# 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) = μ$_0$ 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.
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 m$^2$ 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 m$^3$ | [M][L][T]$^{−2}$[I]$^{−1}$ |
| Magnetic flux | Φ$_B$ | $\Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} \,\!$ | Wb = T m$^2$ | [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 m$^2$ | [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, $\mathbf {H} ={\mathbf {F} \over q_{m}}\,$ | 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 | I$_0$ | $I_{0}=I_{\mathrm {rms} }{\sqrt {2}}\,\!$ | 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 | V$_0$ | $V_{0}=V_{\mathrm {rms} }{\sqrt {2}}\,\!$ | 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 |