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Benedict Irwin edited Electromagnetism.tex
about 10 years ago
Commit id: a995e32d24d7eab6a4fa19c7d11c5f3106450ec2
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diff --git a/Electromagnetism.tex b/Electromagnetism.tex
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I_{\mu \nu}=Im_i[H_{\mu \nu}]=\partial_\mu B_\nu - \partial_\nu B_\mu \\
J_{\mu \nu}=Im_j[H_{\mu \nu}]=\partial_\mu \Gamma_\nu - \partial_\nu \Gamma_\mu \\
K_{\mu \nu}=Im_k[H_{\mu \nu}]=\partial_\mu \Delta_\nu - \partial_\nu \Delta_\mu
\end{equation}
This means that the real component $F_{\mu \nu}$ is equal to th regular EM tensor \begin{equation}
F_{\mu \nu} = \frac{1}{c} \begin{bmatrix}
0 E_x E_y E_z \\
-E_x 0 -cB_z cB_y \\
-E_y cB_z 0 -cB_x \\
-Ez -cB_y cB_x 0
\end{bmatrix}
\end{equation}
However, \begin{equation}
F^{\alpha \beta}=\eta^{\alpha \gamma}\eta^{\beta \delta}F_{\gamma \delta}
\end{equation}
Through explicit calculation this results in \begin{equation}
\end{equation}