Formelsammlung Physik: Elektrodynamik

Im SI-System gilt:

Ampèresches Gesetz	${\displaystyle \nabla \times \mathbf {H} ={\frac {\partial \mathbf {D} }{\partial t}}+\mathbf {j} }$
Induktionsgesetz Faradaysches Gesetz	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
Coulombsches Gesetz	${\displaystyle \nabla \cdot \mathbf {D} =\rho }$
Gaußsches Gesetz des Magnetismus	${\displaystyle \nabla \cdot \mathbf {B} =0}$
Elektrische Flussdichte	${\displaystyle \mathbf {D} =\varepsilon _{0}\,\mathbf {E} +\mathbf {P} }$
Magnetische Flussdichte	${\displaystyle \mathbf {B} =\mu _{0}\,(\mathbf {H} +\mathbf {M} )}$

Wobei ${\displaystyle \mathbf {H} }$ die magnetische Feldstärke, ${\displaystyle \mathbf {B} }$ die magnetische Flussdichte, ${\displaystyle \mathbf {E} }$ das elektrische Feld, ${\displaystyle \mathbf {D} }$ die elektrische Flussdichte,${\displaystyle \mathbf {P} }$ die Polarisation, ${\displaystyle \mathbf {M} }$ die Magnetisierung, ${\displaystyle \mathbf {j} }$ die Stromdichte und ${\displaystyle \rho }$ die Ladungsdichte ist

Für den stationären Fall gilt:

${\displaystyle \nabla \cdot \mathbf {j} =-{\frac {\partial \rho }{\partial t}}}$

Für isotrope Medien mit der Leitfähigkeit ${\displaystyle \sigma }$ gilt:

${\displaystyle \mathbf {j} =\sigma \mathbf {E} }$

${\displaystyle \mathbf {A} }$ ist das Vektorpotential, mit dem man das ${\displaystyle \mathbf {B} }$-Feld berechnen kann.

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$

${\displaystyle \mathbf {A} ={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {j} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}r'}$

Mithilfe des skalaren Potentials ${\displaystyle \mathbf {\Phi } }$, kann man auch das ${\displaystyle \mathbf {E} }$-Feld berechnen.

${\displaystyle \mathbf {E} =-\nabla \Phi -{\frac {\partial \mathbf {A} }{\partial t}}}$

${\displaystyle \Phi ={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}r'}$

${\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}$

unter Vernachlässigung des E-Feldes folgt

${\displaystyle \mathbf {F} =\int d\mathbf {r} \rho \mathbf {v} \times \mathbf {B} =\int d\mathbf {r} \mathbf {j} \times \mathbf {B} }$

${\displaystyle L_{jm}={\frac {\mu _{r}\mu _{0}}{4\pi }}\oint _{C}\oint _{D}{\frac {d\mathbf {r} \cdot d\mathbf {r} '}{|\mathbf {r} -\mathbf {r} '|}}=L_{mj}}$ wobei C und D geschlossene Leiterwege um die Leiterkreise j und m sind

Um die inhomogene Wellengleichung zu lösen, benutzt man retardierende Potentiale. Man erhält dann folgendes: ${\displaystyle \mathbf {A} ={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {j} (\mathbf {r} ',t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}})}{|\mathbf {r} -\mathbf {r} '|}}d^{3}r'}$

${\displaystyle \Phi ={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\mathbf {\rho } (\mathbf {r} ',t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}})}{|\mathbf {r} -\mathbf {r} '|}}d^{3}r'}$

${\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} }$

${\displaystyle -\mathbf {j} \cdot \mathbf {E} ={\frac {\partial w}{\partial t}}+\nabla \cdot \mathbf {S} }$

${\displaystyle w={\frac {1}{2}}(\mathbf {H} \cdot \mathbf {B} +\mathbf {E} \cdot \mathbf {D} )}$

${\displaystyle \nabla \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \Phi }{\partial t}}=0}$

${\displaystyle \Box \mathbf {A} =-\mu _{0}\mathbf {j} }$

${\displaystyle \Box \Phi =-{\frac {\rho }{\varepsilon _{0}}}}$

${\displaystyle \nabla \cdot \mathbf {A} =0}$

${\displaystyle \Box \mathbf {A} =-\mu _{0}\mathbf {j} -{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$

${\displaystyle \nabla ^{2}\Phi =-{\frac {\rho }{\varepsilon _{0}}}}$

wobei ${\displaystyle \Box }$ der d'Alembertoperator ist.