# List of electromagnetism equations

## 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 qm(Wb) = μ0 qm(Am).

### Initial quantities

Quantity (common name/s) (Common) symbol/s SI units Dimension
Electric charge qe, q, Q C = As [I][T]
Monopole strength, magnetic charge qm, 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 , 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 λe for Linear, σe for surface, ρe 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 mn, 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 Jd $\mathbf {J} _{\mathrm {d} }=\epsilon _{0}\left(\partial \mathbf {E} /\partial t\right)=\partial \mathbf {D} /\partial t\,\!$  A m−2 [I][L]−2
Convection current density Jc $\mathbf {J} _{\mathrm {c} }=\rho \mathbf {v} \,\!$  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 $\mathbf {E} =\mathbf {F} /q\,\!$  N C−1 = V m−1 [M][L][T]−3[I]−1
Electric flux ΦE $\Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!$  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 $\mathbf {p} =q\mathbf {a} \,\!$

a = charge separation directed from -ve to +ve charge

C m [I][T][L]
Electric Polarization, polarization density P $\mathbf {P} =\mathrm {d} \langle \mathbf {p} \rangle /\mathrm {d} V\,\!$  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 ΦD $\Phi _{D}=\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} \,\!$  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 ΔφV $\Delta V=-{\frac {\Delta W}{q}}=-{\frac {1}{q}}\int _{r_{1}}^{r_{2}}\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!$  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 λm for Linear, σm for surface, ρm 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 mn

A m(−n + 1),
n = 1, 2, 3

[L]2[M][T]−2 [I]−1 (Wb)

[I][L] (Am)

Monopole current Im $I_{m}=\mathrm {d} q_{m}/\mathrm {d} t\,\!$  Wb s−1

A m s−1

[L]2[M][T]−3 [I]−1 (Wb)

[I][L][T]−1 (Am)

Monopole current density Jm $I=\iint \mathbf {J} _{\mathrm {m} }\cdot \mathrm {d} \mathbf {A}$  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 ΦB $\Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} \,\!$  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, μB, 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,
$\mathbf {H} =\mathbf {F} /q_{m}\,$

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

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Terminal Voltage for Vter V = J C−1 [M] [L]2 [T]−3 [I]−1
Load Voltage for Circuit Vload V = J C−1 [M] [L]2 [T]−3 [I]−1
Internal resistance of power supply Rint $R_{\mathrm {int} }=V_{\mathrm {ter} }/I\,\!$  Ω = V A−1 = J s C−2 [M][L]2 [T]−3 [I]−2
Load resistance of circuit Rext $R_{\mathrm {ext} }=V_{\mathrm {load} }/I\,\!$  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

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Resistive load voltage VR $V_{R}=I_{R}R\,\!$  V = J C−1 [M] [L]2 [T]−3 [I]−1
Capacitive load voltage VC $V_{C}=I_{C}X_{C}\,\!$  V = J C−1 [M] [L]2 [T]−3 [I]−1
Inductive load voltage VL $V_{L}=I_{L}X_{L}\,\!$  V = J C−1 [M] [L]2 [T]−3 [I]−1
Capacitive reactance XC $X_{C}={\frac {1}{\omega _{\mathrm {d} }C}}\,\!$  Ω−1 m−1 [I]2 [T]3 [M]−2 [L]−2
Inductive reactance XL $X_{L}=\omega _{d}L\,\!$  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 Irms $I_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$  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 Vrms $V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$  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 τC $\tau _{C}=RC\,\!$  s [T]
Inductive time constant τL $\tau _{L}=L/R\,\!$  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 $\mathbf {E} ={\frac {q}{4\pi \epsilon _{0}\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!$
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

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

$U=\int _{V}\mathrm {d} \mathbf {m} \cdot \mathbf {B}$

### 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
• Ri = resistance of resistor or conductor i
• Gi = 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
• Ci = capacitance of capacitor i
• qi = 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
• Li = self-inductance of inductor i
• Lij = self-inductance element ij of L matrix
• Mij = mutual inductance between inductors i and j
$L_{\mathrm {net} }=\sum _{i=1}^{N}L_{i}\,\!$  ${1 \over L_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over L_{i}}\,\!$

$V_{i}=\sum _{j=1}^{N}L_{ij}{\frac {\mathrm {d} I_{j}}{\mathrm {d} t}}\,\!$

Circuit DC Circuit equations AC Circuit equations
Series 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 $I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-Rt/L}\,\!$

LC circuits Circuit equation

$L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\,\!$

Circuit equation

$L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$

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

$L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!$

Circuit equation

$L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$

Circuit charge

$q=q_{0}eT^{-Rt/2L}\cos(\omega 't+\phi )\,\!$