Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD
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Der Artikel Polarisation basiert auf der Vorlesungsmitschrift von Franz- Josef Schmitt des 5.Kapitels (Abschnitt 1) der Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD.
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{{#set:Urheber=Prof. Dr. E. Schöll, PhD|Inhaltstyp=Script|Kapitel=5|Abschnitt=1}}
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Materie enthält mikroskopische elektrisch geladene Bausteine
- freie Ladungsträger
Elektronen in Metallen, Elektronen + Löcher in Halbleitern
- Beschleunigung in äußeren Feldern, E- Felder, B- Felder über Ohmsches Gesetz und Lorentz-kraft
![{\displaystyle {\bar {K}}=q\left[{\bar {E}}+\left({\bar {v}}\times {\bar {B}}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9948bfeb14b293fd992fa16cae0bf954a76e5dec)
- gebundene Ladungen ( In Isolatoren)
- Für E =0 vorhandene mikroskopische Dipole p werden zur Minimierung der potenziellen Energie
Wel.=-p E
vorzugsweise ( entgegen der zufälligen thermischen Bewegung) parallel zu E orientiert ( z.B. bei polarisierten Molekülen, Wasser etc... gut zu beobachten !)
- Nicht- polare Atome oder Moleküle werden dann durch E durch Verschiebung der Ladungswolken polarisiert. Es entstehen induzierte elektrische Dipole, die zu E parallel ausgerichtet sind:
![{\displaystyle {\bar {p}}=\int _{}^{}{{{d}^{3}}r}\rho \left({\bar {r}}\right){\bar {r}}\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88ba7a52cfd174735b2a295aa463f7421e70d052)
nach Einschalten des Feldes.
Es werden in den Atomen/ Molekülen positive und negative Ladungen getrennt !
Makroskopische räumliche Mittelung
Netto- Ladungen entstehen dadurch an den Grenzflächen
Dies erzeugt im Inneren ein Polarisationsgegenfeld
![{\displaystyle {\begin{aligned}&{\bar {E}}{\acute {\ }}={\bar {E}}+{{\bar {E}}_{p}}\\&{{\varepsilon }_{0}}\nabla \cdot {\bar {E}}{\acute {\ }}={{\varepsilon }_{0}}\nabla \cdot {\bar {E}}+{{\rho }_{P}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33385c4d95540d308fb9f190c044834ccd52891c)
gemäß
![{\displaystyle {{\varepsilon }_{0}}\nabla \cdot {{\bar {E}}_{p}}={{\rho }_{P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3179021ed8071db04b4d9c977cd249dec6f2e3a5)
Das resultierende Gesamtfeld lautet:
![{\displaystyle {\begin{aligned}&{\bar {E}}{\acute {\ }}={\bar {E}}+{{\bar {E}}_{p}}\\&{{\varepsilon }_{0}}\nabla \cdot {\bar {E}}{\acute {\ }}={{\varepsilon }_{0}}\nabla \cdot {\bar {E}}+{{\rho }_{P}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33385c4d95540d308fb9f190c044834ccd52891c)
Mit der freien Ladungsdichte
![{\displaystyle {{\varepsilon }_{0}}\nabla \cdot {\bar {E}}=\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/41d6856ed4b5df42a59b9f9855fa817a38cb54e6)
Also:
![{\displaystyle {{\varepsilon }_{0}}\nabla \cdot {\bar {E}}{\acute {\ }}=\rho +{{\rho }_{P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a8cabe53fefd2b7622c46ea0a6ae12a70560c9)
Die Polarisation selbst bestimmt sich nach
![{\displaystyle {\bar {P}}\left({\bar {r}},t\right):=-{{\varepsilon }_{0}}{{\bar {E}}_{p}}\left({\bar {r}},t\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f60bad03ea4daf69ddb5b380dbddd2755a5824df)
ein makroskopisches lokales Feld, dessen Quelle Polarisationsladungen sind.
Somit:
![{\displaystyle {\begin{aligned}&\nabla \cdot \left({{\varepsilon }_{0}}{\bar {E}}{\acute {\ }}+{\bar {P}}\right)=\rho \\&\nabla \cdot {\bar {P}}=-{{\rho }_{P}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/555563f3e8bdbe2a346888f76bad07ef4a2c4941)
Als Dielektrische Verschiebung bezeichnen wir
![{\displaystyle {\bar {D}}({\bar {r}},t)=\left({{\varepsilon }_{0}}{\bar {E}}{\acute {\ }}+{\bar {P}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4a18c9a265f4e27710d4e1ae87406e289498dc)
Dies ist die effektive makroskopische Feldgröße, als dessen Quellen nur noch die freien Ladungen ( ohne Polarisationsladungen) auftreten:
![{\displaystyle \nabla \cdot {\bar {D}}=\rho }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0896619da2a7b4982a7046bfb8897a4e88a43025)
Wir bezeichnen mit
![{\displaystyle {\bar {P}}\left({\bar {r}},t\right)d{\bar {f}}=d{{Q}_{P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff01c1c077a4a2b95acf0f8732e9d3783819c94d)
die Polarisationsladung, die beim Übergang vom unpolarisierten zum polarisierten Zustand durch die Fläche df verschoben wird:
Denn ( bei Betrachtung eines Volumens V, das durch df begrenzt ist):
![{\displaystyle \oint _{\partial V}{}{\bar {P}}\left({\bar {r}},t\right)d{\bar {f}}=\int _{V}^{}{{{d}^{3}}r\nabla \cdot {\bar {P}}\left({\bar {r}},t\right)=-\int _{V}^{}{}{{d}^{3}}r}{{\rho }_{P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/760be23854ec9382d89739bdcb0a7ec5b5c212ca)
= Polarisationsladung, die V verläßt !
Zusammenhang mikroskopische elektrische Dipole / makroskopische Größen:
![{\displaystyle {{\rho }_{m}}\left({\bar {r}},t\right)=\sum \limits _{i}{}{{q}_{i}}\delta \left({\bar {r}}-{{\bar {r}}_{i}}(t)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28e762fa7bbdf121fd7992319ce4c734f36eeeb4)
( mikroskopische Ladungsdichte)
![{\displaystyle {{\bar {P}}_{m}}\left({\bar {r}},t\right)=\sum \limits _{i}{}{{\bar {p}}_{i}}\delta \left({\bar {r}}-{{\bar {r}}_{i}}(t)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6184fe0a395954d8d0f44dd15e897a25039b251f)
( mikroskopische Dipoldichte) mit:
![{\displaystyle \int _{V}^{}{{{d}^{3}}r{{\bar {P}}_{m}}\left({\bar {r}},t\right)=\sum \limits _{i}{}{{\bar {p}}_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f64dc8541722d051243053c01c38b564d21da55)
Mittelung über ein kleines makroskopisches Volumen
![{\displaystyle \Delta V:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d611800bf234f4e8687cb596308a673162160e2d)
![{\displaystyle {{\left(\Delta V\right)}^{\frac {1}{3}}}<<}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33f355041bf3aa8a6117e0c673baf8474362ed90)
Längenskala der makroskopischen Dichtevariation
Somit:
![{\displaystyle \rho \left({\bar {r}},t\right)={\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\rho }_{m}}\left({\bar {r}}+{\bar {s}},t\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08a87e83c1a9ab02dc95dcdf8c2c97a0340c9e24)
( makroskopische Ladungsdichte)
![{\displaystyle {\bar {P}}\left({\bar {r}},t\right)={\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\bar {P}}_{m}}\left({\bar {r}}+{\bar {s}},t\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a98e4d22a34a10d6d56bcf479e129db5a0691b69)
Also: Die makroskopische Dipoldichte ist GLEICH DER POLARISATION !!
Beweis:
Betrachten wir das mikroskopische retardierte Potenzial:
![{\displaystyle {{\Phi }_{m}}\left({\bar {r}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {{{\rho }_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b4ccc29c1f09072efa39322d929260ae0830ffa)
wobei unter dem Integral die mikroskopische Ladungsdichte einzusetzen ist !
Das makroskopisch gemittelte Potenzial folgt dann gemäß
![{\displaystyle {\begin{aligned}&\Phi \left({\bar {r}},t\right)={\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\Phi }_{m}}\left({\bar {r}}+{\bar {s}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\frac {{{\rho }_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)}{\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}}\\&{\bar {r}}{\acute {\ }}{\acute {\ }}:={\bar {r}}{\acute {\ }}-{\bar {s}}\\&\Phi \left({\bar {r}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{\frac {{{\rho }_{m}}\left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}\\&={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\rho }_{m}}\left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1c5f21593be2c25088c9c9305737e5207970e9)
Wobei
![{\displaystyle {\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\rho }_{m}}\left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)=\rho \left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1435944bb1fe94b62c22de3df5779b8a9dfccd3f)
Die makroskopische Ladungsdichte ist !
![{\displaystyle {\begin{aligned}&\Rightarrow \Phi \left({\bar {r}},t\right)={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\rho }_{m}}\left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)\\&={\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}\rho \left({\bar {r}}{\acute {\ }}{\acute {\ }}+{\bar {s}},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca8b6c789af9ad5ca473641e142d6169dfc4b658)
Analog:
Das mikroskopische Potenzial der elektrischen Dipole
![{\displaystyle {{\bar {p}}_{i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f547a577ee664c68c485d7667cb67eeacd0ecac2)
![{\displaystyle {{\Phi }_{m}}\left({\bar {r}},t\right)=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}{{\nabla }_{r}}\left\{\sum \limits _{i}{}{\frac {1}{\left|{\bar {r}}-{{\bar {r}}_{i}}\right|}}{{\bar {p}}_{i}}\left(t-{\frac {\left|{\bar {r}}-{{\bar {r}}_{i}}\right|}{c}}\right)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/912019daca89334caca1d5309e8714c3a1f55ce1)
mit dem mikroskopischen Dipolmoment
![{\displaystyle {{\bar {p}}_{i}}\left(t-{\frac {\left|{\bar {r}}-{{\bar {r}}_{i}}\right|}{c}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb7e1777c17e4d7316dd4611069ea4c9a56b412)
Analog:
![{\displaystyle {{\Phi }_{m}}\left({\bar {r}},t\right)=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\bar {P}}_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb79484cd3bfa09c0ad29e58ab4871ec78c8dd0)
mit der mikroskopischen Dipoldichte
![{\displaystyle {{\bar {P}}_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6986067705013cd077cdaa8890f0be6bc78d1df9)
Somit ergibt sich für das makroskopisch gemittelte elektrische Potenzial:
![{\displaystyle {\begin{aligned}&\Phi \left({\bar {r}},t\right)={\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s{{\Phi }_{m}}\left({\bar {r}}+{\bar {s}},t\right)\\&=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}{\frac {1}{\Delta V}}\int _{\Delta V}^{}{}{{d}^{3}}s\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}}{{\bar {P}}_{m}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}+{\bar {s}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}\\&=-{\frac {1}{4\pi {{\varepsilon }_{0}}}}\int _{{R}^{3}}^{}{}{{d}^{3}}r{\acute {\ }}{\acute {\ }}{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}{\acute {\ }}\right|}{c}}\right)\right\}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b08fa850ea3bc27d4daea298f8973ecab04e715f)
Umformung:
![{\displaystyle {{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}=-{{\nabla }_{r{\acute {\ }}}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}+Korrektur}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe2939dad216bb37f7e428377fc354894c90847)
Dabei haben wir das Problem , dass beim Übergang zur gestrichenen Ableitung hier auch nach dem Argument r´ von P abgeleitet wird. Also müssen wir dies wieder abziehen:
![{\displaystyle {\begin{aligned}&{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}=-{{\nabla }_{r{\acute {\ }}}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}+{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\\&t{\acute {\ }}=t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\\&{{\nabla }_{r}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}=-{{\nabla }_{r{\acute {\ }}}}\left\{{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t-{\frac {\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}{c}}\right)\right\}+{\frac {1}{\left|{\bar {r}}-{\bar {r}}{\acute {\ }}\right|}}{{\nabla }_{r{\acute {\ }}}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04d5d119e6f88f63c1b3f432e8368520541766f6)
Also folgt für das Potenzial:
Dies ist das makroskopische Potenzial einer Polarisationsladungsdichte
![{\displaystyle {{\rho }_{p}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)=\left(-{{\nabla }_{r{\acute {\ }}}}{\bar {P}}\left({\bar {r}}{\acute {\ }},t{\acute {\ }}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d497a983297f14cb343ee48eb749b57f584cd4)
Damit können wir die makroskopische Dipoldichte
![{\displaystyle {\bar {P}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d1fdaf8f50ca1bfe522c83b892dd55f87659fe)
mit der durch
![{\displaystyle {\bar {P}}:=-{{\varepsilon }_{0}}{{\bar {E}}_{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afb96e9443985d7505492d5dc9390bb55f5573c0)
bzw.
![{\displaystyle \nabla \cdot {\bar {P}}=-{{\rho }_{p}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4c60a8b8d5411d9f9cf1c38f0f22d43bf077ca)
definierten Polarisation identifizieren.