nLab Maxwell's equations




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In the context of electromagnetism, Maxwell’s equations are the equations of motion for the electromagnetic field strength electric current and magnetic current.

Three dimensional formulation

EE is here the (vector of) strength of electric field and BB the strength of magnetic field; QQ is the charge and j elj_{el} the density of the electrical current; ϵ 0\epsilon_0, cc, μ 0\mu_0 are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: μ 0ϵ 0=1/c 2\mu_0 \epsilon_0 = 1/c^2).

Integral formulation in vacuum

Gauss' law for electric fields

VEdA=Qϵ 0 \textstyle{\int}_{\partial V} E\cdot d A = \frac{Q}{\epsilon_0}

where V\partial V is a closed surface which is a boundary of a 3d domain VV (physicists say “volume”) and Q= VρdVQ = \textstyle{\int}_V \rho d V the charge in the domain VV; \cdot denotes the scalar (dot) product. Surface element dAd A is nd|A|\vec{n} d |A|, i.e. it is the scalar surface measure times the unit vector of normal outwards.

No magnetic monopoles (Gauss’ law for magnetic fields)

ΣBdA=0 \textstyle{\int}_\Sigma B\cdot d A = 0

where Σ\Sigma is any closed surface.

Faraday’s law of induction

ΣEds=ddt ΣBdA \textstyle{\oint}_{\partial \Sigma} E\cdot d s = - \frac{d}{d t} \textstyle{\int}_\Sigma B\cdot d A

The line element dsd s is the differential (or 1-d measure on the boundary) of the length times the unit vector in counter-circle direction (or parametrize the curve with ss being a vector in 3d space, express magnetic field in the same parameter and calculate the integral as a function of parameter: \cdot is a scalar (“dot”) product).

Ampère-Maxwell law (or generalized Ampère’s law; Maxwell added the second term involving derivative of the flux of electric field to the Ampère’s law which described the magnetic field due electric current).

ΣBds=μ 0I+μ 0ϵ 0ddt ΣEdA \textstyle{\oint}_{\partial \Sigma} B\cdot d s = \mu_0 I + \mu_0 \epsilon_0 \frac{d}{d t} \textstyle{\int}_\Sigma E\cdot d A

where Σ\Sigma is a surface and Σ\partial \Sigma its boundary; II is the total current through Σ\Sigma (integral of the component of j elj_{el} normal to the surface).

Differential equations

Here we put units with c=1c = 1. By ρ\rho we denote the density of the charge.

In pregeometric form, Maxwell’s equations are differential equations for four fields (vector fields) called

  • electric field E\vec E

  • magnetic flux density B\vec B

  • magnetic field H\vec H

  • displacement field D\vec D (or similar)

and state:

  • (magnetic Gauss law)

    divB=0div B = 0

    (“there is no source for magnetic flux, hence no magnetic monopoles”)

  • Faraday’s law:

    ddtB+rotE=0\frac{d}{d t} B + rot E = 0

  • (electric Gauss law):

    divD=ρdiv D = \rho

    (“the source of electric flux is electric current”)

  • generalized Ampère’s law

    ddtD+rotH=j el- \frac{d}{d t} D + rot H = j_{el}

In order to complete this pregeometric form to the actual equations of motion of the electromagnetic field, these four fields are to be subjected to a constraint called the constitutive equation which expresses (D,H)(\vec D, \vec H) as a function of (E,B)(\vec E, \vec B).

In vacuum and in the absence of background gravity, this constitutive relation:

  • equates the displacement field D\vec D with the electric field E\vec E times a constant ϵ 0\epsilon_0

    called the “permitivity of the vacuum”,

  • equates the magnetic flux B\vec B with the magnetic field H\vec H times a constant μ 0\mu_0

    called the “permeability of the vacuum”.

But for electromagnetic fields inside dielectric media other constitutive relations appear. For small field strengths these are typically linear functions CC (e.g. de Lange & Raab 2006 (19))

(D,H)=!C((E,B)), (\vec D, \vec H) \;\overset{!}{=}\; C\big((\vec E, \vec B)\big) \,,

but in general the constitutive relation can be a non-linear or even be a “multi-valued function” (namely when there are hysteresis effects in the dielectric medium).

Similarly (interestingly), in the presence of background gravity (such as for electromagnetic fields in and around a star) there is a linear such relation depending on the pseudo-Riemannian metric. This is most transparently expressed in terms of the Hodge star operator acting on the electromagnetic fields re-packaged as a Faraday tensor differential 2-form (see below).

In terms of Faraday tensor FF

This is adapted from electromagnetic field – Maxwell’s equations, for more see the references below.

In modern language, the insight of (Maxwell, 1865) is that locally, when physical spacetime is well approximated by a patch of its tangent space, i.e. by a patch of 4-dimensional Minkowski space U( 4,g=diag(1,1,1,1))U \subset (\mathbb{R}^4, g = diag(-1,1,1,1)), the electric field E=[E 1 E 2 E 3]\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right] and magnetic field B=[B 1 B 2 B 3]\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right] combine into a differential 2-form

F Edt+B E 1dx 1dt+E 2dx 2dt+E 3dx 3dt +B 1dx 2dx 3+B 2dx 3dx 1+B 3dx 1dx 2 \begin{aligned} F & \coloneqq E \wedge d t + B \\ & \coloneqq E_1 d x^1 \wedge d t + E_2 d x^2 \wedge d t + E_3 d x^3 \wedge d t \\ & \phantom{\coloneqq} + B_1 d x^2 \wedge d x^3 + B_2 d x^3 \wedge d x^1 + B_3 d x^1 \wedge d x^2 \end{aligned}

in Ω 2(U)\Omega^2(U) and the electric charge density and current density combine to a differential 3-form

j el jdtρdx 1dx 2dx 3 j 1dx 2dx 3dt+j 2dx 3dx 1dt+j 3dx 1dx 2dtρdx 1dx 2dx 3 \begin{aligned} j_{el} & \coloneqq j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\ & \coloneqq j_1 d x^2 \wedge d x^3 \wedge d t + j_2 d x^3 \wedge d x^1 \wedge d t + j_3 d x^1 \wedge d x^2 \wedge d t - \rho \; d x^1 \wedge d x^2 \wedge d x^3 \end{aligned}

in Ω 3(U)\Omega^3(U) such that the following two equations of differential forms are satisfied

dF=0 dF=j el, \begin{aligned} d F = 0 \\ d \star F = j_{el} \end{aligned} \,,

where dd is the de Rham differential operator and \star the Hodge star operator. If we decompose F\star F into its components as before as

F =D+Hdt =D 1dx 2dx 3D 2dx 3dx 1D 3dx 1dx 2 =+H 1dx 1dt+H 2dx 2dt+H 3dx 3dt \begin{aligned} \star F &= -D + H\wedge dt \\ &= -D_1 \; d x^2 \wedge d x^3 -D_2 \; d x^3 \wedge d x^1 -D_3 \; d x^1 \wedge d x^2 \\ & \phantom{=} + H_1 \; d x^1 \wedge d t + H_2 \; d x^2 \wedge d t + H_3 \; d x^3 \wedge d t \end{aligned}

then in terms of these components the Maxwell equations read as follows:

dF=0d F = 0 gives the magnetic Gauss law and Faraday’s law

dF=j eld \star F = j_{el} gives Gauss's law and Ampère-Maxwell law

In terms of the Faraday bivector FF

In the geometric algebra formalism of electromagnetism, one works in the 4-dimensional Clifford algebra Cl 1,3()\mathrm{Cl}^{1, 3}(\mathbb{R}) representing spacetime, where the orthonormal basis vectors {γ i}\{\gamma_i\} have signature (+,,,)(+, -, -, -) with γ 0\gamma_0 representing the time dimension and the other three basis vectors representing the spacial dimensions. The pseudoscalar of Cl 1,3()\mathrm{Cl}^{1, 3}(\mathbb{R}) is represented by the product of all the basis vectors

I= iγ i=γ 0γ 1γ 2γ 3I = \prod_{i} \gamma_i = \gamma_0 \gamma_1 \gamma_2 \gamma_3

The basis bivectors of Cl 1,3()\mathrm{Cl}^{1, 3}(\mathbb{R}) come in two sets, the timelike bivectors σ 1=γ 1γ 0\sigma_1 = \gamma_1 \gamma_0, σ 2=γ 2γ 0\sigma_2 = \gamma_2 \gamma_0, σ 3=γ 3γ 0\sigma_3 = \gamma_3 \gamma_0, and the spacelike bivectors Iσ 1=γ 2γ 3I \sigma_1 = -\gamma_2 \gamma_3, Iσ 2=γ 1γ 3I \sigma_2 = \gamma_1 \gamma_3, Iσ 3=γ 1γ 2I \sigma_3 = \gamma_1 \gamma_2. The bivector subalgebra of Cl 1,3()\mathrm{Cl}^{1, 3}(\mathbb{R}) is equivalent to the three-dimensional Clifford algebra Cl 3,0()\mathrm{Cl}^{3, 0}(\mathbb{R}) corresponding to the relative space in the rest frame defined by γ 0\gamma_0, where the basis timelike bivectors correspond to the basis relative vectors and the basis spacelike bivectors correspond to the basis relative bivectors of the rest frame.

Let xx be a vector in Cl 1,3()\mathrm{Cl}^{1, 3}(\mathbb{R}). Then we define the coordinates of xx relative to the basis {γ i}\{\gamma_i\} to be x i=γ ixx^i = \gamma_i \cdot x. x 0x^0 is also denoted as tt since it represents the time coordinate. The spacetime vector derivative is defined as the operator

=γ 0t+γ 1x 1+γ 2x 2+γ 3x 3\nabla = \gamma_0 \frac{\partial}{\partial t} + \gamma_1 \frac{\partial}{\partial x^1} + \gamma_2 \frac{\partial}{\partial x^2} + \gamma_3 \frac{\partial}{\partial x^3}

The relative vector derivative is defined as the operator

=σ 1x 1+σ 2x 2+σ 3x 3\mathbf{\nabla} = \sigma_1 \frac{\partial}{\partial x^1} + \sigma_2 \frac{\partial}{\partial x^2} + \sigma_3 \frac{\partial}{\partial x^3}

Assuming the use of natural units where c=ϵ 0=1c = \epsilon_0 = 1 and ignoring the polarization and magnatization fields for the time being, Maxwell’s equations are written as:

  • electric Gauss’s law:

    E=ρ\mathbf{\nabla} \cdot E = \rho
  • Faraday’s law:

    E=t(IB)\mathbf{\nabla} \wedge E = -\frac{\partial}{\partial t}\left(I B\right)
  • magnetic Gauss’s law:

    B=0\mathbf{\nabla} \cdot B = 0
  • Ampère-Maxwell law:

    B=I(J+t(E))\mathbf{\nabla} \wedge B = I\left(\mathbf{J} + \frac{\partial}{\partial t}\left(E\right)\right)

where EE and BB are the relative electric and magnetic relative fields, ρ\rho is the density of the charge, and J\mathbf{J} is the relative current of the charge.

The Faraday bivector is the bivector F=E+IBF = E + I B. By multiplying the last two equations by the pseudoscalar, one gets

(IB)=0\mathbf{\nabla} \cdot (I B) = 0
(IB)=Jt(E)\mathbf{\nabla} \wedge (I B) = -\mathbf{J} - \frac{\partial}{\partial t}\left(E\right)

and by adding the four equations together, one gets the equation

(E+IB)+(E+IB)=ρJt(E+IB)\mathbf{\nabla} \cdot (E + I B) + \mathbf{\nabla} \wedge (E + I B) = \rho -\mathbf{J} -\frac{\partial}{\partial t}\left(E + I B\right)

which then becomes

F+F+t(F)=ρJ\mathbf{\nabla} \cdot F + \mathbf{\nabla} \wedge F + \frac{\partial}{\partial t}\left(F\right) = \rho - \mathbf{J}

Given a timelike bivector vv and a general bivector MM in Cl 1,3()\mathrm{Cl}^{1, 3}(\mathbb{R}), vM=vM+vMv M = v \cdot M + v \wedge M. Thus, the above equation could be simplified even further to

F+t(F)=ρJ\mathbf{\nabla} F + \frac{\partial}{\partial t}\left(F\right) = \rho - \mathbf{J}


(+t)(F)=ρJ\left(\mathbf{\nabla} + \frac{\partial}{\partial t}\right)\left(F\right) = \rho - \mathbf{J}

Now, the spacetime current is given by J=γ 0(ρJ)J = \gamma_0 (\rho - \mathbf{J}), which is a vector in spacetime. The spacetime vector derivative is related to the relative vector derivative by the following equation:

=γ 0(+t)\nabla = \gamma_0 (\mathbf{\nabla} + \frac{\partial}{\partial t})

Thus, by left multiplying each side by γ 0\gamma_0, one gets

γ 0(+t)(F)=γ 0(ρJ)\gamma_0 \left(\mathbf{\nabla} + \frac{\partial}{\partial t}\right)\left(F\right) = \gamma_0 (\rho - \mathbf{J})


F=J\nabla F = J

where \nabla is the spacetime vector derivative, FF is the Faraday bivector, and JJ is the spacetime current.


For more see the references at electromagnetism.


Maxwell’s equations originate in:

Some history and reflection:

For Maxwell’s equations in the generality of dielectric media, see the references there, such as:

  • G. Russakoff, A Derivation of the Macroscopic Maxwell Equations, American Journal of Physics 38 (1970) 1188–1195 [doi:10.1119/1.1976000]

  • O. L. de Lange, R. E. Raab Surprises in the multipole description of macroscopic electrodynamics, American Journal of Physics 74 (2006) 301–312 [doi:10.1119/1.2151213]

On the Maxwell Green's function (propagator) and numerical solutions:

  • Boris Lo, Victor Minden, Phillip Colella, A real-space Green’s function method for the numerical solution of Maxwell’s equations, 11 2 (2016) 143–170 [doi:10.2140/camcos.2016.11.143, pdf]

Maxwell’s equations via differential forms

On the expression of classical electromagnetism, and especially of Maxwell's equations, in terms of differential forms, the de Rham differential and Hodge star operators:

Via “geometric algebra: (Clifford algebra)

Formulation of Maxwell’s equations via “geometric algebra”:

  • Chris Doran, Anthony Lasenby?, Geometric algebra for physicists, Cambridge University Press (2003) (pdf)

  • John W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons Inc. (2011). (ISBN:978-0470941638, doi:10.1002/9781118078549)

Last revised on February 2, 2024 at 08:24:41. See the history of this page for a list of all contributions to it.