Optics, electromagnetic theory

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• TheoremEarnshaw's theorem

  Let $U \subseteq \R^n$ be an open set, $n \ge 2$, and let $\varphi \colon U \to \R$ be a harmonic function. Then $\varphi$ has no strict local minimum and no strict local maximum in $U$. Consequently, consider a charged particle with charge $q$ that moves under the force $F = -q \nabla \varphi$. No equilibrium point in $U$ is a strict local minimum of its potential energy $q\varphi$. In this sense the particle has no stable equilibrium in $U$.

• ProblemThomson problem

  Let $N \ge 2$ be an integer and let $S^2 \subset \R^3$ be the unit sphere. Find the minimum of the energy $$E(x_1, \dots, x_N) = \sum_{1 \le i < j \le N} \frac{1}{\abs{x_i - x_j}}$$ over all $N$-tuples of distinct points $x_1, \dots, x_N \in S^2$, where $\abs{x_i - x_j}$ is the Euclidean distance. Find also the configurations that attain it.

• ConceptMaxwell's equations

  Maxwell's equations are four partial differential equations that govern the electric field $E$ and the magnetic field $B$. They relate the fields to their sources, the electric charge density and the current density. They unify electricity, magnetism and light, since in vacuum each field component solves the wave equation with the speed of light. They form a linear first-order system and are a basic model for hyperbolic systems in mathematical physics.

• TheoremKramers–Kronig relations

  Let $H = \set{z \in \C : \operatorname{Im} z > 0}$ be the upper half-plane. Let $\chi$ be continuous on the closure $\bar H$ and holomorphic on $H$, with $\chi(z) \to 0$ as $\abs{z} \to \infty$ in $\bar H$, uniformly in the direction. Write $\chi(\omega) = \chi_1(\omega) + i \chi_2(\omega)$ for real $\omega$, with $\chi_1$ and $\chi_2$ real. Then for each real $\omega$, $$\chi_1(\omega) = \frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^{\infty} \frac{\chi_2(\omega')}{\omega' - \omega} \, d\omega', \qquad \chi_2(\omega) = -\frac{1}{\pi} \, \mathrm{p.v.} \int_{-\infty}^{\infty} \frac{\chi_1(\omega')}{\omega' - \omega} \, d\omega'.$$ Here $\mathrm{p.v.}$ is the principal value at $\omega$ and at infinity: the limit of the integral over $\varepsilon < \abs{\omega' - \omega} < R$ as $\varepsilon \to 0$ and $R \to \infty$.

• ConceptTelegrapher's equations

  The telegrapher's equations are a pair of linear first-order equations for the voltage and the current along an electrical transmission line. They were developed by Oliver Heaviside to describe signals on telegraph cables. Eliminating the current gives one second-order equation for the voltage, the telegraph equation, which is a damped wave equation. The equations show how resistance and leakage make a signal both decay and spread out.

• FactFermat's principle

  Let $\Omega \subseteq \R^3$ be a region filled with an isotropic optical medium of refractive index $n \colon \Omega \to (0, \infty)$. For a piecewise smooth curve $\gamma \colon [a, b] \to \Omega$ define the optical path length $$\mathcal{L}[\gamma] = \int_a^b n(\gamma(s))\, \abs{\gamma'(s)} \, ds.$$ Let $A$ and $B$ be two points of $\Omega$. In geometric optics, the path of a light ray from $A$ to $B$ is a critical point of $\mathcal{L}$ among curves from $A$ to $B$.