List of electromagnetism equations

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Part of a series of articles about

Electromagnetism

* Electricity
* Magnetism

Electrostatics

* Conductor
* Coulomb's law
* Electric charge
* Electric dipole moment
* Electric field
* Electric flux / potential energy
* Electrostatic discharge
* Gauss's law
* Induction
* Insulator
* Polarization density
* Static electricity
* Triboelectricity

Magnetostatics

* Ampère's law
* Biot–Savart law
* Gauss's law for magnetism
* Magnetic field
* Magnetic flux
* Magnetic dipole moment
* Magnetization
* Magnetomotive force

Electrodynamics

* Lorentz force law
* Electromagnetic induction
* Faraday's law
* Lenz's law
* Displacement current
* Magnetic potential
* Maxwell's equations
* Electromagnetic field
* Electromagnetic pulse
* Electromagnetic radiation
* Maxwell tensor
* Poynting vector
* Liénard–Wiechert potential
* Jefimenko's equations
* Eddy current
* London equations

* Mathematical descriptions of the electromagnetic field

Electrical network

* Alternating current
* Capacitance
* Direct current
* Electric current
* Electric potential
* Electromotive force
* Impedance
* Inductance
* Ohm's law
* Parallel circuit
* Resistance
* Resonant cavities
* Series circuit
* Voltage
* Waveguides

Covariant formulation

* Electromagnetic tensor
(stress–energy tensor)

* Four-current
* Electromagnetic four-potential

Scientists

* Ampère
* Biot
* Coulomb
* Faraday
* Gauss
* Heaviside
* Henry
* Hertz
* Lenz
* Lorentz
* Maxwell
* Ørsted
* Savart
* Tesla
* Volta
* Weber

* v
* t
* e

This article summarizes equations in the theory of electromagnetism.

## Contents

* 1 Definitions
* 1.1 Initial quantities
* 1.2 Electric quantities
* 1.3 Magnetic quantities
* 1.4 Electric circuits
* 1.5 Magnetic circuits
* 2 Electromagnetism
* 2.1 Electric fields
* 2.2 Magnetic fields and moments
* 3 Electric circuits and electronics
* 4 See also
* 5 Footnotes
* 6 Sources
* 7 Further reading

## Definitions

Lorentz force on a charged particle (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 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 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.

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.  | qe=∫λedℓ

qe=∬σedS

qe=∭ρedV

| C m−n, n = 1, 2, 3  | [I][T][L]−n
Capacitance | C | C=dqdV

V = voltage, not volume.

| F = C V−1 | [I]2[T]4[L]−2[M]−1
Electric current | I | I=dqdt | A  | [I]
Electric current density | J | I=𝐉⋅d𝐒 | A m−2 | [I][L]−2
Displacement current density  | Jd | 𝐉d=∂𝐃∂t=ε0(∂𝐄∂t) | A m−2 | [I][L]−2
Convection current density  | Jc | 𝐉c=ρ𝐯 | 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 | 𝐄=𝐅q | N C−1 = V m−1 | [M][L][T]−3[I]−1
Electric flux | ΦE | ΦE=∫S𝐄⋅d𝐀 | N m2 C−1 | [M][L]3[T]−3[I]−1
Absolute permittivity;  | ε | ε=εrε0 | F m−1 | [I]2 [T]4 [M]−1 [L]−3
Electric dipole moment | p | 𝐩=q𝐚

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

| C m  | [I][T][L]
Electric Polarization, polarization density | P | 𝐏=d⟨𝐩⟩dV | C m−2 | [I][T][L]−2
Electric displacement field, flux density  | D | 𝐃=ε𝐄=ε0𝐄+𝐏 | C m−2 | [I][T][L]−2
Electric displacement flux  | ΦD | ΦD=∫S𝐃⋅d𝐀 | C  | [I][T]
Absolute electric potential, EM scalar potential relative to point r0

Theoretical: r0=∞
Practical: r0=Rearth (Earth's radius)

| φ ,V | V=−W∞rq=−1q∫∞r𝐅⋅d𝐫=−∫r1r2𝐄⋅d𝐫 | V = J C−1 | [M] [L]2 [T]−3 [I]−1
Voltage, Electric potential difference  | Δφ,ΔV | ΔV=−ΔWq=−1q∫r1r2𝐅⋅d𝐫=−∫r1r2𝐄⋅d𝐫 | 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.  | qm=∫λmdℓ

qm=∬σmdS

qm=∭ρmdV

| Wb m−n

A m(−n \+ 1),
n = 1, 2, 3

| [L]2[M][T]−2 [I]−1 (Wb)

[I][L] (Am)

Monopole current  | Im | Im=dqmdt | 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=∬𝐉m⋅d𝐀 | 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 | 𝐅=qe(𝐯×𝐁) | T = N A−1 m−1 = Wb m−2 | [M][T]−2[I]−1
Magnetic potential, EM vector potential  | A | 𝐁=∇×𝐀 | T m = N A−1 = Wb m3 | [M][L][T]−2[I]−1
Magnetic flux | ΦB | ΦB=∫S𝐁⋅d𝐀 | Wb = T m2 | [L]2[M][T]−2[I]−1
Magnetic permeability  | μ | μ =μrμ0 | 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,
𝐦=qm𝐚

using currents:
𝐦=NIAn^

a = pole separation

N is the number of turns of conductor

| A m2 | [I][L]2
Magnetization | M | 𝐌=d⟨𝐦⟩dV | A m−1 | [I] [L]−1
Magnetic field intensity, (AKA field strength)  | H | Two definitions are possible:

most common:
𝐁=μ𝐇=μ0(𝐇+𝐌)

using pole strengths,[1]
𝐇=𝐅qm

| A m−1 | [I] [L]−1
Intensity of magnetization, magnetic polarization  | I, J | 𝐈=μ0𝐌 | T = N A−1 m−1 = Wb m−2 | [M][T]−2[I]−1
Self Inductance | L | Two equivalent definitions are possible:

L=N(dΦdI)

L(dIdt)=−NV

| H = Wb A−1 | [L]2 [M] [T]−2 [I]−2
Mutual inductance | M | Again two equivalent definitions are possible:

M1=N(dΦ2dI1)

M(dI2dt)=−NV1

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;

M2=N(dΦ1dI2)
M(dI1dt)=−NV2

| H = Wb A−1 | [L]2 [M] [T]−2 [I]−2
Gyromagnetic ratio (for charged particles in a magnetic field)  | γ | ω=γB | Hz T−1 | [M]−1[T][I]

### Electric circuits

DC circuits, general definitions

Main article: Physics:Direct current

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Terminal Voltage for power supply  | 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 | Rint=VterI | Ω = V A−1 = J s C−2 | [M][L]2 [T]−3 [I]−2
Load resistance of circuit  | Rext | Rext=VloadI | 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 | ℰ=Vter+Vload | V = J C−1 | [M] [L]2 [T]−3 [I]−1

AC circuits

Main articles: Physics:Alternating current and Resonance

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Resistive load voltage  | VR | VR=IRR | V = J C−1 | [M] [L]2 [T]−3 [I]−1
Capacitive load voltage  | VC | VC=ICXC | V = J C−1 | [M] [L]2 [T]−3 [I]−1
Inductive load voltage  | VL | VL=ILXL | V = J C−1 | [M] [L]2 [T]−3 [I]−1
Capacitive reactance  | XC | XC=1ωdC | Ω−1 m−1 | [I]2 [T]3 [M]−2 [L]−2
Inductive reactance  | XL | XL=ωdL | Ω−1 m−1 | [I]2 [T]3 [M]−2 [L]−2
AC electrical impedance | Z | V=IZ

Z=R2+(XL−XC)2

| Ω−1 m−1 | [I]2 [T]3 [M]−2 [L]−2
Phase constant  | δ, φ | tan⁡ϕ=XL−XCR | dimensionless  | dimensionless
AC peak current  | I0 | I0=Irms2 | A  | [I]
AC root mean square current  | Irms | Irms=1T∫0T[I(t)]2dt | A  | [I]
AC peak voltage  | V0 | V0=Vrms2 | V = J C−1 | [M] [L]2 [T]−3 [I]−1
AC root mean square voltage  | Vrms | Vrms=1T∫0T[V(t)]2dt | V = J C−1 | [M] [L]2 [T]−3 [I]−1
AC emf, root mean square  | ℰrms,⟨ℰ⟩ | ℰrms=ℰm/2 | V = J C−1 | [M] [L]2 [T]−3 [I]−1
AC average power  | ⟨P⟩ | ⟨P⟩=ℰIrmscos⁡ϕ | W = J s−1 | [M] [L]2 [T]−3
Capacitive time constant  | τC | τC=RC | s  | [T]
Inductive time constant  | τL | τL=LR | s  | [T]

### Magnetic circuits

Quantity (common name/s)  | (Common) symbol/s  | Defining equation  | SI units  | Dimension

|
|
|
|

Magnetomotive force, mmf  | F, ℱ,ℳ | ℳ=NI

N = number of turns of conductor

| A  | [I]

## Electromagnetism

### Electric fields

Summary of electrostatic relations between electric potential, electric field and charge density. Here, 𝐫=𝐱−𝐱′.

General Classical Equations

Physical situation  | Equations

|

Electric potential gradient and field  | 𝐄=−∇V

ΔV=−∫r1r2𝐄⋅d𝐫

Point charge  | 𝐄(𝐫)=q4πε0𝐫^|𝐫|2=q4πε0𝐫|𝐫|3
At a point in a local array of point charges  | 𝐄(𝐫)=14πε0∑i=1nqi𝐫^i|r𝐢−𝐫|2=14πε0∑i=1nqi𝐫i|r𝐢−𝐫|3
At a point due to a continuum of charge  | 𝐄(𝐫)=14πε0∭ρ(𝐫′)𝐫′|𝐫′|3d3|𝐫′|
Electrostatic torque and potential energy due to non-uniform fields and dipole moments  | τ=∫Vd𝐩×𝐄

U=−∫Vd𝐩⋅𝐄

### Magnetic fields and moments

See also: Physics:Magnetic moment

Summary of magnetostatic relations between magnetic vector potential, magnetic field and current density. Here, 𝐫=𝐱−𝐱′.

General classical equations

Physical situation  | Equations

|

Magnetic potential, EM vector potential  | 𝐁=∇×𝐀
Due to a magnetic moment  | 𝐀=μ04π𝐦×𝐫|𝐫|3

𝐁(𝐫)=∇×𝐀=μ04π(3𝐫(𝐦⋅𝐫)|𝐫|5−𝐦|𝐫|3)

Magnetic moment due to a current distribution  | 𝐦=12∫V𝐫×𝐉dV
Magnetostatic torque and potential energy due to non-uniform fields and dipole moments  | τ=∫Vd𝐦×𝐁

U=−∫Vd𝐦⋅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  |

* Ri = resistance of resistor or conductor i
* Gi = conductance of resistor or conductor i

| Rnet=∑i=1NRi

1Gnet=∑i=1N1Gi

| 1Rnet=∑i=1N1Ri

Gnet=∑i=1NGi

Charge, capacitors, currents  |

* Ci = capacitance of capacitor i
* qi = charge of charge carrier i

| qnet=∑i=1Nqi

1Cnet=∑i=1N1Ci Inet=Ii

| qnet=∑i=1Nqi

Cnet=∑i=1NCi Inet=∑i=1NIi

Inductors  |

* Li = self-inductance of inductor i
* Lij = self-inductance element ij of L matrix
* Mij = mutual inductance between inductors i and j

| Lnet=∑i=1NLi | 1Lnet=∑i=1N1Li

Vi=∑j=1NLijdIjdt

Circuit  | DC Circuit equations  | AC Circuit equations

|
|

Series circuit equations  RC circuits  | Circuit equation

Rdqdt+qC=ℰ

Capacitor charge q=Cℰ(1−e−t/RC)

Capacitor discharge q=Cℰe−t/RC

RL circuits  | Circuit equation

LdIdt+RI=ℰ

Inductor current rise I=ℰR(1−e−Rt/L)

Inductor current fall I=ℰRe−t/τL=I0e−Rt/L

LC circuits | Circuit equation

Ld2qdt2+qC=ℰ

| Circuit equation

Ld2qdt2+qC=ℰsin⁡(ω0t+ϕ)

Circuit resonant frequency ωres=1LC

Circuit charge q=q0cos⁡(ωt+ϕ)

Circuit current I=−ωq0sin⁡(ωt+ϕ)

Circuit electrical potential energy UE=q22C=q02cos2(ωt+ϕ)2C

Circuit magnetic potential energy UB=q02sin2(ωt+ϕ)2C

RLC circuits  | Circuit equation

Ld2qdt2+Rdqdt+qC=ℰ

| Circuit equation

Ld2qdt2+Rdqdt+qC=ℰsin⁡(ω0t+ϕ)

Circuit charge

q=q0eT−Rt/2Lcos⁡(ω′t+ϕ)

## 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

1. ↑ M. Mansfield; C. O'Sullivan (2011). Understanding Physics (2nd ed.). John Wiley & Sons. ISBN 978-0-470-74637-0.

## Sources

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

## Further reading

* L.H. Greenberg (1978). Physics with Modern Applications. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0. https://archive.org/details/physicswithmoder0000gree.
* J.B. Marion; W.F. Hornyak (1984). Principles of Physics. Holt-Saunders International Saunders College. ISBN 4-8337-0195-2.
* A. Beiser (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
* H.D. Young; R.A. Freedman (2008). University Physics – With Modern Physics (12th ed.). Addison-Wesley (Pearson International). ISBN 978-0-321-50130-1.

* v
* t
* e

SI units

Base units|

* ampere
* candela
* kelvin
* kilogram
* {{{sp}}}}}|us|er|re}}
* 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
* {{{sp}}}}}|us|er|re}}
* minute
* minute and second of arc
* neper
* tonne

See also|

* Conversion of units
* Metric prefixes
* 2005–2019 definition
* 2019 redefinition
* Systems of measurement

* Book
* Category

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