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Next: The T.E.M. Field Equations Up: The Method of the Previous: The T.E. Field

The T.M. Field

For the T.M. degrees of freedom the source and the electromagnetic field are also derived from a solution to the same inhomogeneous scalar wave Eq.(5). However, the difference from the T.E. case is that the four-vector components of the source and the vector potential lie in the Lorentz $(t,z)$-plane. Thus, instead of Eqs.(9) and (10), one has the T.M. source

( S_{t}, S_{z}, S_{x}, S_{y})=\left(
\frac{\partial S}{\partial z},
\frac{\partial S}{\partial t},0,0 \right)
\end{displaymath} (11)

and the T.M. vector potential
( A_{t}, A_{z}, A_{x}, A_{y})=\left(
\frac{\partial \psi}{\partial z},
\frac{\partial \psi}{\partial t} ,0,0 \right)~.
\end{displaymath} (12)

All the corresponding T.M. field components are derived from the scalar $\psi(t,z,x,y)$:
\begin{tabular}[t]{\vert l\vert c\vert}
...ial \psi}{\partial t}$\\
\end{tabular}\end{array}\end{displaymath} (13)

These components are guaranteed to satisfy all the Maxwell field equations with the T.M. source, Eq.(11), whenever $\psi$ satisfies the inhomogeneous scalar wave equation, Eq.(5).

Ulrich Gerlach 2001-10-09