Aequationes Maxwellianae

Aequationes Maxwellianae (pro nomine physici Scotiensis Iacobi Maxwell) praeclarae sunt series aequationum quae campos magneticos et electricos atque eorum interactionem cum materia plane describunt. Hae aequationes et aequatio Lorentziana basim physicae electromagneticae coniunctim fundant.Aequationes Maxwellianae in vacuo sunt basis pro theoria electromagnetica lucis in quo velocitas lucis

$c=2.99792458\times 10^{8}m/s$$c=2.99792458\times 10^{8}m/s$

in vacuo esse praecinitur.

Aequationes Maxwellianae Modernae

Multa sunt modi Maxwellianarum aequationum scribendarum. Quamquam Maxwell primitus scripsit 20 aequationes cum viginti variabilibus, in usu moderno vocabulum "Aequationes Maxwellianae" solum quattuor principalibus aequationibus Maxwellianis vectoriali modo scriptis refert. Historia eiusdem aequationium in pagina "physica electromagnetica" iam praesentata, in hac pagina aequationes praesentamus in variis formis modernis sicut in literatura scientifica invenire possumus.

Aequationes Maxwellianae modernae vectorali forma unitatibus MKSA scriptae

Hae sunt aequationes Maxwellianae modernae principales forma vectorali unitatibus MKSA modo scriptae:

$\nabla \cdot {\vec {\mathbf {E} }}=\rho /\epsilon _{o}$$\nabla \cdot {\vec {\mathbf {E}}}=\rho /\epsilon _{o}$

$\nabla \cdot {\vec {\mathbf {B} }}=0$$\nabla \cdot {\vec {\mathbf {B}}}=0$

$\nabla \times {\vec {\mathbf {E} }}=-{\frac {\partial {\vec {\mathbf {B} }}}{\partial t}}$$\nabla \times {\vec {\mathbf {E}}}=-{\frac {\partial {\vec {\mathbf {B}}}}{\partial t}}$

$\nabla \times {\vec {\mathbf {B} }}=\mu _{0}\varepsilon _{0}{\frac {\partial {\vec {\mathbf {E} }}}{\partial t}}+\mu _{0}{\vec {\mathbf {J} }}$$\nabla \times {\vec {\mathbf {B}}}=\mu _{0}\varepsilon _{0}{\frac {\partial {\vec {\mathbf {E}}}}{\partial t}}+\mu _{0}{\vec {\mathbf {J}}}$

ubi

${\vec {\mathbf {B} }}$${\vec {\mathbf {B}}}$ est campus magneticus in Teslis,

${\vec {\mathbf {E} }}$${\vec {\mathbf {E}}}$ campus electricus in Newtonis per Coulomb,

$\rho$$\rho$ densitas oneris electrici in Coulombibus per metrum cubicum,

${\vec {\mathbf {J} }}=\rho {\vec {\mathbf {v} }}$${\vec {\mathbf {J}}}=\rho {\vec {\mathbf {v}}}$ densitas currentis electricae in Amperiis per metrum quadatum,

$\epsilon _{o}=8.85412\times 10^{-12}$$\epsilon _{o}=8.85412\times 10^{{-12}}$ Faradii per metrum,

$\mu _{o}=4\pi \times 10^{-7}$$\mu _{o}=4\pi \times 10^{{-7}}$ Henrii per metrum, et

$\times$$\times$ est productum vectorialis sive productum crucis.

Aequationes Maxwellianae modernae tensorali forma unitatibus MKSA scriptae

Manifestum est ex tempore Alberti Einstein aequationes Maxwellianas praecepta relativitatis specialis obtemperare, qua de causa notatione tensorali exprimi possunt. Aliquas definitiones primum statuimus. Sit $\phi$$\phi$ potentiale electricum et ${\vec {\mathbf {A} }}$${\vec {\mathbf {A}}}$ vector magnetici potentialis, tunc ponamus (unitatibus MKSA)

$A^{\alpha }=\left(\phi /c,{\vec {\mathbf {A} }}\right)$$A^{{\alpha }}=\left(\phi /c,{\vec {\mathbf {A}}}\right)$, $J^{\alpha }=\left(c\rho ,{\vec {\mathbf {J} }}\right)$$J^{{\alpha }}=\left(c\rho ,{\vec {\mathbf {J}}}\right)$, $\partial ^{\alpha }=\left(-{\frac {\partial }{c\partial t}},{\vec {\nabla }}\right)$$\partial ^{\alpha }=\left(-{\frac {\partial }{c\partial t}},{\vec \nabla }\right)$

et tensorem ${F}^{\alpha \beta }$${F}^{{\alpha \beta }}$, Faraday tensor dictum, sic definimus:

$F^{\alpha \beta }=\partial ^{\beta }A^{\alpha }-\partial ^{\alpha }A^{\beta }\,\!$$F^{{\alpha \beta }}=\partial ^{\beta }A^{\alpha }-\partial ^{\alpha }A^{\beta }\,\!$

ex quo sequitur matrix

${F}^{\alpha \beta }=\left({\begin{matrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&B_{z}&-B_{y}\\E_{y}/c&-B_{z}&0&B_{x}\\E_{z}/c&B_{y}&-B_{x}&0\end{matrix}}\right).$${F}^{{\alpha \beta }}=\left({\begin{matrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&B_{z}&-B_{y}\\E_{y}/c&-B_{z}&0&B_{x}\\E_{z}/c&B_{y}&-B_{x}&0\end{matrix}}\right).$

Tunc aequationes Maxwellianae in vacuo scribere sequente modo possumus

$\partial _{\alpha }F^{\alpha \beta }=\mu _{o}J^{\beta }$$\partial _{\alpha }F^{{\alpha \beta }}=\mu _{o}J^{\beta }$,

atque

$\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }=0$$\partial _{\gamma }F_{{\alpha \beta }}+\partial _{\beta }F_{{\gamma \alpha }}+\partial _{\alpha }F_{{\beta \gamma }}=0$

Hoc tensorali modo manifestum est ut phaenomena electromagnetica principio relativitatis pareat, i.e., omnes aequationes physicae sunt identicae in omnibus systematibus coordinatium inertialibus. Etiam videmus separationem inter magneticum electricumque campum artificialem esse, quod ab uno spectatore in uno systemate referentiale campum magneticum vocatum alius spectator in systemate alio potest campum electricum vocari et vice versa.

Aequationes Maxwellianae modernae differentiali forma unitatibus MKSA scriptae

Aequationes Maxwellianae etiam differentiali forma exprimi possunt, quae modus est omnium simplissimus. Mathematice definimus 1-forma

$\mathbf {A} =\sum _{\alpha }A_{\alpha }dx^{\alpha }=-\phi dt+A_{x}dx+A_{y}dy+A_{z}dz$${\mathbf {A}}=\sum _{\alpha }A_{\alpha }dx^{\alpha }=-\phi dt+A_{x}dx+A_{y}dy+A_{z}dz$

ex quo 2-formam "Faraday" $\mathbf {F}$${\mathbf {F}}$ obtinemus applicando derivativum exteriore $d=\sum _{\alpha }{\frac {\partial }{\partial x^{\alpha }}}\;dx^{\alpha }\wedge$$d=\sum _{\alpha }{\frac {\partial }{\partial x^{\alpha }}}\;dx^{\alpha }\wedge$

$\mathbf {F} =d\mathbf {A} =-dt\wedge (E_{x}dx+E_{y}dy+E_{z}dz)+B_{x}dy\wedge dz+B_{y}dz\wedge dx+B_{z}dx\wedge dy$${\mathbf {F}}=d{\mathbf {A}}=-dt\wedge (E_{x}dx+E_{y}dy+E_{z}dz)+B_{x}dy\wedge dz+B_{y}dz\wedge dx+B_{z}dx\wedge dy$.

Bianchi identitas $d^{2}=0$$d^{2}=0$ bene nota plene satisfacta obtinemus recta

$d\mathbf {F} =0$$d{\mathbf {F}}=0$

qui duas aequationes Maxwellianas continet. Duas autem alias aequationes Maxwellianas a sequente aequatione differentialibus formis modo dantur,

$d*{\mathbf {F}}=\mu _{o}\mathbf {J}$$d*{\mathbf {F}}=\mu _{o}{\mathbf {J}}$

si definimus 3-formam de densitate currentis

$\mathbf {J} =\rho dx\wedge dy\wedge dz-dt\wedge (J_{x}dy\wedge dz+J_{y}dz\wedge dx+J_{z}dx\wedge dy)$${\mathbf {J}}=\rho dx\wedge dy\wedge dz-dt\wedge (J_{x}dy\wedge dz+J_{y}dz\wedge dx+J_{z}dx\wedge dy)$

et per Hodge operator * 2-formam "Maxwell"

$\mathbf {M} =*\mathbf {F} =dt\wedge (B_{x}dx+B_{y}dy+B_{z}dz)+E_{x}dy\wedge dz+E_{y}dz\wedge dx+E_{z}dx\wedge dy$${\mathbf {M}}=*{\mathbf {F}}=dt\wedge (B_{x}dx+B_{y}dy+B_{z}dz)+E_{x}dy\wedge dz+E_{y}dz\wedge dx+E_{z}dx\wedge dy$.

Aequationes Maxwellianae vectorali forma unitatibus Gaussianis (CGSF)

Unitates Gaussianae CGSF permittent aequationes Maxwellinas scribere ut symmetria inter campos magneticum et electrum manifestum sit. Haec sunt aequationes Maxwellianae forma vectorali unitatibus Gaussianis (CGSF) modo scriptae:

$\nabla \cdot {\vec {\mathbf {E} }}=4\pi \rho$$\nabla \cdot {\vec {\mathbf {E}}}=4\pi \rho$

$\nabla \cdot {\vec {\mathbf {B} }}=0$$\nabla \cdot {\vec {\mathbf {B}}}=0$

$\nabla \times {\vec {\mathbf {E} }}=-{\frac {1}{c}}{\frac {\partial {\vec {\mathbf {B} }}}{\partial t}}$$\nabla \times {\vec {\mathbf {E}}}=-{\frac {1}{c}}{\frac {\partial {\vec {\mathbf {B}}}}{\partial t}}$

$\nabla \times {\vec {\mathbf {B} }}={\frac {1}{c}}{\frac {\partial {\vec {\mathbf {E} }}}{\partial t}}+{\frac {4\pi }{c}}{\vec {\mathbf {J} }}$$\nabla \times {\vec {\mathbf {B}}}={\frac {1}{c}}{\frac {\partial {\vec {\mathbf {E}}}}{\partial t}}+{\frac {4\pi }{c}}{\vec {\mathbf {J}}}$

ubi

${\vec {\mathbf {B} }}$${\vec {\mathbf {B}}}$ est campus magneticus in Gaussibus vel dyniis per Franklin,

${\vec {\mathbf {E} }}$${\vec {\mathbf {E}}}$ campus electricus in Gaussibus vel dyniis per Franklin,

$\rho$$\rho$ densitas oneris electrici in Franklinibus per centimetrum cubicum,

${\vec {\mathbf {J} }}=\rho {\vec {\mathbf {v} }}$${\vec {\mathbf {J}}}=\rho {\vec {\mathbf {v}}}$ densitas currentis electricae in Franklinibus per secundum per centimetrum quadatum.

Aequationes Maxwellianae tensorali forma unitatibus Gaussianis (CGSF)

Et haec sunt aequationes Maxwellianae forma tensorali unitatibus Gaussianis (CGSF) modo scriptae:

$A^{\alpha }=\left(\phi ,{\vec {\mathbf {A} }}\right)$$A^{{\alpha }}=\left(\phi ,{\vec {\mathbf {A}}}\right)$, $J^{\alpha }=\left(c\rho ,{\vec {\mathbf {J} }}\right)$$J^{{\alpha }}=\left(c\rho ,{\vec {\mathbf {J}}}\right)$, $\partial ^{\alpha }=\left(-{\frac {\partial }{c\partial t}},{\vec {\nabla }}\right)$$\partial ^{\alpha }=\left(-{\frac {\partial }{c\partial t}},{\vec \nabla }\right)$

et tensorem ${F}^{\alpha \beta }$${F}^{{\alpha \beta }}$, Faraday tensor dictum, sic definimus:

$F^{\alpha \beta }=\partial ^{\beta }A^{\alpha }-\partial ^{\alpha }A^{\beta }\,\!$$F^{{\alpha \beta }}=\partial ^{\beta }A^{\alpha }-\partial ^{\alpha }A^{\beta }\,\!$

ex quo sequitur matrix

${F}^{\alpha \beta }=\left({\begin{matrix}0&-E_{x}&-E_{y}&-E_{z}\\E_{x}&0&B_{z}&-B_{y}\\E_{y}&-B_{z}&0&B_{x}\\E_{z}&B_{y}&-B_{x}&0\end{matrix}}\right).$${F}^{{\alpha \beta }}=\left({\begin{matrix}0&-E_{x}&-E_{y}&-E_{z}\\E_{x}&0&B_{z}&-B_{y}\\E_{y}&-B_{z}&0&B_{x}\\E_{z}&B_{y}&-B_{x}&0\end{matrix}}\right).$

Tunc aequationes Maxwellianae in vacuo scribere sequente modo possumus

$\partial _{\alpha }F^{\alpha \beta }={\frac {4\pi }{c}}J^{\beta }$$\partial _{\alpha }F^{{\alpha \beta }}={\frac {4\pi }{c}}J^{\beta }$,

atque

$\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma }=0$$\partial _{\gamma }F_{{\alpha \beta }}+\partial _{\beta }F_{{\gamma \alpha }}+\partial _{\alpha }F_{{\beta \gamma }}=0$

Fontes

• Jackson, John David. 1998. Classical Electrodynamics. New York: John Wiley & Sons.
• Griffiths, David. 1987. Introduction to Elementary Particle Physics. New York: John Wiley & Sons. ISBN 0-471-60386-4.