View source for List of electromagnetism equations

{{short description|None}}

{{electromagnetism}}
This article summarizes [[equation]]s in the theory of [[electromagnetism]].

==Definitions==
[[File:Lorentz force particle.svg|200px|thumb|Lorentz force on a [[charged particle]] (of [[electric charge|charge]] ''q'') in motion (velocity '''v'''), used as the definition of the [[electric field|'''E''' field]] and [[magnetic field|'''B''' field]].]]

Here subscripts ''e'' and ''m'' are used to differ between electric and [[magnetic charge]]s. 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<sub>m</sub>''(Wb) = ''μ''<sub>0</sub> ''q<sub>m</sub>''(Am).

===Initial quantities===

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s)
! scope="col" width="125" | (Common) symbol/s
! scope="col" width="125" | SI units
! scope="col" width="100" | Dimension
|-
![[Electric charge]]
| ''q<sub>e</sub>'', ''q'', ''Q''
| C = As
| [I][T] 
|-
![[Magnetic monopole|Monopole]] strength, [[magnetic charge]]
| ''q<sub>m</sub>'', ''g'', ''p''
| Wb or Am
| [L]<sup>2</sup>[M][T]<sup>−2</sup> [I]<sup>−1</sup> (Wb)<br />
[I][L] (Am)
|-
|}

===Electric quantities===

[[File:Universal charge distribution.svg|200px|right|thumb|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 [[electric dipole moment|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&prime;''' 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.{{Citation needed|date=August 2024}}

'''Electric transport'''

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s)
! scope="col" width="125" | (Common) symbol/s
! scope="col" width="200" | Defining equation
! scope="col" width="125" | SI units
! scope="col" width="100" | Dimension
|-
! Linear, surface, volumetric charge density
| ''λ<sub>e</sub>'' for Linear, ''σ<sub>e</sub>'' for surface, ''ρ<sub>e</sub>'' for volume. 
| <math> q_e = \int \lambda_e \mathrm{d}\ell </math>

<math> q_e = \iint \sigma_e \mathrm{d} S </math>

<math> q_e = \iiint \rho_e \mathrm{d}V </math>
| C m<sup>&minus;''n''</sup>, ''n'' = 1, 2, 3 
| [I][T][L]<sup>&minus;''n''</sup>
|-
![[Capacitance]]
| ''C''
|<math>C = {\mathrm{d}q\over\mathrm{d}V}\,\!</math>
''V'' = voltage, ''not'' volume.
| F = C V<sup>&minus;1</sup>
| [I]<sup>2</sup>[T]<sup>4</sup>[L]<sup>&minus;2</sup>[M]<sup>&minus;1</sup>    
|-
![[Electric current]]
| ''I''
|<math> I = {\mathrm{d}q\over\mathrm{d}t} \,\!</math>
| A
| [I]
|-
!Electric [[current density]]
| '''J'''
|<math>I = \mathbf{J} \cdot \mathrm{d} \mathbf{S}</math>
| A m<sup>&minus;2</sup>
| [I][L]<sup>&minus;2</sup>
|-
![[Displacement current]] density
| '''J'''<sub>d</sub>
| <math> \mathbf{J}_\mathrm{d} = {\partial\mathbf{D}\over\partial t} = \varepsilon_0 \left ({\partial\mathbf{E}\over\partial t}\right) \,\!</math>
| A m<sup>&minus;2</sup>
| [I][L]<sup>&minus;2</sup>
|-
!Convection current density
| '''J'''<sub>c</sub>
| <math> \mathbf{J}_\mathrm{c} = \rho \mathbf{v} \,\!</math>
| A m<sup>−2</sup>
| [I][L]<sup>−2</sup>
|-
|}

'''Electric fields'''

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s)
! scope="col" width="125" | (Common) symbol/s
! scope="col" width="200" | Defining equation 
! scope="col" width="125" | SI units
! scope="col" width="100" | Dimension
|-
![[Electric field]], field strength, flux density, potential gradient
| '''E'''
| <math>\mathbf{E} ={\mathbf{F}\over q}\,\!</math>
| N C<sup>−1</sup> = V m<sup>−1</sup>
| [M][L][T]<sup>−3</sup>[I]<sup>−1</sup>
|-
![[Electric flux]]
| ''Φ<sub>E</sub>''
|<math>\Phi_E = \int_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}\,\!</math>
| N m<sup>2</sup> C<sup>−1</sup>
| [M][L]<sup>3</sup>[T]<sup>−3</sup>[I]<sup>−1</sup>
|-
!Absolute [[permittivity]];
| ''ε''
| <math> \varepsilon = \varepsilon_r \varepsilon_0\,\!</math>
| F m<sup>−1</sup>
| [I]<sup>2</sup> [T]<sup>4</sup> [M]<sup>−1</sup> [L]<sup>−3</sup> 
|-
![[Electric dipole moment]]
|| '''p'''
||<math>\mathbf{p} = q\mathbf{a}\,\!</math>

'''a''' = charge separation
directed from -ve to +ve charge
|| C m
|| [I][T][L]
|-
!Electric Polarization, [[polarization density]]
| '''P'''
|<math>\mathbf{P} = {\mathrm{d}\langle\mathbf{p}\rangle\over\mathrm{d} V} \,\!</math>
| C m<sup>−2</sup>
| [I][T][L]<sup>−2</sup>
|-
![[Electric displacement field]], flux density
| '''D'''
| <math> \mathbf{D} = \varepsilon\mathbf{E} = \varepsilon_0 \mathbf{E} + \mathbf{P}\,</math>
| C m<sup>−2</sup>
| [I][T][L]<sup>−2</sup>
|-
!Electric displacement flux
|| ''Φ<sub>D</sub>''
||<math>\Phi_D = \int_S \mathbf{D} \cdot \mathrm{d} \mathbf{A}\,\!</math>
|| C
|| [I][T]
|-
!Absolute [[electric potential]], EM scalar potential relative to point <math> r_0 \,\!</math>

Theoretical: <math> r_0 = \infty \,\!</math><br />
Practical: <math> r_0 = R_\mathrm{earth} \,\!</math> (Earth's radius)<br />
| ''φ ,V'' 
| <math> 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}\,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
![[Voltage]], Electric potential difference
| Δ''φ'',Δ''V''
|<math>\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} \,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
|}

===Magnetic quantities===

'''Magnetic transport'''

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-
! Linear, surface, volumetric pole density
| ''λ<sub>m</sub>'' for Linear, ''σ<sub>m</sub>'' for surface, ''ρ<sub>m</sub>'' for volume. 
| <math> q_m = \int \lambda_m \mathrm{d}\ell </math>

<math> q_m = \iint \sigma_m \mathrm{d} S </math>

<math> q_m = \iiint \rho_m \mathrm{d}V </math>
| Wb m<sup>&minus;''n''</sup> <br />
A m<sup>(&minus;''n'' + 1)</sup>, <br />
''n'' = 1, 2, 3 
|[L]<sup>2</sup>[M][T]<sup>−2</sup> [I]<sup>−1</sup> (Wb)<br />
[I][L] (Am)
|-
!Monopole current
| ''I<sub>m</sub>''
|<math> I_m = {\mathrm{d}q_m\over\mathrm{dt}} \,\!</math>
| Wb s<sup>&minus;1</sup> <br />
A m s<sup>&minus;1</sup>
|[L]<sup>2</sup>[M][T]<sup>&minus;3</sup> [I]<sup>&minus;1</sup> (Wb)<br />
[I][L][T]<sup>&minus;1</sup> (Am)
|-
!Monopole current density
| '''J'''<sub>m</sub>
|<math> I = \iint \mathbf{J}_\mathrm{m} \cdot \mathrm{d} \mathbf{A} </math>
| Wb s<sup>&minus;1</sup> m<sup>&minus;2</sup> <br />
A m<sup>&minus;1</sup> s<sup>&minus;1</sup>
|[M][T]<sup>&minus;3</sup> [I]<sup>&minus;1</sup> (Wb)<br />
[I][L]<sup>&minus;1</sup>[T]<sup>&minus;1</sup> (Am)
|-
|}

'''Magnetic fields'''

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-
![[Magnetic field]], field strength, flux density, induction field
| '''B'''
| <math>\mathbf{F} =q_e \left ( \mathbf{v}\times\mathbf{B} \right ) \,\!</math>
| T = N A<sup>−1</sup> m<sup>−1</sup> = Wb m<sup>−2</sup>
| [M][T]<sup>−2</sup>[I]<sup>−1</sup> 
|-
! [[Magnetic vector potential|Magnetic potential]], EM vector potential 
| '''A'''
| <math> \mathbf{B} = \nabla \times \mathbf{A} </math>
| T m = N A<sup>−1</sup> = Wb m<sup>3</sup>
| [M][L][T]<sup>−2</sup>[I]<sup>−1</sup>
|-
! [[Magnetic flux]]
| ''Φ<sub>B</sub>''
| <math>\Phi_B = \int_S \mathbf{B} \cdot \mathrm{d} \mathbf{A}\,\!</math>
| Wb = T m<sup>2</sup>
| [L]<sup>2</sup>[M][T]<sup>−2</sup>[I]<sup>−1</sup>
|-
![[Magnetic permeability]]
| <math>\mu \,\!</math>
| <math>\mu \ = \mu_r \,\mu_0 \,\!</math>
| V·s·A<sup>&minus;1</sup>·m<sup>&minus;1</sup> = N·A<sup>&minus;2</sup> = T·m·A<sup>&minus;1</sup> = Wb·A<sup>&minus;1</sup>·m<sup>&minus;1</sup>
| [M][L][T]<sup>−2</sup>[I]<sup>−2</sup> 
|-
![[Magnetic moment]], [[magnetic dipole moment]]
| '''m''', '''μ<sub>B</sub>''', '''Π'''
|
Two definitions are possible:

using pole strengths, <br />
<math>\mathbf{m} = q_m \mathbf{a}\,\!</math>

using currents: <br />
<math>\mathbf{m} = NIA\mathbf{\hat{n}}\,\!</math>

'''a''' = pole separation

''N'' is the number of turns of conductor 
| A m<sup>2</sup>
| [I][L]<sup>2</sup>
|-
![[Magnetization]]
| '''M'''
|<math>\mathbf{M} = {\mathrm{d} \langle\mathbf{m}\rangle\over\mathrm{d}V} \,\!</math>
| A m<sup>−1</sup>
| [I] [L]<sup>−1</sup>
|-
![[Magnetic field]] intensity, (AKA field strength)
| '''H'''
| Two definitions are possible:

most common:<br /> 
<math>\mathbf{B} = \mu \mathbf{H} = \mu_0 \left ( \mathbf{H} + \mathbf{M} \right ) \,</math>

using pole strengths,<ref>{{cite book|title=Understanding Physics|edition=2nd |author1=M. Mansfield |author2=C. O'Sullivan |publisher=John Wiley & Sons|year=2011|isbn=978-0-470-74637-0}}</ref> <br />
<math>\mathbf{H} = {\mathbf{F}\over q_m} \,</math>
| A m<sup>−1</sup>
| [I] [L]<sup>−1</sup>
|-
![[Magnetic susceptibility|Intensity of magnetization]], magnetic polarization
| '''I''', '''J'''
|<math>\mathbf{I} = \mu_0 \mathbf{M} \,\!</math>
| T = N A<sup>−1</sup> m<sup>−1</sup> = Wb m<sup>−2</sup>
| [M][T]<sup>−2</sup>[I]<sup>−1</sup> 
|-
! Self [[Inductance]]
| ''L''
| Two equivalent definitions are possible: 
<math>L = N\left ( {\mathrm{d}\Phi\over\mathrm{d} I}\right )\,\!</math>

<math>L\left({\mathrm{d} I\over\mathrm{d}t}\right) = -NV\,\!</math>
| H = Wb A<sup>−1</sup>
| [L]<sup>2</sup> [M] [T]<sup>−2</sup> [I]<sup>−2</sup>
|-
! Mutual [[inductance]]
| ''M''
| Again two equivalent definitions are possible: 
<math>M_1 = N\left ({\mathrm{d} \Phi_2\over\mathrm{d}I_1}\right)\,\!</math>

<math>M\left({\mathrm{d} I_2\over\mathrm{d}t}\right ) = -NV_1\,\!</math>

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;

<math>M_2 = N\left({\mathrm{d} \Phi_1\over\mathrm{d}I_2}\right)\,\!</math> <br />
<math>M\left({\mathrm{d} I_1\over\mathrm{d}t} \right ) = -NV_2\,\!</math>
| H = Wb A<sup>−1</sup>
| [L]<sup>2</sup> [M] [T]<sup>−2</sup> [I]<sup>−2</sup>
|-
![[Gyromagnetic ratio]] (for charged particles in a magnetic field) 
| ''γ''
| <math>\omega = \gamma B \,\!</math> 
| Hz T<sup>−1</sup>
| [M]<sup>−1</sup>[T][I]
|-
|}

===Electric circuits===

'''DC circuits, general definitions'''

{{main|Direct current}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-
!Terminal Voltage for
[[Power Supply]]
| ''V''<sub>ter</sub>
|
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!Load Voltage for Circuit
|''V''<sub>load</sub>
|
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!Internal resistance of power supply
| ''R''<sub>int</sub>
| <math> R_\mathrm{int} = {V_\mathrm{ter}\over I} \,\!</math>
| Ω = V A<sup>−1</sup> = J s C<sup>−2</sup>
| [M][L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−2</sup>
|-
!Load resistance of circuit
| ''R''<sub>ext</sub>
| <math> R_\mathrm{ext} = {V_\mathrm{load}\over I} \,\!</math>
| Ω = V A<sup>−1</sup> = J s C<sup>−2</sup>
| [M][L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−2</sup>
|-
![[Electromotive force]] (emf), voltage across entire circuit including power supply, external components and conductors
| ''E''
| <math>\mathcal{E} = V_\mathrm{ter} + V_\mathrm{load} \,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
|}

'''AC circuits'''

{{main|Alternating current|Resonance}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|-  
!Resistive load voltage
| ''V<sub>R</sub>''
|<math> V_R = I_R R \,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!Capacitive load voltage
| ''V<sub>C</sub>''
|<math> V_C = I_C X_C\,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!Inductive load voltage
|''V<sub>L</sub>''
|<math>V_L = I_L X_L\,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
![[Capacitive reactance]]
|''X<sub>C</sub>''
| <math>X_C = \frac{1}{\omega_\mathrm{d} C} \,\!</math>
| Ω<sup>−1</sup> m<sup>−1</sup>
| [I]<sup>2</sup> [T]<sup>3</sup> [M]<sup>−2</sup> [L]<sup>−2</sup> 
|-
![[Inductive reactance]]
|''X<sub>L</sub>''
|<math> X_L = \omega_d L \,\!</math>
| Ω<sup>−1</sup> m<sup>−1</sup>
| [I]<sup>2</sup> [T]<sup>3</sup> [M]<sup>−2</sup> [L]<sup>−2</sup> 
|-
!AC electrical [[Electrical impedance|impedance]]
| ''Z''
|<math>V = I Z\,\!</math>
<math>Z = \sqrt{R^2 + \left ( X_L - X_C \right )^2 } \,\!</math>
| Ω<sup>−1</sup> m<sup>−1</sup>
| [I]<sup>2</sup> [T]<sup>3</sup> [M]<sup>−2</sup> [L]<sup>−2</sup> 
|-
!Phase constant
| ''δ, φ''
|<math>\tan\phi= \frac{X_L - X_C}{R}\,\!</math>
|dimensionless
|dimensionless
|-
!AC peak current
| ''I''<sub>0</sub>
|<math>I_0 = I_\mathrm{rms} \sqrt{2}\,\!</math>
| A
| [I]
|-
!AC root mean square current
|''I''<sub>rms</sub>
|<math> I_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} \left [ I \left ( t \right ) \right ]^2 \mathrm{d} t}  \,\!</math>
| A
| [I]
|-
!AC peak voltage
|''V''<sub>0</sub>
| <math> V_0 = V_\mathrm{rms} \sqrt{2} \,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!AC root mean square voltage
|''V''<sub>rms</sub>
|<math> V_\mathrm{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} \left [ V \left ( t \right ) \right ]^2 \mathrm{d} t}  \,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!AC emf, root mean square
|<math>\mathcal{E}_\mathrm{rms}, \sqrt{\langle \mathcal{E} \rangle} \,\!</math>
|<math>\mathcal{E}_\mathrm{rms}=\mathcal{E}_\mathrm{m}/\sqrt{2}\,\!</math>
| V = J C<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup> [I]<sup>−1</sup>
|-
!AC average power
|<math> \langle P \rangle \,\!</math>
|<math> \langle P \rangle =\mathcal{E}I_\mathrm{rms}\cos\phi\,\!</math>
| W = J s<sup>−1</sup>
| [M] [L]<sup>2</sup> [T]<sup>−3</sup>
|-
!Capacitive time constant
| ''τ<sub>C</sub>''
|<math>\tau_C = RC\,\!</math>
| s
| [T]
|-
!Inductive time constant
| ''τ<sub>L</sub>''
|<math>\tau_L = {L\over R}\,\!</math>
| s
| [T]
|-
|}

===Magnetic circuits===

{{Main|Magnetic circuits}}

{| class="wikitable"
|-
! scope="col" width="100" | Quantity (common name/s) 
! scope="col" width="100" | (Common) symbol/s 
! scope="col" width="300" | Defining equation 
! scope="col" width="125" | SI units 
! scope="col" width="100" | Dimension
|- 
! [[Magnetomotive force]], mmf
| ''F'', <math>\mathcal{F}, \mathcal{M} </math>
| <math>\mathcal{M} = NI</math>

''N'' = number of turns of conductor
| A
| [I]
|-
|}

==Electromagnetism==

===Electric fields===
[[File:Electrostatics_relation_triangle.svg|thumb|Summary of electrostatic relations between electric potential, electric field and charge density. Here, <math>\mathbf r = \mathbf x - \mathbf{x'}</math>.]]

General Classical Equations

{| class="wikitable"
|-
! scope="col" width="200" | Physical situation 
! scope="col" width="10" | Equations
|-
!Electric potential gradient and field
| <math> \mathbf{E} = - \nabla V </math>

<math> \Delta V = -\int_{r_1}^{r_2} \mathbf{E} \cdot d\mathbf{r}\,\!</math>
|-
!Point charge
| <math> \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}\,\!</math>
|-
!At a point in a local array of point charges
| <math>\mathbf E(\mathbf r) = {1\over4\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}</math>
|-
!At a point due to a continuum of charge
| <math> \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'|</math>
|-
!Electrostatic torque and potential energy due to non-uniform fields and dipole moments
| <math> \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{p} \times \mathbf{E} </math>

<math> U = - \int_V  \mathrm{d} \mathbf{p} \cdot \mathbf{E} </math>
|-
|}

===Magnetic fields and moments===
{{See also|Magnetic moment}}
[[File:Magnetostatics_relation_triangle.svg|thumb|Summary of magnetostatic relations between magnetic vector potential, magnetic field and current density. Here, <math>\mathbf r = \mathbf x - \mathbf{x'}</math>.]]
General classical equations

{| class="wikitable"
|-
! scope="col" width="200" | Physical situation 
! scope="col" width="10" | Equations
|-
!Magnetic potential,  EM vector potential
| <math> \mathbf{B} = \nabla \times \mathbf{A} </math> 
|-
!Due to a magnetic moment
| <math> \mathbf{A} = \frac{\mu_0}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{\left | \mathbf{r} \right |^3} </math>

<math>\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)</math>
|-
!Magnetic moment due to a current distribution
| <math> \mathbf{m} = \frac{1}{2}\int_V \mathbf{r}\times\mathbf{J} \mathrm{d} V </math>
|-
!Magnetostatic torque and potential energy due to non-uniform fields and dipole moments
| <math> \boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{m} \times \mathbf{B} </math>

<math> U = - \int_V \mathrm{d} \mathbf{m} \cdot \mathbf{B} </math>
|-
|}

==Electric circuits and electronics==

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

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Series
! scope="col" width="10" | Parallel
|-
!Resistors and conductors
|{{plainlist}}
*''R<sub>i</sub>'' = resistance of resistor or conductor ''i''
*''G<sub>i</sub>'' = conductance of resistor or conductor ''i''
{{endplainlist}}
|<math>R_\mathrm{net} = \sum_{i=1}^{N} R_i\,\!</math>

<math>{1\over G_\mathrm{net}} = \sum_{i=1}^{N} {1\over G_i}\,\!</math>
| <math>{1\over R_\mathrm{net}} = \sum_{i=1}^{N} {1\over R_i}\,\!</math>

<math>G_\mathrm{net} = \sum_{i=1}^{N} G_i \,\!</math>
|-
!Charge, capacitors, currents
| {{plainlist}}
*''C<sub>i</sub>'' = capacitance of capacitor ''i''
*''q<sub>i</sub>'' = charge of charge carrier ''i''
 {{endplainlist}}
| <math>q_\mathrm{net} = \sum_{i=1}^N q_i \,\!</math>
<math>{1\over C_\mathrm{net}} = \sum_{i=1}^N {1\over C_i} \,\!</math>
<math>I_\mathrm{net} = I_i \,\!</math>
|<math>q_\mathrm{net} = \sum_{i=1}^N q_i \,\!</math>
<math>C_\mathrm{net} = \sum_{i=1}^N C_i \,\!</math>
<math>I_\mathrm{net} = \sum_{i=1}^N I_i \,\!</math>
|-
!Inductors
| {{plainlist}}
*''L<sub>i</sub>'' = self-inductance of inductor ''i''
*''L<sub>ij</sub>'' = self-inductance element ''ij'' of ''L'' matrix
*''M<sub>ij</sub>'' = mutual inductance between inductors ''i'' and ''j''
{{endplainlist}}
|<math>L_\mathrm{net} = \sum_{i=1}^N L_i \,\!</math>
|<math>{1\over L_\mathrm{net}} = \sum_{i=1}^N {1\over L_i} \,\!</math>
<math>V_i = \sum_{j=1}^N L_{ij} \frac{\mathrm{d}I_j}{\mathrm{d}t} \,\!</math>
|-
|}
{| class="wikitable"
! scope="col" width="100" | Circuit 
! scope="col" width="200" | DC Circuit equations 
! scope="col" width="200" | AC Circuit equations
|+Series circuit equations
|-
![[RC circuits]]
|Circuit equation
<math>R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E}\,\!</math>

Capacitor charge
<math> q = C\mathcal{E}\left ( 1 - e^{-t/RC} \right )\,\!</math>

Capacitor discharge
<math> q = C\mathcal{E}e^{-t/RC}\,\!</math>
|
|-
![[RL circuits]]
|Circuit equation
<math>L{\mathrm{d}I\over \mathrm{d}t}+RI=\mathcal{E}\,\!</math>

Inductor current rise
<math>I = \frac{\mathcal{E}}{R}\left ( 1-e^{-Rt/L}\right )\,\!</math>

Inductor current fall
<math>I=\frac{\mathcal{E}}{R}e^{-t/\tau_L}=I_0e^{-Rt/L}\,\!</math>
|
|-
![[LC circuit]]s
|Circuit equation
<math>L{\mathrm{d}^2q\over \mathrm{d}t^2} + {q\over C} = \mathcal{E}\,\!</math>
|Circuit equation
<math>L{\mathrm{d}^2q\over \mathrm{d}t^2} + {q\over C} = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!</math>

Circuit resonant frequency
<math>\omega_\mathrm{res} = {1\over\sqrt{LC}}\,\!</math>

Circuit charge
<math>q = q_0 \cos(\omega t + \phi)\,\!</math>

Circuit current
<math>I=-\omega q_0 \sin(\omega t + \phi)\,\!</math>

Circuit electrical potential energy
<math>U_E = {q^2\over2C} = {q_0^2\cos^2(\omega t + \phi)\over2C}\,\!</math>

Circuit magnetic potential energy
<math>U_B={q_0^2\sin^2(\omega t + \phi)\over2C}\,\!</math>
|-
![[RLC circuits]]
|Circuit equation

<math>L{\mathrm{d}^2q\over \mathrm{d}t^2} + R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E} \,\!</math>

|Circuit equation

<math> L{\mathrm{d}^2q\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) \,\!</math>

Circuit charge

<math>q = q_0 eT^{-Rt/2L}\cos(\omega't+\phi)\,\!</math>
|-
|}

==See also==
*[[Defining equation (physical chemistry)]]
*[[Fresnel equations]]
*[[List of equations in classical mechanics]]
*[[List of equations in fluid mechanics]]
*[[List of equations in gravitation]]
*[[List of equations in nuclear and particle physics]]
*[[List of equations in quantum mechanics]]
*[[List of equations in wave theory]]
*[[List of photonics equations]]
*[[List of relativistic equations]]
*[[SI electromagnetism units]]
*[[Table of thermodynamic equations]]

==Footnotes==

{{Reflist}}

==Sources==

* {{cite book|author1=P.M. Whelan |author2=M.J. Hodgeson | title=Essential Principles of Physics| publisher=John Murray|edition=2nd| year=1978 | isbn=0-7195-3382-1}}
* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| url=https://archive.org/details/cambridgehandboo0000woan| url-access=registration| publisher=Cambridge University Press| year=2010| isbn=978-0-521-57507-2}}
* {{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=Mc Graw Hill| year=1988| isbn=978-0-07-025734-4}}
* {{cite book|pages=12–13|author1=R.G. Lerner|author1-link=Rita G. Lerner |author2=G.L. Trigg | title=Encyclopaedia of Physics| publisher=VHC Publishers, Hans Warlimont, Springer|edition=2nd| year=2005| isbn=978-0-07-025734-4}}
* {{cite book| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| year=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park}}
* {{cite book|author1=P.A. Tipler |author2=G. Mosca | title=Physics for Scientists and Engineers: With Modern Physics| publisher=W.H. Freeman and Co|edition=6th| year=2008| isbn=978-1-4292-0265-7}}
* {{cite book|title=Analytical Mechanics|author1=L.N. Hand |author2=J.D. Finch |publisher=Cambridge University Press |year=2008|isbn=978-0-521-57572-0}}
* {{cite book|title=Mechanics, Vibrations and Waves|author1=T.B. Arkill |author2=C.J. Millar |publisher=John Murray |year=1974|isbn=0-7195-2882-8}}
* {{cite book|title=The Physics of Vibrations and Waves|edition=3rd|author=H.J. Pain|publisher=John Wiley & Sons |year=1983|isbn=0-471-90182-2}}
* {{cite book|title=Dynamics and Relativity|author1=J.R. Forshaw |author2=A.G. Smith |publisher=Wiley |year=2009|isbn=978-0-470-01460-8}}
* {{cite book|title=Electricity and Modern Physics |edition=2nd|author=G.A.G. Bennet|publisher=Edward Arnold (UK)|year=1974|isbn=0-7131-2459-8}}
* {{cite book|title=Electromagnetism|url=https://archive.org/details/electromagnetism0000gran|url-access=registration|edition=2nd|author1=I.S. Grant |author2=W.R. Phillips |author3=Manchester Physics |publisher=John Wiley & Sons|year=2008|isbn=978-0-471-92712-9}}
* {{cite book|title=Introduction to Electrodynamics|edition=3rd |author=D.J. Griffiths|publisher=Pearson Education, Dorling Kindersley |year=2007|isbn=978-81-7758-293-2}}

==Further reading==
* {{cite book|title=Physics with Modern Applications|author=L.H. Greenberg|publisher=Holt-Saunders International W.B. Saunders and Co|year=1978|isbn=0-7216-4247-0|url-access=registration|url=https://archive.org/details/physicswithmoder0000gree}}
* {{cite book|title=Principles of Physics|author1=J.B. Marion |author2=W.F. Hornyak |publisher=Holt-Saunders International Saunders College|year=1984|isbn=4-8337-0195-2}}
* {{cite book|title=Concepts of Modern Physics|edition=4th|author=A. Beiser|publisher=McGraw-Hill (International)|year=1987|isbn=0-07-100144-1}}
* {{cite book|title=University Physics – With Modern Physics|edition=12th|author1=H.D. Young |author2=R.A. Freedman |publisher=Addison-Wesley (Pearson International)|year=2008|isbn=978-0-321-50130-1}}

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