List of electromagnetism equations#Definitions

This article summarizes equations in the theory of electromagnetism.

Definitions

File:Lorentz force particle.svg (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=

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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)

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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=

File:Universal charge distribution.svg 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.{{Citation needed|date=August 2024}}

Electric transport

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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

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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

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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

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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

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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)

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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

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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,{{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}} $\mathbf{H} = {\mathbf{F}\over q_m} \,$ \| 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]

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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,{{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}}

$\mathbf{H} = {\mathbf{F}\over q_m} \,$

| 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

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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

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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

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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 \| $\tau_L = {L\over R}\,\!$ \| s \| [T]

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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

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=

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, $\mathcal{F}, \mathcal{M}$ \| $\mathcal{M} = NI$ N = number of turns of conductor \| A \| [I]

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, $\mathcal{F}, \mathcal{M}$

| $\mathcal{M} = NI$

N = number of turns of conductor

| A

| [I]

Electromagnetism

=Electric fields=

File:Electrostatics_relation_triangle.svg

General Classical Equations

class="wikitable"
scope="col" width="200" \| Physical situation ! scope="col" width="10" \| 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 \| $\mathbf E(\mathbf r) = {q\over 4\pi \varepsilon_0}{\hat\mathbf{r}\over{\|\mathbf r$

class="wikitable"

scope="col" width="200" | Physical situation

! scope="col" width="10" | 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

| $\mathbf E(\mathbf r) = {1\over4\pi\varepsilon_0} \sum_{i=1}^n q_i {\hat\mathbf r_i\over$

\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}

!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=

File:Magnetostatics_relation_triangle.svg

General classical equations

class="wikitable"
scope="col" width="200" \| Physical situation ! scope="col" width="10" \| 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 \| $\boldsymbol{\tau} = \int_V \mathrm{d} \mathbf{m} \times \mathbf{B}$ $U = - \int_V \mathrm{d} \mathbf{m} \cdot \mathbf{B}$

class="wikitable"

scope="col" width="200" | Physical situation

! scope="col" width="10" | 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.

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_i = resistance of resistor or conductor i G_i = conductance of resistor or conductor i {{endplainlist}} \| $R_\mathrm{net} = \sum_{i=1}^{N} R_i\,\!$ ${1\over G_\mathrm{net}} = \sum_{i=1}^{N} {1\over G_i}\,\!$ \| ${1\over R_\mathrm{net}} = \sum_{i=1}^{N} {1\over R_i}\,\!$ $G_\mathrm{net} = \sum_{i=1}^{N} G_i \,\!$
Charge, capacitors, currents \| {{plainlist}} C_i = capacitance of capacitor i q_i = charge of charge carrier i {{endplainlist}} \| $q_\mathrm{net} = \sum_{i=1}^N q_i \,\!$ ${1\over C_\mathrm{net}} = \sum_{i=1}^N {1\over C_i} \,\!$ $I_\mathrm{net} = I_i \,\!$ \| $q_\mathrm{net} = \sum_{i=1}^N q_i \,\!$ $C_\mathrm{net} = \sum_{i=1}^N C_i \,\!$ $I_\mathrm{net} = \sum_{i=1}^N I_i \,\!$
Inductors \| {{plainlist}} 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 {{endplainlist}} \| $L_\mathrm{net} = \sum_{i=1}^N L_i \,\!$ \| ${1\over L_\mathrm{net}} = \sum_{i=1}^N {1\over L_i} \,\!$ $V_i = \sum_{j=1}^N L_{ij} \frac{\mathrm{d}I_j}{\mathrm{d}t} \,\!$

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_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

| {{plainlist}}

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

| {{plainlist}}

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

| $L_\mathrm{net} = \sum_{i=1}^N L_i \,\!$

| ${1\over L_\mathrm{net}} = \sum_{i=1}^N {1\over L_i} \,\!$

$V_i = \sum_{j=1}^N L_{ij} \frac{\mathrm{d}I_j}{\mathrm{d}t} \,\!$

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 $R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E}\,\!$ 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 $I=\frac{\mathcal{E}}{R}e^{-t/\tau_L}=I_0e^{-Rt/L}\,\!$ \|
LC circuits \|Circuit equation $L{\mathrm{d}^2q\over \mathrm{d}t^2} + {q\over C} = \mathcal{E}\,\!$ \|Circuit equation $L{\mathrm{d}^2q\over \mathrm{d}t^2} + {q\over C} = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!$ 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 $U_E = {q^2\over2C} = {q_0^2\cos^2(\omega t + \phi)\over2C}\,\!$ Circuit magnetic potential energy $U_B={q_0^2\sin^2(\omega t + \phi)\over2C}\,\!$
RLC circuits \|Circuit equation $L{\mathrm{d}^2q\over \mathrm{d}t^2} + R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E} \,\!$ \|Circuit equation $L{\mathrm{d}^2q\over \mathrm{d}t^2} + R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!$ Circuit charge $q = q_0 eT^{-Rt/2L}\cos(\omega't+\phi)\,\!$

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

$R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E}\,\!$

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

$I=\frac{\mathcal{E}}{R}e^{-t/\tau_L}=I_0e^{-Rt/L}\,\!$

LC circuits

|Circuit equation

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

|Circuit equation

$L{\mathrm{d}^2q\over \mathrm{d}t^2} + {q\over C} = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!$

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

$U_E = {q^2\over2C} = {q_0^2\cos^2(\omega t + \phi)\over2C}\,\!$

Circuit magnetic potential energy

$U_B={q_0^2\sin^2(\omega t + \phi)\over2C}\,\!$

RLC circuits

|Circuit equation

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

|Circuit equation

$L{\mathrm{d}^2q\over \mathrm{d}t^2} + R{\mathrm{d}q\over \mathrm{d}t} + {q\over C} = \mathcal{E} \sin\left(\omega_0 t + \phi \right) \,\!$

Circuit charge

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

Footnotes

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}}

List of electromagnetism equations#Definitions

Definitions

=Initial quantities=

=Electric quantities=

=Magnetic quantities=

=Electric circuits=

=Magnetic circuits=

Electromagnetism

=Electric fields=

=Magnetic fields and moments=

Electric circuits and electronics

See also

Footnotes

Sources

Further reading