Maxwell- Gleichungen in Materie

Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD

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Der Artikel Maxwell- Gleichungen in Materie basiert auf der Vorlesungsmitschrift von Franz- Josef Schmitt des 5.Kapitels (Abschnitt 3) der Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD.

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{{#set:Urheber=Prof. Dr. E. Schöll, PhD|Inhaltstyp=Script|Kapitel=5|Abschnitt=3}} Kategorie:Elektrodynamik __SHOWFACTBOX__

Die vollständigen Potenziale enthalten

• die freie Ladungs- und Stromdichten
• $\rho ,{\bar {j}}$
• die Polarisations- und Magnetisierungsbeiträge
• ${{\rho }_{p}},{{\bar {j}}_{p}},{{\bar {j}}_{m}}$

Somit folgt für die vollständigen Potenziale:

${\begin{aligned}&t{\acute {\ }}=t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\\&{\bar {A}}\left({\bar {r}},t\right)={\frac {{\mu }_{{\acute {\ }}0}}{4\pi }}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\left[{\bar {j}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{{\bar {j}}_{P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{{\bar {j}}_{M}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right]\\&\Phi \left({\bar {r}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}\left[\rho \left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)+{{\rho }_{P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right]\\&\\\end{aligned}}$

Diese Potenziale sind Lösungen der inhomogenen Wellengleichung in Lorentz- Eichung

${\begin{aligned}&\#{\bar {A}}\left({\bar {r}},t\right)=-{{\mu }_{{\acute {\ }}0}}\left[{\bar {j}}+{{\bar {j}}_{P}}+{{\bar {j}}_{M}}\right]\\&\#\Phi \left({\bar {r}},t\right)=-{\frac {1}{{\varepsilon }_{0}}}\left[\rho +{{\rho }_{P}}\right]\\&\\\end{aligned}}$

Für die Felder in Materie folgt:

${\begin{aligned}&{\bar {E}}=-\nabla \Phi \left({\bar {r}},t\right)-{\frac {\partial }{\partial t}}{\bar {A}}\left({\bar {r}},t\right)\\&{\bar {B}}=\nabla \times {\bar {A}}\left({\bar {r}},t\right)\\\end{aligned}}$

Daraus folgen die Maxwell- Gleichungen:

${\begin{aligned}&1)\nabla \times {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \times {\bar {A}}\left({\bar {r}},t\right)=-{\frac {\partial }{\partial t}}{\bar {B}}\\&2)\nabla \cdot {\bar {B}}=0\\\end{aligned}}$

• Wie im Vakuum

${\begin{aligned}&3)\nabla \cdot {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)-\nabla \cdot \nabla \Phi \\&\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)=-{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\Phi \\&\Rightarrow \nabla \cdot {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)-\nabla \cdot \nabla \Phi ={\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}}{\partial {{t}^{2}}}}\Phi -\Delta \Phi =-\#\Phi \\\end{aligned}}$

In Lorentz Eichung!

$\nabla \cdot {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)-\nabla \cdot \nabla \Phi ={\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}}{\partial {{t}^{2}}}}\Phi -\Delta \Phi =-\#\Phi ={\frac {1}{{\varepsilon }_{0}}}\left(\rho +{{\rho }_{p}}\right)={\frac {1}{{\varepsilon }_{0}}}\left(\rho -\nabla \cdot {\bar {P}}\right)$

per Definition von

${{\rho }_{p}}$.

${\begin{aligned}&\Rightarrow 3)\nabla \cdot \left({{\varepsilon }_{0}}{\bar {E}}\left({\bar {r}},t\right)+{\bar {P}}\left({\bar {r}},t\right)\right)=\rho \left({\bar {r}},t\right)\\&\left({{\varepsilon }_{0}}{\bar {E}}\left({\bar {r}},t\right)+{\bar {P}}\left({\bar {r}},t\right)\right):={\bar {D}}\left({\bar {r}},t\right)\\&\Rightarrow \nabla \cdot {\bar {D}}\left({\bar {r}},t\right)=\rho \left({\bar {r}},t\right)\\\end{aligned}}$

Die Dielektrische Verschiebung

4) Letzte Gleichung:

${\begin{aligned}&\nabla \times {\bar {B}}\left({\bar {r}},t\right)=\nabla \times \left(\nabla \times {\bar {A}}\left({\bar {r}},t\right)\right)=\nabla \left(\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)\right)-\Delta {\bar {A}}\left({\bar {r}},t\right)\\&\nabla \cdot {\bar {A}}\left({\bar {r}},t\right)=-{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\Phi \\&\Rightarrow \nabla \times {\bar {B}}\left({\bar {r}},t\right)=-\Delta {\bar {A}}\left({\bar {r}},t\right)-{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\nabla \Phi \\&\nabla \Phi =-{\bar {E}}-{\frac {\partial }{\partial t}}{\bar {A}}\left({\bar {r}},t\right)\\&\Rightarrow \nabla \times {\bar {B}}\left({\bar {r}},t\right)=-\Delta {\bar {A}}\left({\bar {r}},t\right)+{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}{\bar {E}}+{\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}}{\partial {{t}^{2}}}}{\bar {A}}\left({\bar {r}},t\right)=-\#{\bar {A}}\left({\bar {r}},t\right)+{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}{\bar {E}}\\&={{\mu }_{0}}\left({\bar {j}}+{{\bar {j}}_{P}}+{{\bar {j}}_{M}}\right)+{{\varepsilon }_{0}}{{\mu }_{0}}{\frac {\partial }{\partial t}}{\bar {E}}\\&{{\bar {j}}_{P}}={\dot {\bar {P}}}\\&{{\bar {j}}_{M}}=\nabla \times {\bar {M}}\\&\Rightarrow \nabla \times {\bar {B}}\left({\bar {r}},t\right)={{\mu }_{0}}{\frac {\partial }{\partial t}}\left({\bar {P}}+{{\varepsilon }_{0}}{\bar {E}}\right)+{{\mu }_{0}}\nabla \times {\bar {M}}+{{\mu }_{0}}{\bar {j}}\\&\Rightarrow 4)\\&\Rightarrow \nabla \times \left({\frac {1}{{\mu }_{0}}}{\bar {B}}\left({\bar {r}},t\right)-{\bar {M}}\right)={\bar {j}}+{\frac {\partial }{\partial t}}{\bar {D}}\\&\left({\frac {1}{{\mu }_{0}}}{\bar {B}}\left({\bar {r}},t\right)-{\bar {M}}\right)=H\left({\bar {r}},t\right)\\&\Rightarrow \nabla \times H\left({\bar {r}},t\right)={\bar {j}}+{\frac {\partial }{\partial t}}{\bar {D}}\\\end{aligned}}$

Mit dem Magnetfeld

$H\left({\bar {r}},t\right)$,

welches so definiert wurde, dass es nur durch die FREIEN Ströme erzeugt wird:

Zusammenfassung:

${\begin{aligned}&1)\nabla \times {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \times {\bar {A}}\left({\bar {r}},t\right)=-{\frac {\partial }{\partial t}}{\bar {B}}\\&2)\nabla \cdot {\bar {B}}=0\\\end{aligned}}$

$3)\nabla \cdot {\bar {D}}\left({\bar {r}},t\right)=\rho \left({\bar {r}},t\right)$

$4)\nabla \times H\left({\bar {r}},t\right)={\bar {j}}+{\frac {\partial }{\partial t}}{\bar {D}}$

$4)\nabla \times H\left({\bar {r}},t\right)-{\frac {\partial }{\partial t}}{\bar {D}}={\bar {j}}$

Dabei beschreibt

${\begin{aligned}&1)\nabla \times {\bar {E}}=-{\frac {\partial }{\partial t}}\nabla \times {\bar {A}}\left({\bar {r}},t\right)=-{\frac {\partial }{\partial t}}{\bar {B}}\\&2)\nabla \cdot {\bar {B}}=0\\\end{aligned}}$

die Wechselwirkung der Felder mit Probeladungen und

$3)\nabla \cdot {\bar {D}}\left({\bar {r}},t\right)=\rho \left({\bar {r}},t\right)$

$4)\nabla \times H\left({\bar {r}},t\right)-{\frac {\partial }{\partial t}}{\bar {D}}={\bar {j}}$

die Erzeugung der Felder durch FREIE Ladungen und Ströme

Weiter:

${\begin{aligned}&{\bar {D}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}{\bar {E}}\left({\bar {r}},t\right)+{\bar {P}}\left({\bar {r}},t\right)\\&{\bar {H}}\left({\bar {r}},t\right)={\frac {1}{{\mu }_{0}}}{\bar {B}}\left({\bar {r}},t\right)-{\bar {M}}\left({\bar {r}},t\right)\\\end{aligned}}$

Im Gauß System (weil so oft in diesem angegeben, vergl. Jackson):

${\begin{aligned}&1)\nabla \times {\bar {E}}+{\frac {1}{c}}{\frac {\partial }{\partial t}}{\bar {B}}=0\\&2)\nabla \cdot {\bar {B}}=0\\\end{aligned}}$

$3)\nabla \cdot {\bar {D}}\left({\bar {r}},t\right)=4\pi \rho \left({\bar {r}},t\right)$

$4)\nabla \times H\left({\bar {r}},t\right)-{\frac {1}{c}}{\frac {\partial }{\partial t}}{\bar {D}}={\frac {4\pi }{c}}{\bar {j}}$

die Erzeugung der Felder durch FREIE Ladungen und Ströme

Weiter:

${\begin{aligned}&5){\bar {D}}\left({\bar {r}},t\right)={\bar {E}}\left({\bar {r}},t\right)+4\pi {\bar {P}}\left({\bar {r}},t\right)\\&6){\bar {H}}\left({\bar {r}},t\right)={\bar {B}}\left({\bar {r}},t\right)-4\pi {\bar {M}}\left({\bar {r}},t\right)\\\end{aligned}}$

Unsere 6 Feldgleichungen (wenn man so will, also (es kann nicht oft genug gezeigt werden):

${\begin{aligned}&1)\nabla \times {\bar {E}}+{\frac {1}{c}}{\frac {\partial }{\partial t}}{\bar {B}}=0\\&2)\nabla \cdot {\bar {B}}=0\\\end{aligned}}$

$3)\nabla \cdot {\bar {D}}\left({\bar {r}},t\right)=4\pi \rho \left({\bar {r}},t\right)$

$4)\nabla \times H\left({\bar {r}},t\right)-{\frac {1}{c}}{\frac {\partial }{\partial t}}{\bar {D}}={\frac {4\pi }{c}}{\bar {j}}$

${\begin{aligned}&5){\bar {D}}\left({\bar {r}},t\right)={\bar {E}}\left({\bar {r}},t\right)+4\pi {\bar {P}}\left({\bar {r}},t\right)\\&6){\bar {H}}\left({\bar {r}},t\right)={\bar {B}}\left({\bar {r}},t\right)-4\pi {\bar {M}}\left({\bar {r}},t\right)\\\end{aligned}}$

sind nicht vollständig. Es muss noch der Zusammenhang zwischen Polarisation und E- Feld, bzw. B- Feld und Magnetisierung angegeben werden. Dies sind die sogenannten " Materialgleichungen".

Einfachster Fall:

1. isotrope Materie:

${\bar {E}}\left({\bar {r}},t\right)||{\bar {P}}\left({\bar {r}},t\right)$

und für paramagnetische Stoffe

${\bar {B}}\left({\bar {r}},t\right)\uparrow \uparrow {\bar {M}}\left({\bar {r}},t\right)$

für diamagnetische Stoffe:

${\bar {B}}\left({\bar {r}},t\right)\uparrow \downarrow {\bar {M}}\left({\bar {r}},t\right)$,

also ein skalarer Zusammenhang

1. bei nicht zu hohen Feldern:

${\bar {E}}{\tilde {\ }}{\bar {P}}$

${\bar {B}}{\tilde {\ }}{\bar {M}}$

also ein linearer Zusammenhang

1. ohne Gedächtniseffekte, keine nichtlokale Wechselwirkung (keine Phasenkohärenzen):

${\bar {E}}\left({\bar {r}},t\right){\tilde {\ }}{\bar {P}}\left({\bar {r}},t\right)$

${\bar {B}}\left({\bar {r}},t\right){\tilde {\ }}{\bar {M}}\left({\bar {r}},t\right)$

neben der Linearität also ein INSTANTANER, LOKALER Zusammenhang!

Dann kann man schreiben:

${\bar {P}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}{{\chi }_{e}}{\bar {E}}\left({\bar {r}},t\right)$

${\bar {M}}\left({\bar {r}},t\right)={{\chi }_{M}}{\bar {H}}\left({\bar {r}},t\right)$

Mit den Suszeptibilitäten, der elektrischen Suszeptibilität

${{\chi }_{e}}$

und der magnetischen Suszeptibilität

${{\chi }_{M}}$

(Materialkonstanten). Die Materialkonstanten müssen aus den mikroskopischen Theorien (z.B. Quantentheorie, Festkörperphysik) abgeleitet werden.

${\bar {D}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}{\bar {E}}\left({\bar {r}},t\right)+{\bar {P}}={{\varepsilon }_{0}}\left(1+{{\chi }_{e}}\right){\bar {E}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}\varepsilon {\bar {E}}\left({\bar {r}},t\right)$ mit $\varepsilon =\left(1+{{\chi }_{e}}\right)$,

der relativen Dielektrizitätskonstante (permittivity)

${\bar {B}}={{\mu }_{0}}\left({\bar {H}}+{\bar {M}}\right)={{\mu }_{0}}\left(1+{{\chi }_{M}}\right){\bar {H}}\left({\bar {r}},t\right)={{\mu }_{0}}\mu {\bar {H}}$ mit $\left(1+{{\chi }_{M}}\right)=\mu$,

der relativen Permeabilität

$\Rightarrow {\bar {M}}={{\chi }_{M}}{\bar {H}}={\frac {1}{{\mu }_{0}}}{\frac {{\chi }_{M}}{\mu }}{\bar {B}}={\frac {1}{{\mu }_{0}}}{\frac {{\chi }_{M}}{\left(1+{{\chi }_{M}}\right)}}{\bar {B}}$

Man sagt: Ein Stoff ist paramagnetisch für

${\frac {{\chi }_{M}}{\left(1+{{\chi }_{M}}\right)}}>0$

diamagnetisch für

${\frac {{\chi }_{M}}{\left(1+{{\chi }_{M}}\right)}}<0$

paramagnetisch:

${{\chi }_{M}}>0\Rightarrow \mu >1$ diamagnetisch $0>{{\chi }_{M}}>-1\Rightarrow 0<\mu <1$

Bemerkungen

${\bar {E}}\left({\bar {r}},t\right)=0\Rightarrow {\bar {P}}=0$

beschreibt kein Ferroelektrikum

${\bar {B}}=0\Rightarrow {\bar {M}}=0$

kein Ferromagnet

Es gilt stets

${{\chi }_{e}}>0$

(Dielektrischer Effekt, Polarisierbarkeit → es existiert keine negative Polarisierbarkeit)

${{\chi }_{M}}{\begin{matrix}>\\<\\\end{matrix}}0$

Para- ODER Diamagnet

Ein Term

${\tilde {\ }}{\bar {B}}$ in ${\bar {P}}$ oder ${\tilde {\ }}{\bar {E}}$ in ${\bar {M}}$

kann gar nicht auftreten, schon wegen des falschen Raumspiegelverhaltens!

${\bar {E}}$

ist polarer Vektor,

${\bar {B}}$

ist axialer Vektor!

${{\rho }_{P}}\left({\bar {r}},t\right)=-\nabla \cdot {\bar {P}}\left({\bar {r}},t\right)$

ist ein Skalar

${{\bar {j}}_{M}}=rot{\bar {M}}$

ist ein polarer Vektor.

Abweichungen

1)Für anisotrope Kristalle :

${\bar {P}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}{{\bar {\bar {\chi }}}_{e}}{\bar {E}}$

drückt den anisotropen Charakter aus mit einem symmetrischen Tensor

${{\bar {\bar {\chi }}}_{e}}$.

2) für starke Felder gibt es nichtlineare Effekte, die ebenfalls tensoriellen Charakter der Suszeptibilität bedingen:

${\bar {P}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}\left({{\bar {\bar {\chi }}}_{e}}^{(1)}{\bar {E}}+{{\bar {\bar {\chi }}}_{e}}^{(2)}{{\bar {E}}^{2}}+{{\bar {\bar {\chi }}}_{e}}^{(3)}{{\bar {E}}^{3}}+...\right)$

Anwendung: optische Nichtlinearität, Beispiel: optische Bistabilität, optische Schalter:

Für hochfrequente Felder folgt:

${\bar {P}}\left({\bar {r}},t\right)={{\varepsilon }_{0}}\int _{}^{}{}{{d}^{3}}r{\acute {\ }}dt{\acute {\ }}{{\chi }_{e}}\left({\bar {r}},{\bar {r}}{\acute {\ }},t,t{\acute {\ }}\right){\bar {E}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)$

(räumliche bzw. zeitliche Dispersion):

${\hat {\bar {P}}}\left({\bar {k}},\omega \right)={{\varepsilon }_{0}}{{\hat {\chi }}_{e}}\left({\bar {k}},\omega \right){\hat {\bar {E}}}\left({\bar {k}},\omega \right)$