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, in SI units

Gauss’ law for electric fields

VEdA=Qϵ 0 \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 = \int_V \rho d V the charge in the domain VV; \cdot denotes the scalar (dot) product. Surface element dAd A is ndA\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 \int_\Sigma B\cdot d A = 0

where Σ\Sigma is any closed surface.

Faraday’s law of induction

ΣBds=ddt ΣBdA \oint_{\partial \Sigma} B\cdot d s = - \frac{d}{d t} \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 \oint_{\partial \Sigma} B\cdot d s = \mu_0 I + \mu_0 \epsilon_0 \frac{d}{d t} \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.

  • no magnetic charges (magnetic Gauss law): divB=0div B = 0

  • Faraday’s law: ddtB+rotE=0\frac{d}{d t} B + rot E = 0

  • Gauss’ law: divD=ρdiv D = \rho

  • generalized Ampère’s law ddtD+rotH=j el- \frac{d}{d t} D + rot H = j_{el}

Equations in terms of Faraday tensor FF

This is adapted from electromagnetic field – Maxwell’s equations.

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 & := E \wedge d t + B \\ &:= E_1 d x^1 \wedge d t + E_2 d x^2 \wedge d t + E_3 d x^3 \wedge d t \\ & + 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} &:= j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\ & := 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 \\ & + 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 field equations – called Maxwell’s equations – read as follows.

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

dF=0d \star F = 0 gives Gauss’s law and Ampère-Maxwell law


Maxwell's equations originate in

Discussion in terms of differential forms is for instance in

Revised on September 27, 2012 20:31:46 by Zoran Škoda (