment of a Force about a position point R, Torque	τ, M	$\boldsymbol{\tau} = \left ( \mathbf{r} - \mathbf{R} \right ) \times \mathbf{F} = \frac{\mathrm{d} \mathbf{L}}{\mathrm{d} t } \,\!$	N m = kg m² s⁻²	[M][L]²[T]⁻²
Angular Impulse	ΔL No common symbol	ΔL = ∫_t₁^t₂Ldt	kg m² s⁻¹	[M][L]²[T]⁻¹
Coefficient of Restitution	e, ε	$e = \frac{\left \| \mathbf{v} \right \| _\mathrm{separation}}{\left \| \mathbf{v} \right \| _\mathrm{approach}} \,\!$ usually 0 ≤ e ≤ 1 but it is possible that e ≥ 1	Dimensionless	Dimensionless

General Energy Definitions

Main Article: Mechanical energy

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Mechanical Work due to a Resultant Force	W	W = ∫_CF ⋅ dr	J = N m = kg m² s⁻²	[M][L]²[T]⁻²
Work done ON mechanical system, Work done BY	W_ON, W_BY	ΔW_ON = − ΔW_BY	J = N m = kg m² s⁻²	[M][L]²[T]⁻²
Potential Energy	φ, Φ, U, V, E_p	ΔW = − ΔV	J = N m = kg m² s⁻²	[M][L]²[T]⁻²
Mechanical Power	P	$P = \frac{\mathrm{d}E}{\mathrm{d}t} \,\!$	W = J s⁻¹	[M][L]²[T]⁻³
Lagrangian	L	L = T − V	J	[M][L]²[T]⁻²
Action	S	S = ∫Ldt	J s	[M][L]²[T]⁻¹

Potential Energy and Work

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 n̂ is a unit vector normal to the cross-section surface at the cross section considered.

Quantity (Common Name/s)	'''(Common) Symbol/s'''	Defining Equation	SI Units	Dimension
Flow Velocity Vector Field	u	u = u(x,t)	m s⁻¹	[L][T]⁻¹
Mass Current, Mass flow rate	I_m	I_m = ∂m/∂t	kg s⁻¹	[M][T]⁻¹
Mass Current Density	j_m	j_m = n̂(∂I_m/∂A) = n̂(∂²m/∂A∂t)	kg m⁻² s⁻¹	[M][L]⁻²[T]⁻¹
Momentum Current	I_p	I_p = ∂\|p\|/∂t	kg m s⁻²	[M][L][T]⁻²
Momentum Current Density	j_p	j_p = n̂(∂I_p/∂A) = ∂²p/∂A∂t	kg m s⁻²	[M][L][T]⁻²

Properties of Matter

Stress and strain

Quantity (Common Name/s)	(Common) Symbol/s	Definining Equation	SI Units	Dimension
General Stress	σ	σ = F/A F may be any force applied to area A	Pa = N m⁻²	[M] [T] [L]⁻¹
General Strain	ϵ	ϵ = ΔD/D D = dimension (length, area, volume) ΔD = change in dimension	dimensionless	dimensionless
General Elastic Modulus	E_mod	E_mod = σ/ϵ	Pa = N m⁻²	[M] [T] [L]⁻¹
Yield Strength/
Ultimate Tensile Strength
Young's Modulus	E, Y	$Y = \frac{F L}{A \Delta L} \,\!$	Pa = N m⁻²	[M] [T] [L]⁻¹
Shear Modulus	G	G = Δx/L	Pa = N m⁻²	[M] [T] [L]⁻¹
Bulk Modulus	B	$B = \frac{P}{\Delta V / V}\,\!$	Pa = N m⁻²	[M] [T] [L]⁻¹

Thermodynamics

Thermodynamic Quantities

State Properties for Systems

Main articles: List of thermodynamic properties, Thermodynamic Potential, Free entropy

Quantity (Common Name/s)	(Common Symbol/s)	Defining Equation	SI Units	Dimension
Number of Molecules	N		dimensionless	dimensionless
Temperature	T		K	[Θ]
Heat Energy	Q, q		J	[M][L]²[T]⁻²
Latent Heat	Q_L		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
Internal Energy Sum of all total energies which constitute the system	U	U = ∑_iE_i	J	[M][L]²[T]⁻²
Enthalpy	H	H = U + pV	J	[M][L]²[T]⁻²
Partition Function	Z		dimensionless	dimensionless
Inverse Temperature	β	β = 1/k_BT	J^-1	[T]²[M]^-1[L]^-2
Gibbs Free Energy	G	G = H − TS	J	[M][L]²[T]⁻²
Chemical Potential (of component i in a mixture)	μ_i	μ_i = (∂U/∂N_i)_{N_i ≠ j, S, V} (N_i, S, V must all be constant)	J	[M][L]²[T]⁻²
Electrochemical Potential (of component i in a mixture)	μ̄_i	μ̄_i = μ − zeN_Aϕ φ = local electrostatic potential (see below also) z_i = valency (charge) of the ion i	J	[M][L]²[T]⁻²
Helmholtz Free Energy	A, F	A = U − TS	J	[M][L]²[T]⁻²
Landeu Potential, Landau Free Energy	Ω	Ω = U − TS − μN	J	[M][L]²[T]⁻²
Grand Potential	Φ_G	Φ_G = E − TS − μN	J	[M][L]²[T]⁻²
Massieu Potential, Helmholtz free entropy	Φ	Φ = S − U/T	J K⁻¹	[M][L]²[T]⁻² [Θ]⁻¹
Planck potential, Gibbs free entropy	Ξ	Ξ = Φ − pV/T	J K⁻¹	[M][L]²[T]⁻² [Θ]⁻¹

Thermal Properties of Matter

Main Articles: Heat capacity, Thermal expansion,

Quantity (Common Name/s)	(Common Symbol/s)	Defining Equation	SI Units	Dimension
General Heat Capacity	C	C = ∂Q/∂T	J K ⁻¹	[M][L]²[T]⁻² [Θ]⁻¹
Heat Capacity (isobaric)	C_p	C_p = ∂Q/∂T	J K ⁻¹	[M][L]²[T]⁻² [Θ]⁻¹
Specific Heat Capacity (isobaric)	C_mp	C_mp = ∂²Q/∂m∂T	J kg⁻¹ K⁻¹	[L]²[T]⁻² [Θ]⁻¹
Molar Specific Heat Capacity (isobaric)	C_np	C_np = ∂²Q/∂n∂T	J K ⁻¹ mol⁻¹	[M][L]²[T]⁻² [Θ]⁻¹ [N]⁻¹
Heat Capacity (isochoric)	C_V	C_V = ∂Q/∂T	J K ⁻¹	[M][L]²[T]⁻² [Θ]⁻¹
Specific Heat Capacity (isochoric)	C_mV	C_mV = ∂²Q/∂m∂T	J kg⁻¹ K⁻¹	[L]²[T]⁻² [Θ]⁻¹
Molar Specific Heat Capacity (isochoric)	C_nV	C_nV = ∂²Q/∂n∂T	J K ⁻¹ mol⁻¹	[M][L]²[T]⁻² [Θ]⁻¹ [N]⁻¹
Specific Latent Heat	L	L = ∂Q/∂m	J kg⁻¹	[L]²[T]⁻²
Ratio of Isobaric to Isochoric Heat Capacity, Adiabatic Index	γ	γ = C_p/C_V = c_p/c_V = C_mp/C_mV	dimensionless	dimensionless
Linear Coefficient of Thermal Expansion	α	∂L/∂t = αL	K⁻¹	[Θ]⁻¹
Volume Coefficient of Thermal Expansion	3α, γ	∂V/∂t = 3αV	K⁻¹	[Θ]⁻¹

Thermal Transfer

Main articles: Thermal conductivity

Quantity (Common Name/s)	(Common Symbol/s)	Defining Equation	SI Units	Dimension
Temperature Gradient	T (No standard symbol)	T = ∇T	K m⁻¹	[Θ][L]⁻¹
Thermal Conduction Rate, Thermal Current, Thermal/Heat Flux	P	P = ∂q/∂t	W = J s⁻¹	[M] [L]² [T]⁻²
Thermal Intensity	I	I = ∂P/∂A = ∂²q/∂A∂t	W m⁻²	[M] [T]⁻³
Thermal/Heat Flux Density (vector analogue of Thermal Intensity above)	q	q = IT̂ = T̂∂²q/∂A∂t	W m⁻²	[M] [T]⁻³
Thermal Conductance	U	U = λ/δx	W m⁻² K⁻¹	[M] [T]⁻³ [Θ]
Thermal Conductivity	κ, K, λ	λ = − P/(A⋅∇T)	W m⁻¹ K⁻¹	[M] [L] [T]⁻² [Θ]⁻¹
Thermal Resistance	R	R = 1/U = Δx/λ	m² K W⁻¹	[L] [T]² [Θ]¹ [M]⁻¹
Emissivity Coefficient	ε	Can only be found from experiment 0 ≤ ϵ ≤ 1 ϵ = 0 for perfect reflector ϵ = 1 for perfect absorber (true black body)	dimensionless	dimensionless

Waves

General Fundamental Quantities

Main articles: Transverse wave, Longitudinal wave

Quantity (common name/s)	(Common) symbol/s	SI units	Dimension
Number of wave cycles	N	dimensionless	dimensionless
(Transverse) Displacement	y, r_⊥	m	[L]
(Transverse) Displacement amplitude	A, B, C, x₀, x_m	m	[L]
(Transverse) Velocity amplitude	V, v₀ , v_m	m s⁻¹	[L][T]⁻¹
(Transverse) Acceleration Amplitude	A, a₀, a_m	m s⁻²	[L][T]⁻²
(Longnitudinal) Displacement	x, r_∥	m	[L]
Period	T	s	[T]
Wavelength	λ	m	[L]
Phase angle	δ, ε, φ	rad	dimensionless

General Derived Quantities

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
(Transverse) Velocity	v_⊥ , v_t	v_⊥ = ∂r_⊥/∂t	m s⁻¹	[L][T]⁻¹
(Transverse) Acceleration	a_⊥ , a_t	v_⊥ = ∂v_⊥/∂t = ∂²r_⊥/∂t²	m s⁻²	[L][T]⁻²
Path Length Difference	L, ΔL, Δx, Δr_∥	Δr_∥ = (r_∥)₂ − (r_∥)₁	m	[L]
(Longnitudinal) Velocity	v_∥, v_p	v_∥ = Δr_∥/Δt	m s⁻¹	[L][T]⁻¹
Frequency	f	General definition: f = ΔN/Δt Commonly N is set to 1 cycle, seting t = T = time period for 1 cycle gives the more useful definition: f = 1/T	Hz = s⁻¹	[T]⁻¹
Angular Frequency/ Pulsatance	ω	ω = 2πf = 2π/T	Hz = s⁻¹	[T]⁻¹
Time Delay, Time Lag/Lead	Δt	Δt = t₂ − t₁	s	[T]
Scalar Wavenumber	k	Two definitions are used: k = 2π/λ k = 1/λ In the formalism which follows, only the first definition is used.	m⁻¹	[L]⁻¹
Vector Wavenumber, k-vector, Wave vector	k	Again two definitions are possible: k = r̂_∥(2π/λ) k = r̂_∥(1/λ) In the formalism which follows, only the first definition is used.	m⁻¹	[L]⁻¹
Phase Differance	δ, Δε, Δϕ	Δϕ = ϕ₂ − ϕ₁	rad	dimensionless
Phase	Φ (No standard symbol)	Φ = r − vt + Δr/N = λ Φ = kr − ωt + ϕ = 2πN Physically; wave popagation in +r direction r > 0 ⇒ ω < 0 wave popagation in -r direction r < 0 ⇒ ω > 0 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 Φ:

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

Damping Parameters, Forces and Torques

Main articles: Damping

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Spring Constant (Hooke's Law)	k, K, κ, k_H	k_H = ΔF_⊥/Δx_⊥	N m⁻¹	[M][T]⁻²
Damping Coefficient	b	b = − ΔF_d/Δv_⊥	N s m⁻¹	[L][T]⁻¹
Damping Force	F_d		N	[M][L][T]⁻²
Damping Ratio	ζ	ζ = b/b_critical	dimensionless	dimensionless
Logarithmic decrement	δ	δ = (1/n)ln \|A₀/A_n\| A₀ is any amplitude, A_n is the amplitude n successive peaks later from A₀ , where A₀ > A_n	dimensionless	dimensionless
Torsion Constant	κ	κ = − Δτ/Δθ	N m rad⁻¹	[M][L]²[T]⁻²
Damping Torque	τ_d		N m	[M][L]²[T]⁻²
Rotational Damping Coefficient	β	β = − τ_d/(dθ/dt)	N m s rad⁻¹	[M][L]²[T]⁻¹

Acoustics

Quantity (Common Name/s)	(Common) Symbol/s	Definining Equation	SI Units	Dimension
Linear Mass Density	μ	$\mu = \frac{\partial m}{\partial t}\,\!$	kg m^-1	[M] [L]^-1
Sound Level	β	$\beta = \left ( 10 \mathrm{dB} \right ) \log \left \| \frac{I}{I_0} \right \| \,\!$	dimensionless	dimensionless

Gravitation

A common misconseption occurs between centre of mass and centre of gravity. They are defined in simalar 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.

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

Quantity Name	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Centre of Gravity	r_cog (Symbols can vary enourmously)	i^th moment of mass m_i = r_im_i Centre of gravity for a descrete masses $\mathbf{r}_\mathrm{cog} = \frac{1}{M \left \| \mathbf{g} \left ( \mathbf{r}_i \right ) \right \|}\sum_i \mathbf{m}_i \left \| \mathbf{g} \left ( \mathbf{r}_i \right ) \right \| \,\!$ $= \frac{1}{M \left \| \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right \|}\sum_i \mathbf{r}_i m_i \left \| \mathbf{g} \left ( \mathbf{r}_i \right ) \right \| \,\!$ Centre of a gravity for a continuum of mass $\mathbf{r}_\mathrm{cog} = \frac{1}{M \left \| \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right \|}\int \left \| \mathbf{g} \left ( \mathbf{r} \right ) \right \|\mathrm{d}\mathbf{m} \,\!$ $= \frac{1}{M \left \| \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right \|}\int \mathbf{r} \left \| \mathbf{g} \left ( \mathbf{r} \right ) \right \| \mathrm{d}^n m \,\!$ $= \frac{1}{M \left \| \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right \|}\int \mathbf{r} \rho_n \left \| \mathbf{g} \left ( \mathbf{r} \right ) \right \| \mathrm{d}^n x \,\!$	m	[L]
Standard GravitationParameter of a Mass	μ	μ = Gm	N m² kg⁻¹	[L]³ [T]⁻²
Gravitational Field, Field Strength, Potential Gradient, Acceleration	g	$\mathbf{g} = \frac{\mathbf{F}}{m} \,\!$	N kg⁻¹ = m s⁻²	[L][T]⁻²
Gravitational Flux	Φ_G	Φ_G = ∫_Sg ⋅ dA	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 Differance	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	E_p	E_p = − W_∞r	J	[M][L]²[T]⁻²
Gravitational Torsion Field	Ω	$\nabla \times \mathbf{g} = \frac{\partial \boldsymbol{\Omega}}{\partial t} \,\!$ F = m(v×Ω)	Hz = s⁻¹	[T]⁻¹
Gravitational Torsion Flux	Φ_Ω	Φ_Ω = ∫_SΩ ⋅ dA	N m s kg⁻¹ = m² s⁻¹	[M]² [T]⁻¹
Gravitomagnetic Field	ξ		Hz = s⁻¹	[T]⁻¹
Gravitomagnetic Flux	Φ_ξ	Φ_ξ = ∫_Sξ ⋅ dA	N m s kg⁻¹ = m² s⁻¹	[M]² [T]⁻¹
Gravitomagnetic Vector Potential	h	ξ = ∇ × h	m s⁻¹	[M] [T]⁻¹

Electromagnetism

Electric Quantities

Charge

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Electric Charge (any amount)	q, Q		C = A s	[I][T]
Electric charge density of dimension n (V_n = n-space) n = 1 for linear mass density, n = 2 for surface mass density, n = 3 for volume mass density, etc	linear charge density λ, surface charge density σ, volume charge density ρ, no general symbol for any dimension	n-space charge density: ρ_n = ∂ⁿm/∂x_n⋯∂x₂∂x₁ = ∂m/∂V_n special cases are: λ = ∂m/∂x σ = ∂²m/∂x₂∂x₁ = ∂²m/∂S ρ = ∂³m/∂x₃∂x₂∂x₁ = ∂m/∂V	C m^-n	[I][T][L]^-n
Total descrete charge	Q	Q = ∑_iq_i	C	[I][T]
Total continuum charge	Q	n-space charge density Q = ∫ρ_ndⁿx = ∫⋯∫∫ρ_ndx₁dx₂⋯dx_n special cases are: Q = ∫λdx Q = ∫σdA = ∬σdx₁dx₂ Q = ∫ρdV = ∭ρdx₁dx₂dx₃	C	[I][T]
Capacitance	C	C = ∂q/∂V	F = C V⁻¹
Electric Current	I	I = dq/dt	A	[I]
Current Density	J	J = n̂(∂I/∂A)	A m⁻²	[I][L]⁻²
Displacement current	I_d	I_d = ϵ₀(∂Φ_E/∂t)	A	[I]
Charge Carrier Drift Speed	v_d		m s⁻¹	[L][T]⁻¹

Electric Fields

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Electric Field, Field Strength, Flux Density, Potential Gradient	E	E = F/q	N C⁻¹ = V m⁻¹	[M][L][T]⁻³[I]⁻¹
Electric Flux	Φ_E	Φ_E = ∫_SE ⋅ dA	N m² C⁻¹	[M][L]³[T]⁻³[I]⁻¹
Relative Permittivity (AKA Dielectric constant)	ε_r		F m⁻¹	[I]² [T]⁴ [M]⁻¹ [L]⁻³).
Absolute Electric Permittivity;	ε	ϵ = ϵ_rϵ₀	F m⁻¹	[I]² [T]⁴ [M]⁻¹ [L]⁻³).
Electric Displacement Field	D	D = E/ϵ	C m⁻²	[I][T][L]⁻²
Electric Displacement Flux	Φ_D	Φ_D = ∫_SD ⋅ dA	C	[I][T]
Electric Dipole Moment	p	p = 2qa a is the charge separation directed from -ve to +ve charge	C m	[I][T][L]
Electric Polarization	P	P = ∂⟨p⟩/∂V	C m⁻²	[I][T][L]⁻²
Absolute Electric Potential relative to point r₀ Theoretical: r₀ = ∞ Practical: R₀ = R_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
Electric Potential Difference	Δφ ,ΔV	$\Delta V = -\frac{\Delta W}{q} = -\frac{1}{q}\int_{r_1}^{r_2} \mathbf{F} \cdot \mathrm{d} \mathbf{r} = -\int_{r_1}^{r_2} \mathbf{E} \cdot \mathrm{d} \mathbf{r} \,\!$
Electric Potential Energy	U	U = − W	J	[M][L]²[T]²

Magnetic Quantities

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Magnetic Field, Field Strength, Flux Density, Induction Field	B	F = q(v×B)	T = N A⁻¹ m⁻¹
Magnetic Flux	Φ_B	Φ_B = ∫_SB ⋅ dA	Wb = T m^-2
Relative Permeability	μ_r		H m⁻¹
Absolute Magnetic Permeability	μ	μ = μ_rμ₀	H m⁻¹
Magnetic Field Intensity, (AKA field strength)	H	$\mathbf{H} =\frac{\mathbf{B}}{\mu}\,\!$
Magnetic Dipole Moment	m, μ, Π	m = NIA N is the number of turns of conductor	A m²	[I][L]²
Magnetization	M	$\mathbf{M} = \frac{\partial \langle \mathbf{m} \rangle }{\partial V} \,\!$
Self Inductance	L	Two equivalent definitions are in fact possible: $L=N\frac{\partial \Phi}{\partial I}\,\!$ $L\frac{\partial I}{\partial t}=-NV\,\!$	H = Wb A⁻¹
Mutual Inductance	M	Again two equivalent definitions are in fact possible: $M_{X}=N\frac{\partial \Phi_Y}{\partial I_X}\,\!$ $M\frac{\partial I_Y}{\partial t}=-NV_X\,\!$ X,Y subscripts refer to two conductors mutually inducing voltage/ linking magnetic flux though each other	H = Wb A⁻¹

Electric Circuits

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Electrical Resistance	R	R = V/I	Ω = V A⁻¹ = J s C⁻²	[M][L]² [T]⁻³ [I]⁻²
Resistivity, Scalar	ρ	$\rho = \frac{RA}{l} \,\!$	Ω m	[M]² [L]² [T]⁻³ [I]⁻²
Resistivity Temperature Coefficient, Linear Temperature Dependance	α	ρ − ρ₀ = ρ₀α(T−T₀)	K⁻¹	[Θ]⁻¹
Terminal Voltage for Power Supply	V_ter		V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹
Load Voltage for Circuit	V_load		V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹
Internal Resistance of Power Supply	R_int	$R_\mathrm{int} = \frac{V_\mathrm{ter}}{I} \,\!$	Ω = V A⁻¹ = J s C⁻²	[M][L]² [T]⁻³ [I]⁻²
Load Resistance of Circuit	R_ext	$R_\mathrm{ext} = \frac{V_\mathrm{load}}{I} \,\!$	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	ℰ = V_ter + V_load	V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹
Electrical Conductance	G	G = 1/R	S = Ω⁻¹	[T]³ [I]² [M]⁻¹ [L]⁻²
Electrical Conductivity, Scalar	σ	σ = 1/ρ	Ω⁻¹ m⁻¹	[I]² [T]³ [M]⁻² [L]⁻²
Electrical Power	P	P = VI	W = J s⁻¹	[M] [L]² [T]⁻³

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
Encyclopedia of Physics, R.G. Lerner, G.L. Trigg, 2nd Edition, VHC Publishers, Hans Warlimont, Springer, 2005, pp 12-13

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