Dipolstrahlung Matrix

113 days ago by jg42

omega,k,epsilon,mu,c,f1,f2 = var("omega,k,epsilon,mu,c,f1,f2") 
       
a=i*k;b=i*omega/c;d=epsilon*mu 
       
A = matrix([[0,-a,-b,0],[a,0,0,-b],[b*d,0,0,-a],[0,b*d,a,0]]);A 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
0 & -i \, k & -\frac{i \, \omega}{c} & 0 \\
i \, k & 0 & 0 & -\frac{i \, \omega}{c} \\
\frac{i \, \epsilon \mu \omega}{c} & 0 & 0 & -i \, k \\
0 & \frac{i \, \epsilon \mu \omega}{c} & i \, k & 0
\end{array}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
0 & -i \, k & -\frac{i \, \omega}{c} & 0 \\
i \, k & 0 & 0 & -\frac{i \, \omega}{c} \\
\frac{i \, \epsilon \mu \omega}{c} & 0 & 0 & -i \, k \\
0 & \frac{i \, \epsilon \mu \omega}{c} & i \, k & 0
\end{array}\right)
A.det() 
       
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, \epsilon k^{2} \mu \omega^{2}}{c^{2}} + \frac{\epsilon^{2} \mu^{2} \omega^{4}}{c^{4}} + k^{4}
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{2 \, \epsilon k^{2} \mu \omega^{2}}{c^{2}} + \frac{\epsilon^{2} \mu^{2} \omega^{4}}{c^{4}} + k^{4}
f = vector([f1,f2,0,0]);f 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left(f_{1},\,f_{2},\,0,\,0\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(f_{1},\,f_{2},\,0,\,0\right)
(A.inverse()*f).simplify() 
       
\newcommand{\Bold}[1]{\mathbf{#1}}\left(-{\left(\frac{\epsilon \mu \omega^{2}}{{\left(\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} - i \, k\right)} c^{2} k^{2}} + \frac{i}{k}\right)} f_{2},\,-{\left(\frac{\epsilon \mu \omega^{2}}{{\left(-\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} + i \, k\right)} c^{2} k^{2}} - \frac{i}{k}\right)} f_{1},\,\frac{\epsilon f_{1} \mu \omega}{{\left(-\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} + i \, k\right)} c k},\,-\frac{\epsilon f_{2} \mu \omega}{{\left(\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} - i \, k\right)} c k}\right)
\newcommand{\Bold}[1]{\mathbf{#1}}\left(-{\left(\frac{\epsilon \mu \omega^{2}}{{\left(\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} - i \, k\right)} c^{2} k^{2}} + \frac{i}{k}\right)} f_{2},\,-{\left(\frac{\epsilon \mu \omega^{2}}{{\left(-\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} + i \, k\right)} c^{2} k^{2}} - \frac{i}{k}\right)} f_{1},\,\frac{\epsilon f_{1} \mu \omega}{{\left(-\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} + i \, k\right)} c k},\,-\frac{\epsilon f_{2} \mu \omega}{{\left(\frac{i \, \epsilon \mu \omega^{2}}{c^{2} k} - i \, k\right)} c k}\right)