List of electromagnetism equations

This article summarizes equations in the theory of electromagnetism.

Definitions

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

Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.^{[citation needed]}

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	$C={\mathrm {d} q \over \mathrm {d} V}\,\!$ V = voltage, not volume.	F = C V⁻¹	[I]²[T]⁴[L]⁻²[M]⁻¹
Electric current	I	$I={\mathrm {d} q \over \mathrm {d} t}\,\!$	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	$\mathbf {J} _{\mathrm {d} }={\partial \mathbf {D} \over \partial t}=\varepsilon _{0}\left({\partial \mathbf {E} \over \partial t}\right)\,\!$	A m⁻²	[I][L]⁻²
Convection current density	J_c	$\mathbf {J} _{\mathrm {c} }=\rho \mathbf {v} \,\!$	A m⁻²	[I][L]⁻²

Electric fields

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Electric field, field strength, flux density, potential gradient	E	$\mathbf {E} ={\mathbf {F} \over q}\,\!$	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	$\varepsilon =\varepsilon _{r}\varepsilon _{0}\,\!$	F m⁻¹	[I]² [T]⁴ [M]⁻¹ [L]⁻³
Electric dipole moment	p	$\mathbf {p} =q\mathbf {a} \,\!$ a = charge separation directed from -ve to +ve charge	C m	[I][T][L]
Electric Polarization, polarization density	P	$\mathbf {P} ={\mathrm {d} \langle \mathbf {p} \rangle \over \mathrm {d} V}\,\!$	C m⁻²	[I][T][L]⁻²
Electric displacement field, flux density	D	$\mathbf {D} =\varepsilon \mathbf {E} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \,$	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	$I_{m}={\mathrm {d} q_{m} \over \mathrm {dt} }\,\!$	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	$\mathbf {M} ={\mathrm {d} \langle \mathbf {m} \rangle \over \mathrm {d} V}\,\!$	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] $\mathbf {H} ={\mathbf {F} \over q_{m}}\,$	A m⁻¹	[I] [L]⁻¹
Intensity of magnetization, magnetic polarization	I, J	$\mathbf {I} =\mu _{0}\mathbf {M} \,\!$	T = N A⁻¹ m⁻¹ = Wb m⁻²	[M][T]⁻²[I]⁻¹
Self Inductance	L	Two equivalent definitions are possible: $L=N\left({\mathrm {d} \Phi \over \mathrm {d} I}\right)\,\!$ $L\left({\mathrm {d} I \over \mathrm {d} t}\right)=-NV\,\!$	H = Wb A⁻¹	[L]² [M] [T]⁻² [I]⁻²
Mutual inductance	M	Again two equivalent definitions are possible: $M_{1}=N\left({\mathrm {d} \Phi _{2} \over \mathrm {d} I_{1}}\right)\,\!$ $M\left({\mathrm {d} I_{2} \over \mathrm {d} t}\right)=-NV_{1}\,\!$ 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; $M_{2}=N\left({\mathrm {d} \Phi _{1} \over \mathrm {d} I_{2}}\right)\,\!$ $M\left({\mathrm {d} I_{1} \over \mathrm {d} t}\right)=-NV_{2}\,\!$	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	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	$R_{\mathrm {int} }={V_{\mathrm {ter} } \over I}\,\!$	Ω = V A⁻¹ = J s C⁻²	[M][L]² [T]⁻³ [I]⁻²
Load resistance of circuit	R_ext	$R_{\mathrm {ext} }={V_{\mathrm {load} } \over I}\,\!$	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	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	$V=IZ\,\!$ $Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}\,\!$	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	$\tau _{L}={L \over R}\,\!$	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]

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	$\mathbf {E} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}}{{\hat {\mathbf {r} }} \over {\|\mathbf {r} \|}^{2}}={q \over 4\pi \varepsilon _{0}}{\mathbf {r} \over {\|\mathbf {r} \|}^{3}}\,\!$
At a point in a local array of point charges	$\mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {\|\mathbf {r_{i}-r} \|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {\|\mathbf {r_{i}-r} \|}^{3}}$
At a point due to a continuum of charge	$\mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint \,\rho (\mathbf {r} '){\mathbf {r} ' \over {\|\mathbf {r} '\|}^{3}}\mathrm {d} ^{3}\|\mathbf {r} '\|$
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

General classical equations

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	${\boldsymbol {\tau }}=\int _{V}\mathrm {d} \mathbf {m} \times \mathbf {B}$ $U=-\int _{V}\mathrm {d} \mathbf {m} \cdot \mathbf {B}$

Electric circuits and electronics

Below N = number of conductors or circuit components. Subscript 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 resistor 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	C_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 $R{\mathrm {d} q \over \mathrm {d} t}+{q \over C}={\mathcal {E}}\,\!$ 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 $L{\mathrm {d} I \over \mathrm {d} t}+RI={\mathcal {E}}\,\!$ Inductor current rise $I={\frac {\mathcal {E}}{R}}\left(1-e^{-Rt/L}\right)\,\!$ Inductor current fall $I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-Rt/L}\,\!$
LC circuits	Circuit equation $L{\mathrm {d} ^{2}q \over \mathrm {d} t^{2}}+{q \over C}={\mathcal {E}}\,\!$	Circuit equation $L{\mathrm {d} ^{2}q \over \mathrm {d} t^{2}}+{q \over C}={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$ Circuit resonant frequency $\omega _{\mathrm {res} }={1 \over {\sqrt {LC}}}\,\!$ Circuit charge $q=q_{0}\cos(\omega t+\phi )\,\!$ Circuit current $I=-\omega q_{0}\sin(\omega t+\phi )\,\!$ Circuit electrical potential energy $U_{E}={q^{2} \over 2C}={q_{0}^{2}\cos ^{2}(\omega t+\phi ) \over 2C}\,\!$ Circuit magnetic potential energy $U_{B}={q_{0}^{2}\sin ^{2}(\omega t+\phi ) \over 2C}\,\!$
RLC circuits	Circuit equation $L{\mathrm {d} ^{2}q \over \mathrm {d} t^{2}}+R{\mathrm {d} q \over \mathrm {d} t}+{q \over C}={\mathcal {E}}\,\!$	Circuit equation $L{\mathrm {d} ^{2}q \over \mathrm {d} t^{2}}+R{\mathrm {d} q \over \mathrm {d} t}+{q \over C}={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$ 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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v t e 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 dalton day decibel degree of arc electronvolt hectare hour litre minute minute and second of arc neper tonne
See also	Conversion of units Metric prefixes Historical definitions of the SI base units 2019 revision of the SI System of units of measurement
Category

Electromagnetism

Electricity Magnetism Optics History Computational Textbooks Phenomena
Electrostatics Charge density Conductor Coulomb law Electret Electric charge Electric dipole Electric field Electric flux Electric potential Electrostatic discharge Electrostatic induction Gauss's law Insulator Permittivity Polarization Potential energy Static electricity Triboelectricity
Magnetostatics Ampère's law Biot–Savart law Gauss's law for magnetism Magnetic dipole Magnetic field Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule
Electrodynamics Bremsstrahlung Cyclotron radiation Displacement current Eddy current Electromagnetic field Electromagnetic induction Electromagnetic pulse Electromagnetic radiation Faraday's law Jefimenko equations Larmor formula Lenz's law Liénard–Wiechert potential London equations Lorentz force Maxwell's equations Maxwell tensor Poynting vector Synchrotron radiation
Electrical network Alternating current Capacitance Current density Direct current Electric current Electric power Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff's laws Network analysis Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Watt Waveguides
Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer
Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor
Scientists Ampère Biot Coulomb Davy Einstein Faraday Fizeau Gauss Heaviside Helmholtz Henry Hertz Hopkinson Jefimenko Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert
v t e