nLab
Maxwell's equations

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 $\stackrel{\rightharpoonup }{n}d\mid A\mid$, 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 $U\subset ({ℝ}^4,g=\mathrm{diag}(-1,1,1,1))$, the electric field $\stackrel{\rightharpoonup }{E}=\left[\begin{array}{c}{E}_1\\ E_2\\ E_3\end{array}\right]$ and magnetic field $\stackrel{\rightharpoonup }{B}=\left[\begin{array}{c}{B}_1\\ B_2\\ B_3\end{array}\right]$ 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 ${\Omega }^2(U)$ and the electric charge density and current density combine to a differential 3-form

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

$$\begin{array}{r}dF=0\\ d\star F=j_{\mathrm{el}}\end{array},$$\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

$$\begin{array}{rl}\star F& =-D+H\wedge \mathrm{dt}\\ & =-D_{1}d{x}^2\wedge d{x}^3-D_{2}d{x}^3\wedge d{x}^1-D_{3}d{x}^1\wedge d{x}^2\\ & +H_{1}d{x}^1\wedge dt+H_{2}d{x}^2\wedge dt+H_{3}d{x}^3\wedge dt\end{array}$$\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

Related concepts

• Kirchhoff's laws

• Yang-Mills equations

References

Maxwell's equations originate in

• James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, Philosophical Transactions of the Royal Society of London 155, 459–512 (1865).

Discussion in terms of differential forms is for instance in

• Theodore Frankel, Maxwell’s equations, The American Mathematical Monthly, Vol 81, No 4 (1974) (pdf, JSTOR)
• section 3.5 in Theodore Frankel, The Geometry of Physics - An Introduction
• Gregory L. Naber, Topology, geometry and gauge fields, Appl. Math. Sciences vol. 141, Springer 2000