Aequationes Maxwellianae in vacuo

Electromagnetismus spatii vacui quasdam proprietates habet quae commemorandae sunt. Fit ut aequationes Maxwellianae quae campos electricos et magneticos describunt, in vacuo hanc formam accipiant:

\begin{array}{rlr} \vec{\nabla} \cdot \vec{E} &= 0 & [1]\\ \vec{\nabla} \cdot \vec{B} &= 0 & [2]\\ \vec{\nabla} \times \vec{E} &\displaystyle = -\frac{\partial \vec{B}}{\partial t} & [3]\\ \vec{\nabla} \times \vec{B} &\displaystyle = \mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} & [4] \end{array}

Ex his confirmari potest perturbationem quamlibet in campis electricis et magneticis propagari velut undam in spatio vacuo. Quomodo hoc scimus? Quia ex analysi harum expressionum aequatio undae pro ambobus campis obtinetur.

Propagatio undarum electromagneticarum

Ex [4] et [5] habetur campum electricum sequentem relationem explere:

\begin{array}{llr} \vec{\nabla} \times (\vec{\nabla} \times \vec{E}) &= \displaystyle \vec{\nabla} \times \left( -\frac{\partial \vec{B}}{\partial t} \right) &\\ &=\displaystyle -\frac{\partial}{\partial t}\left(\vec{\nabla} \times \vec{B}\right) = -\frac{\partial}{\partial t} \left(\mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \right) = -\mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}& [6] \end{array}

Deinde, sicut omne campus vectorialis relationem explere solet:

\vec{\nabla} \times \left(\vec{\nabla} \times \vec{A} \right) = \vec{\nabla}(\vec{\nabla} \cdot \vec{A}) - \nabla^2 \vec{A},\;\;\;[7]

Ex [2, 6] et [7] scribi potest:

\begin{array}{rll} &\displaystyle \vec{\nabla}(\underbrace{\vec{\nabla} \cdot \vec{E}}_{=0}) - \nabla^2 \vec{E} = \vec{\nabla} \times \left(\vec{\nabla} \times \vec{E} \right) = -\mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2} & \\ \equiv &\displaystyle \color{blue}{\nabla^2 \vec{E} = \mu_0\epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}} & [8] \end{array}

Hoc quod caeruleo notatum apparet est ipsa aequatio propagationis undarum pro campo electrico.

Similiter omnino fit pro campo magnetico.

\begin{array}{ll} \vec{\nabla} \times (\vec{\nabla} \times \vec{B}) &= \displaystyle \vec{\nabla} \times \left(\mu_0\epsilon_0 \frac{\partial \vec{E}}{\partial t} \right)\\ &=\displaystyle \mu_0\epsilon_0 \frac{\partial}{\partial t}\left(\vec{\nabla} \times \vec{E}\right) = \mu_0\epsilon_0 \frac{\partial}{\partial t} \left(- \frac{\partial \vec{B}}{\partial t} \right) = -\mu_0\epsilon_0 \displaystyle \frac{\partial^2 \vec{B}}{\partial t^2} \end{array}

et deinde

\begin{array}{rll} & \displaystyle \vec{\nabla}(\underbrace{\vec{\nabla} \cdot \vec{B}}_{=0}) - \nabla^2 \vec{B} = \vec{\nabla} \times \left(\vec{\nabla} \times \vec{B} \right) = -\mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2} &\\ \equiv &\color{blue}{\nabla^2 \vec{B} = \displaystyle \mu_0\epsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2}}& [9] \end{array}

Ex hoc dicitur campos electromagneticos in vacuo multos modos posse habere, et una familia horum modorum formam undae electromagneticae accipit quae per spatium et tempus propagatur.

Celeritas lucis est constans universalis

Aliter dictum, perturbationes in campis electromagneticis semper propagantur celeritate c = 1/\sqrt{\epsilon_0 \mu_0}\approx 3\cdot 10^8[m/s], quae est celeritas lucis in vacuo. Experimentis observatur hanc celeritatem eandem esse omnibus referentialibus inertialibus, quod non congruit cum iis quae obtinerentur si transformationes Galilaeanae adhiberentur, ut ostenditur in Transformationes Galilaeanae et earum Limitationes; nam secundum has, ipsa quoque structura undae mutatur cum ab uno referentiali inertiali ad alterum transitum fit. Haec resultata sunt pars clavissima ad transformationes Galilaeanas reiciendas, locum facientes transformationibus Lorentzianis relativitatis specialis, quia: transformatio coordinatarum recte formata debebit leges physicae omnibus observatoribus inertialibus servare.