Feldstärketensor, Phänomenologische Maxwellgleichungen

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pat
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Feldstärketensor, Phänomenologische Maxwellgleichungen

Beitrag von pat »

\partial^\mu = \left(\begin{matrix}
\frac{1}{c}\partial_t\\ 
-\partial_x\\ 
-\partial_y\\ 
-\partial_z
\end{matrix}\right)
\:\:\:\:
A^\mu = \left(\begin{matrix}
\phi \\
A_x \\
A_y \\
A_z
\end{matrix}\right)
\:\:\:\:
g_{\mu\nu}=\left(\begin{matrix} 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 \end{matrix}\right)

\varepsilon^{\mu\nu\sigma\tau} = \begin{cases}
 & +1 \text{ bei geraden Permutationen von 0123 }\\ 
 & -1 \text{ bei ungeraden Permutationen von 0123 }\\ 
 & 0 \text{ sonst }
\end{cases}

F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu

\color{red}F^{\mu\nu} = \left(\begin{matrix}
0 &\:\: \frac{1}{c}\partial_t A_x + \partial_x \phi \:\: & \:\: \frac{1}{c}\partial_t A_y + \partial_y \phi \:\: & \:\:\frac{1}{c}\partial_t A_z + \partial_z \phi\\ 
-\partial_x \phi - \frac{1}{c}\partial_t A_x \:\:& 0 & \partial_y A_x -\partial_x A_y& \partial_z A_x -\partial_x A_z \\ 
-\partial_y \phi - \frac{1}{c}\partial_t A_y \:\:&  \partial_x A_y -\partial_y A_x & 0 &  \partial_z A_y-\partial_y A_z \\ 
-\partial_z \phi - \frac{1}{c}\partial_t A_z \:\:&  \partial_x A_z -\partial_z A_x & \partial_y A_z -\partial_z A_y & 0
\end{matrix}\right)

F_{\mu\nu} = g_{\mu\alpha} g_{\nu\beta} F^{\alpha\beta}

F_{\mu\nu}=\left(\begin{matrix}
0 & \:\:-\partial_x \phi - \frac{1}{c}\partial_t A_x  \:\:&\:\:  -\partial_y \phi - \frac{1}{c}\partial_t A_y \:\:&\:\:  -\partial_z \phi -\frac{1}{c}\partial_t A_z\:\:\\ 
\frac{1}{c}\partial_t A_x + \partial_x \phi& 0 \:\:&  \partial_y A_x -\partial_x A_y  &  \partial_z A_x -\partial_x A_z \\ 
\frac{1}{c}\partial_t A_y +\partial_y \phi &  \partial_x A_y -\partial_y A_x & 0 & \partial_z A_y-\partial_y A_z \\ 
\frac{1}{c}\partial_t A_z + \partial_z \phi&  \partial_x A_z +\partial_z A_x & \partial_y A_z -\partial_z A_y & 0
\end{matrix}\right)

\widehat{F}^{\mu\nu} = \frac{1}{2}\varepsilon ^{\mu\nu\sigma\tau} F_{\sigma\tau}

\widehat{F}^{\mu\nu} = \frac{1}{2}\left(\begin{matrix} 0 & \:\:F_{23} - F_{32}\:\: & \:\:F_{31} - F_{13}\:\: & \:\:F_{12} - F_{21}\:\:\\ F_{32} - F_{23} & 0 & F_{03} - F_{30} & F_{20} - F_{02}\\ F_{13} - F_{31} & F_{30} - F_{03} & 0 & F_{01} - F_{10}\\ \:\:F_{21} - F_{12}\:\: & F_{02} - F_{20} & F_{10} - F_{01} & 0 \end{matrix}\right)

\color{red}\widehat{F}^{\mu\nu}=\left(\begin{matrix} 0 & \:\: \partial_z A_y-\partial_y A_z \:\: & \:\: \partial_x A_z -\partial_z A_x \:\: & \:\: \partial_y A_x -\partial_x A_y \:\:\\ \:\: \partial_y A_z -\partial_z A_y \:\: & 0 & -\partial_z \phi -\frac{1}{c}\partial_t A_z & \partial_y \phi + \frac{1}{c}\partial_t A_y \\ \partial_z A_x-\partial_x A_z  & \partial_z \phi +\frac{1}{c}\partial_t A_z & 0 & -\partial_x \phi - \frac{1}{c}\partial_t A_x \\ \partial_x A_y -\partial_y A_x & -\partial_y \phi - \frac{1}{c}\partial_t A_y & \partial_x \phi +\frac{1}{c}\partial_t A_x  & 0 \end{matrix}\right)



\color{red}H^{\mu\nu}=\left\langle F^{\mu\nu}_{frei} \right\rangle + \widehat{F}^{\mu\nu}_{hilf} \:\:\: , \:\:\:
4\pi M^{\mu\nu}=\left\langle F^{\mu\nu}_{mat} \right\rangle - \widehat{F}^{\mu\nu}_{hilf}
F^{\mu\nu}=F^{\mu\nu}(\vec{E},\vec{B}) \:\:\: , \:\:\: H^{\mu\nu}=H^{\mu\nu}(\vec{D},\vec{H}) \:\:\: , \:\:\: M^{\mu\nu}=M^{\mu\nu}(-\vec{P},\vec{M})

E_i = F^{i0}=\left(\begin{matrix} -\partial_x \phi - \frac{1}{c}\partial_t A_x \\
-\partial_y \phi - \frac{1}{c}\partial_t A_y \\
-\partial_z \phi - \frac{1}{c}\partial_t A_z
\end{matrix}\right) = -\vec{\nabla}\phi - \frac{1}{c}\frac{\partial}{\partial t}\vec{A}= \left(\begin{matrix} E_x \\ E_y \\ E_z \end{matrix}\right) = \vec{E}

\color{red}D_i = \left\langle F^{i0}_{frei}\right\rangle + \left\langle\widehat{F}^{i0}\right\rangle = \left\langle\vec{E}_{frei}\right\rangle + rot \vec{A}

\color{red}-4\pi P_i = \left\langle F^{i0}_{mat}\right\rangle + \left\langle\widehat{F}^{i0}\right\rangle = \left\langle\vec{E}_{mat}\right\rangle + rot \vec{A}

-\varepsilon_{ijk}B_k = F^{ij} = \left(\begin{matrix}
0 &\:\:  \partial_y A_x -\partial_x A_y  \:\:&\:\:  \partial_z A_x -\partial_x A_z \\ 
\partial_x A_y -\partial_y A_x \:\:& 0 & \partial_z A_y-\partial_y A_z \\ 
\partial_x A_z +\partial_z A_x & \partial_y A_z -\partial_z A_y & 0
\end{matrix}\right)=\left(\begin{matrix}0 & -B_z & B_y \\
B_z & 0 & -B_x \\ -B_y & B_x & 0\end{matrix}\right)

-\varepsilon_{ijk}H_k = \left\langle F_{frei}^{ij}\right\rangle + \widehat{F}_{hilf}^{ij}= \left(\begin{matrix}0 & -B_z & B_y \\
B_z & 0 & -B_x \\ -B_y & B_x & 0\end{matrix}\right) + 
\left(\begin{matrix}0 &\:\:  -\partial_z \phi -\frac{1}{c}\partial_t A_z \:\:&\:\: \partial_y \phi +\frac{1}{c}\partial_t A_y \\
\partial_z \phi + \frac{1}{c}\partial_t A_z \:\: & 0 & - \partial_x \phi - \frac{1}{c}\partial_t A_x \\
- \partial_y \phi - \frac{1}{c}\partial_t A_y & \partial_x \phi + \frac{1}{c}\partial_t A_x & 0\end{matrix}\right)

-\varepsilon_{ijk}4\pi M_k = \left\langle F_{mat}^{ij}\right\rangle - \widehat{F}_{hilf}^{ij}

\color{red}\vec{H} = \left\langle \vec{B}_{frei} \right\rangle + \vec\nabla\phi + \frac{1}{c}\frac{\partial}{\partial}t \vec{A}

\color{red}4\pi\vec{M} = \left\langle \vec{B}_{mat} \right\rangle - \vec\nabla\phi - \frac{1}{c}\frac{\partial}{\partial}t \vec{A}
Ich hab das Forum lieb, weil es schon so lange da ist und man auch Infos von höheren Semestern bekommt :)

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