# nLab Maxwell's equations

Contents

## Surveys, textbooks and lecture notes

#### Differential geoemtry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 $j_{el}$ 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

Maxwell's equations originate in

Discussion in terms of differential forms is for instance in

Some history and reflection is in