nLab Maxwell's equations

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Differential geoemtry

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Contents

Idea

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

$E$ is here the (vector of) strength of electric field and $B$ the strength of magnetic field; $Q$ is the charge and ${j}_{\mathrm{el}}$ the density of the electrical current; ${ϵ}_{0}$, $c$, ${\mu }_{0}$ are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: ${\mu }_{0}{ϵ}_{0}=1/{c}^{2}$).

Integral formulation in vacuum, in SI units

Gauss’ law for electric fields

${\int }_{\partial V}E\cdot dA=\frac{Q}{{ϵ}_{0}}$\int_{\partial V} E\cdot d A = \frac{Q}{\epsilon_0}

where $\partial V$ is a closed surface which is a boundary of a 3d domain $V$ (physicists say “volume”) and $Q={\int }_{V}\rho dV$ the charge in the domain $V$; $\cdot$ denotes the scalar (dot) product. Surface element $dA$ is $\stackrel{⇀}{n}d\mid A\mid$, i.e. it is the scalar surface measure times the unit vector of normal outwards.

No magnetic monopoles (Gauss’ law for magnetic fields)

${\int }_{\Sigma }B\cdot dA=0$\int_\Sigma B\cdot d A = 0

where $\Sigma$ is any closed surface.

Faraday’s law of induction

${\oint }_{\partial \Sigma }B\cdot ds=-\frac{d}{dt}{\int }_{\Sigma }B\cdot dA$\oint_{\partial \Sigma} B\cdot d s = - \frac{d}{d t} \int_\Sigma B\cdot d A

The line element $ds$ 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 $s$ 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).

${\oint }_{\partial \Sigma }B\cdot ds={\mu }_{0}I+{\mu }_{0}{ϵ}_{0}\frac{d}{dt}{\int }_{\Sigma }E\cdot dA$\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 $\Sigma$ is a surface and $\partial \Sigma$ its boundary; $I$ is the total current through $\Sigma$ (integral of the component of ${j}_{\mathrm{el}}$ normal to the surface).

Differential equations

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

• no magnetic charges (magnetic Gauss law): $\mathrm{div}B=0$

• Faraday’s law: $\frac{d}{dt}B+\mathrm{rot}E=0$

• Gauss’ law: $\mathrm{div}D=\rho$

• generalized Ampère’s law $-\frac{d}{dt}D+\mathrm{rot}H={j}_{\mathrm{el}}$

Equations in terms of Faraday tensor $F$

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 $U\subset \left({ℝ}^{4},g=\mathrm{diag}\left(-1,1,1,1\right)\right)$, the electric field $\stackrel{⇀}{E}=\left[\begin{array}{c}{E}_{1}\\ {E}_{2}\\ {E}_{3}\end{array}\right]$ and magnetic field $\stackrel{⇀}{B}=\left[\begin{array}{c}{B}_{1}\\ {B}_{2}\\ {B}_{3}\end{array}\right]$ combine into a differential 2-form

$\begin{array}{rl}F& :=E\wedge dt+B\\ & :={E}_{1}d{x}^{1}\wedge dt+{E}_{2}d{x}^{2}\wedge dt+{E}_{3}d{x}^{3}\wedge dt\\ & +{B}_{1}d{x}^{2}\wedge d{x}^{3}+{B}_{2}d{x}^{3}\wedge d{x}^{1}+{B}_{3}d{x}^{1}\wedge d{x}^{2}\end{array}$\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 ${\Omega }^{2}\left(U\right)$ and the electric charge density and current density combine to a differential 3-form

$\begin{array}{rl}{j}_{\mathrm{el}}& :=j\wedge \mathrm{dt}-\rho d{x}^{1}\wedge d{x}^{2}\wedge d{x}^{3}\\ & :={j}_{1}d{x}^{2}\wedge d{x}^{3}\wedge dt+{j}_{2}d{x}^{3}\wedge d{x}^{1}\wedge dt+{j}_{3}d{x}^{1}\wedge d{x}^{2}\wedge dt-\rho \phantom{\rule{thickmathspace}{0ex}}d{x}^{1}\wedge d{x}^{2}\wedge d{x}^{3}\end{array}$\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 ${\Omega }^{3}\left(U\right)$ such that the following two equations of differential forms are satisfied

$\begin{array}{r}dF=0\\ d\star F={j}_{\mathrm{el}}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\begin{aligned} d F = 0 \\ d \star F = j_{el} \end{aligned} \,,

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

$\begin{array}{rl}\star F& =-D+H\wedge \mathrm{dt}\\ & =-{D}_{1}\phantom{\rule{thickmathspace}{0ex}}d{x}^{2}\wedge d{x}^{3}-{D}_{2}\phantom{\rule{thickmathspace}{0ex}}d{x}^{3}\wedge d{x}^{1}-{D}_{3}\phantom{\rule{thickmathspace}{0ex}}d{x}^{1}\wedge d{x}^{2}\\ & +{H}_{1}\phantom{\rule{thickmathspace}{0ex}}d{x}^{1}\wedge dt+{H}_{2}\phantom{\rule{thickmathspace}{0ex}}d{x}^{2}\wedge dt+{H}_{3}\phantom{\rule{thickmathspace}{0ex}}d{x}^{3}\wedge dt\end{array}$\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.

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

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

References

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 (161.53.130.104)