List of electromagnetism equations explained

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) = μ₀ q_m(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

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 λ _{e} d \ell q _{e} = \iint \sigma _{e} d S q _{e} = \iiint \rho _{e} d V	C m^-n, n = 1, 2, 3	[I][T][L]^-n
Capacitance	C	C = d q/ d V V = voltage, not volume.	F = C V^-1	[I]^2[T]^4[L]^-2[M]^-1
Electric current	I	I = d q/ d t	A	[I]
Electric current density	J	I = J ⋅ d S	A m^-2	[I][L]^-2
Displacement current density	Jd	J _{d} = \epsilon _{0} \left ( \partial E / \partial t \right ) = \partial D / \partial t	A m^-2	[I][L]^-2
Convection current density	Jc	J _{c} = \rho 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	E = F /q	N C^−1 = V m^−1	[M][L][T]^−3[I]^−1
Electric flux	ΦE	\Phi _{E} = \int _{S} E ⋅ d A	N m^2 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	p = q a a = charge separationdirected from -ve to +ve charge	C m	[I][T][L]
Electric Polarization, polarization density	P	P = d \langle p \rangle / d V	C m^−2	[I][T][L]^−2
Electric displacement field, flux density	D	D = \epsilon E = \epsilon _{0} E + P	C m^−2	[I][T][L]^−2
Electric displacement flux	ΦD	\Phi _{D} = \int _{S} D ⋅ d A	C	[I][T]
Absolute electric potential, EM scalar potential relative to point r _{0} Theoretical: r _{0} = inf ty \	Practical: r _{0} = R _{earth} (Earth's radius)	φ,V	V =- (W _{infty})/(q) =- (1)/(q) \int ^{r} _{inf ty} F ⋅ d r = - \int ^{r _{2}} _{r _{1}} E ⋅ d r	V = J C^−1	[M] [L]^2 [T]^−3 [I]^−1
Voltage, Electric potential difference	Δφ,ΔV	\Delta V =- (\Delta W)/(q) =- (1)/(q) \int ^{r _{2}} _{r _{1}} F ⋅ d r = - \int ^{r _{2}} _{r _{1}} E ⋅ d 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 λ _{m} d \ell q _{m} = \iint \sigma _{m} d S q _{m} = \iiint \rho _{m} 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	Im	I _{m} = d q _{m/dt}	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 J _{m} ⋅ d 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	F = q _{e} \left ( v x B \right )	T = N A^−1 m^−1 = Wb m^−2	[M][T]^−2[I]^−1
Magnetic potential, EM vector potential	A	B = \nabla x A	T m = N A^−1 = Wb m^3	[M][L][T]^−2[I]^−1
Magnetic flux	ΦB	\Phi _{B} = \int _{S} B ⋅ d 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, μB, Π	Two definitions are possible: using pole strengths, m = q _{m} a using currents: m = NIA \hat{n }\,\	a = pole separation N is the number of turns of conductor	A m^2	[I][L]^2
Magnetization	M	M = d \langle m \rangle / d V	A m^−1	[I] [L]^−1
Magnetic field intensity, (AKA field strength)	H	Two definitions are possible: most common: B = \mu H = \mu _{0} \left ( H + M \right ) using pole strengths,^[1] H = F / q _{m}	A m^−1	[I] [L]^−1
Intensity of magnetization, magnetic polarization	I, J	I = \mu _{0} 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 ( d \Phi/ d I \right ) L\left ( d I/ 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} ( d \Phi _{2/d} I _{1} \right ) M\left ( d I _{2/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} ( d \Phi _{1/d} I _{2} \right ) M\left ( d I _{1/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

See main article: Direct current.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Terminal Voltage forPower 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	R _{int} = V _{ter/I}	Ω = V A^−1 = J s C^−2	[M][L]^2 [T]^−3 [I]^−2
Load resistance of circuit	Rext	R _{ext} = V _{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	l{E} = V _{ter} + V _{load}	V = J C^−1	[M] [L]^2 [T]^−3 [I]^−1

AC circuits

See main article: Alternating current and Resonance.

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} = (1)/(\omega _{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 = 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 = (X _{L} - X _{C})/(R)	dimensionless	dimensionless
AC peak current	I0	I _{0} = I _{rms} \sqrt{ 2 }	A	[I]
AC root mean square current	Irms	I _{rms} = \sqrt{ (1)/(T) \int ^{T} _{0} \left [ I \left ( t \right ) \right ] ^{2} d t}	A	[I]
AC peak voltage	V0	V _{0} = V _{rms} \sqrt{ 2 }	V = J C^−1	[M] [L]^2 [T]^−3 [I]^−1
AC root mean square voltage	Vrms	V _{rms} = \sqrt{ (1)/(T) \int ^{T} _{0} \left [ V \left ( t \right ) \right ] ^{2} d t}	V = J C^−1	[M] [L]^2 [T]^−3 [I]^−1
AC emf, root mean square	l{E} _{rms,} \sqrt{\langle l{E} \rangle}	l{E} _{rms=l{E}} _{m/\sqrt{2}}	V = J C^−1	[M] [L]^2 [T]^−3 [I]^−1
AC average power	\langle P \rangle	\langle P \rangle = l{E}I _{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

See main article: Magnetic circuits.

Electromagnetism

Electric fields

General Classical Equations

Physical situation	Equations
Electric potential gradient and field	E = - \nabla V \Delta V = - \int ^{r _{2}} _{r _{1}} E ⋅ d r
Point charge	E = (q)/(4 \pi \epsilon _{0} \left | r \right | ^{2}) \hat{r } \,\!
At a point in a local array of point charges	E = \sum E _{i} = (1)/(4 \pi \epsilon _{0}) \sum _{i} (q _{i})/(\left | r _{i} - r \right | ^{2}) \hat{r }_i \,\!
At a point due to a continuum of charge	E = (1)/(4 \pi\epsilon _{0}) \int _{V} (r \rho d V)/(\left | r \right | ^{3})
Electrostatic torque and potential energy due to non-uniform fields and dipole moments	\boldsymbol{\tau} = \int _{V} d p x E U = - \int _{V} d p ⋅ E

Magnetic fields and moments

See also: Magnetic moment.

General classical equations

Physical situation	Equations
Magnetic potential, EM vector potential	B = \nabla x A
Due to a magnetic moment	A = (\mu _{0})/(4 \pi) (m x r)/(\left | r \right | ^{3}) B ({ r })=\nabla\times=\frac\left(\frac-\frac\right)
Magnetic moment due to a current distribution	m = (1)/(2) \int _{V} r x J d V
Magnetostatic torque and potential energy due to non-uniform fields and dipole moments	\boldsymbol{\tau} = \int _{V} d m x B U = - \int _{V} d m ⋅ 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	R _{net} = \sum ^{N} _{i =1} R _{i} { 1 \over G _{net} } = \sum_^ \,\		{ 1 \over R _{net} } = \sum_^ \,\!G _{net} = \sum ^{N} _{i =1} G _{i} \
Charge, capacitors, currents	Ci = capacitance of capacitor i qi = charge of charge carrier i	q _{net} = \sum ^{N} _{i =1} q _{i} { 1 \over C _{net} } = \sum_^N \,\	I _{net} = I _{i}	q _{net} = \sum ^{N} _{i =1} q _{i} C _{net} = \sum ^{N} _{i =1} C _{i} \	I _{net} = \sum ^{N} _{i =1} 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 _{net} = \sum ^{N} _{i =1} L _{i}	{ 1 \over L _{net} } = \sum_^N \,\!V _{i} = \sum ^{N} _{j =1} L _{ij} (d I _{j})/(d t) \

Circuit	DC Circuit equations	AC Circuit equations	+Series circuit equations
RC circuits	Circuit equationR (d q)/(d t) + (q)/(C) = l{E} Capacitor charge q = C l{E}\left ( 1 - e ^{-t/RC} \right )\	Capacitor discharge q = C l{E}e ^{-t/RC}
RL circuits	Circuit equationL (d I)/(d t) + RI = l{E} Inductor current rise I = (l{E)/() }\left (1-e^\right)\,\	Inductor current fall I = (l{E)/() }e^=I_0e^\,\!
LC circuits	Circuit equationL (d ^{2q})/(d t ^{2}) + q/C = l{E}	Circuit equationL (d ^{2q})/(d t ^{2}) + q/C = l{E} \sin\left(\omega _{0} t + \phi \right) Circuit resonant frequency \omega _{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 ^{2 / 2 C = Q} _{E = q} ^{2\cos} ^{2(\omega} t + \phi)/ 2 C Circuit magnetic potential energy U ^{2 \sin} _{B = Q} ^{2(\omega} t + \phi)/ 2 C\
RLC Circuits	Circuit equationL (d ^{2q})/(d t ^{2}) + R (d q)/(d t) + (q)/(C) = l{E}	Circuit equationL (d ^{2q})/(d t ^{2}) + R (d q)/(d t) + (q)/(C) = l{E} \sin\left(\omega _{0} t + \phi \right) Circuit charge q = q _{0} eT ^{-Rt/2L} \cos(\omega't + \phi)\

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

Sources

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

Further reading

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

Notes and References

1. Book: Understanding Physics. 2nd . M. Mansfield . C. O'Sullivan . John Wiley & Sons. 2011. 978-0-470-74637-0.