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 $j_{el}$ the density of the electrical current; $\epsilon_0$, $c$, $\mu_0$ are constants (electrical permeability, speed of the light, and magnetic permeability; all of/in vacuum: $\mu_0 \epsilon_0 = 1/c^2$).

Integral formulation in vacuum, in SI units

Gauss’ law for electric fields

$$\int_{\partial V} E\cdot d A = \frac{Q}{\epsilon_0}$$

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

No magnetic monopoles (Gauss’ law for magnetic fields)

$$\int_\Sigma B\cdot d A = 0$$

where $\Sigma$ is any closed surface.

Faraday’s law of induction

$$\oint_{\partial \Sigma} E\cdot d s = - \frac{d}{d t} \int_\Sigma B\cdot d A$$

The line element $d s$ 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 $s$ 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).

$$\oint_{\partial \Sigma} B\cdot d s = \mu_0 I + \mu_0 \epsilon_0 \frac{d}{d t} \int_\Sigma E\cdot d A$$

where $\Sigma$ is a surface and $\partial \Sigma$ its boundary; $I$ is the total current through $\Sigma$ (integral of the component of $j_{el}$ 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): $div B = 0$

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

• Gauss’ law: $div D = \rho$

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

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 $U \subset (\mathbb{R}^4, g = diag(-1,1,1,1))$, the electric field $\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right]$ and magnetic field $\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]$ combine into a differential 2-form

$$\begin{aligned} F & := E \wedge d t + B \\ &:= E_1 d x^1 \wedge d t + E_2 d x^2 \wedge d t + E_3 d x^3 \wedge d t \\ & + B_1 d x^2 \wedge d x^3 + B_2 d x^3 \wedge d x^1 + B_3 d x^1 \wedge d x^2 \end{aligned}$$

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

$$\begin{aligned} j_{el} &:= j\wedge dt - \rho d x^1 \wedge d x^2 \wedge d x^3 \\ & := j_1 d x^2 \wedge d x^3 \wedge d t + j_2 d x^3 \wedge d x^1 \wedge d t + j_3 d x^1 \wedge d x^2 \wedge d t - \rho \; d x^1 \wedge d x^2 \wedge d x^3 \end{aligned}$$

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

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

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

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

then in terms of these components the field equations – called Maxwell’s equations – read as follows.

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

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

Related concepts

• Hodge-Maxwell theorem

• 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)

• Theodore Frankel, section 3.5 in The Geometry of Physics - An Introduction

• Gregory L. Naber, Topology, geometry and gauge fields, Appl. Math. Sciences vol. 141, Springer 2000

Some history and reflection is in

• Freeman Dyson, Why is Maxwell’s Theory so hard to understand?, Proceedings of The Second European Conference on Antennas and Propagation, EuCAP 2007 (doi: 10.1049/ic.2007.1146)