List of electromagnetism equations

Electromagnetism

Electricity Magnetism
Electrostatics Electric charge Static electricity Electric field Conductor Insulator Triboelectricity Electrostatic discharge Induction Coulomb's law Gauss's law Electric flux / potential energy Electric dipole moment Polarization density
Magnetostatics Ampère's law Magnetic field Magnetization Magnetic flux Biot–Savart law Magnetic dipole moment Gauss's law for magnetism
Electrodynamics Lorentz force law Electromagnetic induction Faraday's law Lenz's law Displacement current Magnetic potential Maxwell's equations Electromagnetic field Electromagnetic radiation Maxwell tensor Poynting vector Liénard–Wiechert potential Jefimenko's equations Eddy current London equations Mathematical descriptions of the electromagnetic field
Electrical network Electric current Electric potential Voltage Resistance Ohm's law Series circuit Parallel circuit Direct current Alternating current Charged current Neutral current Electromotive force Capacitance Inductance Impedance Resonant cavities Waveguides
Covariant formulation Electromagnetic tensor (stress–energy tensor) Four-current Electromagnetic four-potential
Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted

This article summarizes equations in the theory of electromagnetism.

Definitions

Lorentz force on a charged particle (of charge q) in motion (velocity v), used as the definition of the E field and B field.

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 q_m(Wb) = μ₀ q_m(Am).

Initial quantities

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

Electric quantities

Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P. Position vector r is a point to calculate the electric field; r′ is a point in the charged object.

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	λ_e for Linear, σ_e for surface, ρ_e for volume.	$q_e = \int \lambda_e \mathrm{d}\ell$ $q_e = \iint \sigma_e \mathrm{d} S$ $q_e = \iiint \rho_e \mathrm{d}V$	C m⁻ⁿ, 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]²[T]⁴[L]⁻²[M]⁻¹
Electric current	I	$I = \mathrm{d}q/\mathrm{d}t \,\!$	A	[I]
Electric current density	J	$I = \mathbf{J} \cdot \mathrm{d} \mathbf{S}$	A m⁻²	[I][L]⁻²
Displacement current density	J_d	$\mathbf{J}_\mathrm{d} = \epsilon_0 \left ( \partial \mathbf{E} / \partial t \right ) = \partial \mathbf{D} / \partial t \,\!$	Am⁻²	[I][L]m⁻²
Convection current density	J_c	$\mathbf{J}_\mathrm{c} = \rho \mathbf{v} \,\!$	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	Φ_E	$\Phi_E = \int_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}\,\!$	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 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
Electric displacement field	D	$\mathbf{D} = \epsilon\mathbf{E} = \epsilon_0 \mathbf{E} + \mathbf{P}\,$	C m⁻²	[I][T][L]⁻²
Electric displacement flux	Φ_D	$\Phi_D = \int_S \mathbf{D} \cdot \mathrm{d} \mathbf{A}\,\!$	C	[I][T]
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]² [T]⁻³ [I]⁻¹
Voltage, Electric potential difference	Δφ,ΔV	$\Delta V = -\frac{\Delta W}{q} = -\frac{1}{q}\int_{r_1}^{r_2} \mathbf{F} \cdot \mathrm{d} \mathbf{r} = -\int_{r_1}^{r_2} \mathbf{E} \cdot \mathrm{d} \mathbf{r} \,\!$	V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹

Magnetic quantities

Magnetic transport

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Linear, surface, volumetric pole density	λ_m for Linear, σ_m for surface, ρ_m for volume.	$q_m = \int \lambda_m \mathrm{d}\ell$ $q_m = \iint \sigma_m \mathrm{d} S$ $q_m = \iiint \rho_m \mathrm{d}V$	Wb m⁻ⁿ A m^{−(n + 1)}, n = 1, 2, 3	[L]²[M][T]⁻² [I]⁻¹ (Wb) [I][L] (Am)
Monopole current	I_m	$I_m = \mathrm{d}q_m/\mathrm{d}t \,\!$	Wb s⁻¹ A m s⁻¹	[L]²[M][T]⁻³ [I]⁻¹ (Wb) [I][L][T]⁻¹ (Am)
Monopole current density	J_m	$I = \iint \mathbf{J}_\mathrm{m} \cdot \mathrm{d} \mathbf{A}$	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 m⁻²	[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	Φ_B	$\Phi_B = \int_S \mathbf{B} \cdot \mathrm{d} \mathbf{A}\,\!$	Wb = T m²	[L]²[M][T]⁻²[I]⁻¹
Magnetic permeability	$\mu \,\!$	$\mu \ = \mu_r \,\mu_0 \,\!$	V·s·A⁻¹·m⁻¹ = N·A⁻² = T·m·A⁻¹ = Wb·A⁻¹·m⁻¹	[M][L][T]⁻²[I]⁻²
Magnetic moment, magnetic dipole moment	m, μ_B, 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 m⁻¹	[I] [L]⁻¹
Magnetic field intensity, (AKA field strength)	H	Two definitions are possible: most common: $\mathbf{B} = \mu \mathbf{H} = \mu_0 \left ( \mathbf{H} + \mathbf{M} \right ) \,$ using pole strengths,^[1] $\mathbf{H} = \mathbf{F} / q_m \,$	A m⁻¹	[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 through 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

Main article: Direct current

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
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	$R_\mathrm{int} = V_\mathrm{ter}/I \,\!$	Ω = V A⁻¹ = J s C⁻²	[M][L]² [T]⁻³ [I]⁻²
Load resistance of circuit	R_ext	$R_\mathrm{ext} = V_\mathrm{load}/I \,\!$	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

Main articles: Alternating current and Resonance

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Resistive load voltage	V_R	$V_R = I_R R \,\!$	V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹
Capacitive load voltage	V_C	$V_C = I_C X_C\,\!$	V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹
Inductive load voltage	V_L	$V_L = I_L X_L\,\!$	V = J C⁻¹	[M] [L]² [T]⁻³ [I]⁻¹
Capacitive reactance	X_C	$X_C = \frac{1}{\omega_\mathrm{d} C} \,\!$	Ω⁻¹ m⁻¹	[I]² [T]³ [M]⁻² [L]⁻²
Inductive reactance	X_L	$X_L = \omega_d L \,\!$	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	I_rms	$I_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} \left [ I \left ( t \right ) \right ]^2 \mathrm{d} t} \,\!$	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	V_rms	$V_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} \left [ V \left ( t \right ) \right ]^2 \mathrm{d} t} \,\!$	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	τ_C	$\tau_C = RC\,\!$	s	[T]
Inductive time constant	τ_L	$\tau_L = L/R\,\!$	s	[T]

Magnetic circuits

Main article: 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]

Electromagnetism

Electric fields

General Classical Equations

Physical situation	Equations
Electric potential gradient and field	$\mathbf{E} = - \nabla V$ $\Delta V = -\int_{r_1}^{r_2} \mathbf{E} \cdot d\mathbf{r}\,\!$
Point charge	$\mathbf{E} = \frac{q}{4 \pi \epsilon_0 \left \| \mathbf{r} \right \|^2 }\mathbf{\hat{r}} \,\!$
At a point in a local array of point charges	$\mathbf{E} = \sum \mathbf{E}_i = \frac{1}{4 \pi \epsilon_0} \sum_i \frac{q_i}{\left \| \mathbf{r}_i - \mathbf{r} \right \|^2}\mathbf{\hat{r}}_i \,\!$
At a point due to a continuum of charge	$\mathbf{E} = \frac{1}{4\pi\epsilon_0} \int_V \frac{\mathbf{r} \rho \mathrm{d}V}{\left \| \mathbf{r} \right \|^3} \,\!$
Electrostatic torque and potential energy due to non-uniform fields and dipole moments	$\boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{p} \times \mathbf{E}$ $U = \int_V \mathrm{d} \mathbf{p} \cdot \mathbf{E}$

Magnetic fields and moments

Physical situation	Equations
Magnetic potential, EM vector potential	$\mathbf{B} = \nabla \times \mathbf{A}$
Due to a magnetic moment	$\mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{\left \| \mathbf{r} \right \|^3}$ $\mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left(\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{\left \| \mathbf{r} \right \|^{5}}-\frac{{\mathbf{m}}}{\left \| \mathbf{r} \right \|^{3}}\right)$
Magnetic moment due to a current distribution	$\mathbf{m} = \frac{1}{2}\int_V \mathbf{r}\times\mathbf{J} \mathrm{d} V$
Magnetostatic torque and potential energy due to non-uniform fields and dipole moments	$\boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{m} \times \mathbf{B}$ $U = \int_V \mathrm{d} \mathbf{m} \cdot \mathbf{B}$

Electromagnetic induction

Physical situation	Nomenclature	Equations
Transformation of voltage	N = number of turns of conductor η = energy efficiency	$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s} = \eta \,\!$

Electric circuits and electronics

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

Physical situation	Nomenclature	Series	Parallel
Resistors and conductors	R_i = resistance of resistor or conductor i G_i = conductance of conductor or conductor i	$R_\mathrm{net} = \sum_{i=1}^{N} R_i\,\!$ ${1 \over G_{{\mathrm {net}}}}=\sum _{{i=1}}^{{N}}{1 \over G_{i}}\,\!$	${1 \over R_{{\mathrm {net}}}}=\sum _{{i=1}}^{{N}}{1 \over R_{i}}\,\!$ $G_{{\mathrm {net}}}=\sum _{{i=1}}^{{N}}G_{i}\,\!$
Charge, capacitors, currents	q_i = capacitance of capacitor i q_i = charge of charge carrier i	$q_\mathrm{net} = \sum_{i=1}^N q_i \,\!$ ${1 \over C_{{\mathrm {net}}}}=\sum _{{i=1}}^{N}{1 \over C_{i}}\,\!$ $I_{{\mathrm {net}}}=I_{i}\,\!$	$q_\mathrm{net} = \sum_{i=1}^N q_i \,\!$ $C_\mathrm{net} = \sum_{i=1}^N C_i \,\!$ $I_\mathrm{net} = \sum_{i=1}^N I_i \,\!$
Inductors	L_i = self-inductance of inductor i L_ij = self-inductance element ij of L matrix M_ij = mutual inductance between inductors i and j	$L_{{\mathrm {net}}}=\sum _{{i=1}}^{N}L_{i}\,\!$	${1 \over L_{{\mathrm {net}}}}=\sum _{{i=1}}^{N}{1 \over L_{i}}\,\!$ $V_i = \sum_{j=1}^N L_{ij} \frac{\mathrm{d}I_j}{\mathrm{d}t} \,\!$

Series circuit equations
Circuit	DC Circuit equations	AC Circuit equations
RC circuits	Circuit equation $R \frac{\mathrm{d}q}{\mathrm{d}t} + \frac{q}{C} = \mathcal{E}\,\!$ Capacitor charge $q = C\mathcal{E}\left ( 1 - e^{-t/RC} \right )\,\!$ Capacitor discharge $q = C\mathcal{E}e^{-t/RC}\,\!$
RL circuits	Circuit equation $L\frac{\mathrm{d}I}{\mathrm{d}t}+RI=\mathcal{E}\,\!$ Inductor current rise $I = \frac{\mathcal{E}}{R}\left ( 1-e^{-Rt/L}\right )\,\!$ Inductor current fall $I=\frac{\mathcal{E}}{R}e^{-t/\tau_L}=I_0e^{-Rt/L}\,\!$
LC circuits	Circuit equation $L\frac{\mathrm{d}^2q}{\mathrm{d}t^2} + q/C = \mathcal{E}\,\!$	Circuit equation $L\frac{\mathrm{d}^2q}{\mathrm{d}t^2} + q/C = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!$ Circuit resonant frequency $\omega_\mathrm{res} = 1/\sqrt{LC}\,\!$ Circuit charge $q = q_0 \cos(\omega t + \phi)\,\!$ Circuit current $I=-\omega q_0 \sin(\omega t + \phi)\,\!$ Circuit electrical potential energy $U_E=q^2/2C=Q^2\cos^2(\omega t + \phi)/2C\,\!$ Circuit magnetic potential energy $U_B=Q^2\sin^2(\omega t + \phi)/2C\,\!$
RLC Circuits	Circuit equation $L\frac{\mathrm{d}^2q}{\mathrm{d}t^2} + R\frac{\mathrm{d}q}{\mathrm{d}t} + \frac{q}{C} = \mathcal{E} \,\!$	Circuit equation $L\frac{\mathrm{d}^2q}{\mathrm{d}t^2} + R\frac{\mathrm{d}q}{\mathrm{d}t} + \frac{q}{C} = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!$ Circuit charge $q = q_0 eT^{-Rt/2L}\cos(\omega't+\phi)\,\!$

Footnotes

↑ M. Mansfield; C. O'Sullivan (2011). Understanding Physics (2nd ed.). John Wiley & Sons. ISBN 978-0-470-74637-0.

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SI units

Base units	ampere candela kelvin kilogram metre mole second

Derived units with special names	becquerel coulomb degree Celsius farad gray henry hertz joule katal lumen lux newton ohm pascal radian siemens sievert steradian tesla volt watt weber

Other accepted units	astronomical unit bar dalton day decibel degree of arc electronvolt hectare hour litre minute minute of arc neper second of arc tonne atomic units natural units

See also	Conversion of units History of the metric system Metric prefixes Proposed redefinitions Systems of measurement

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