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The T.E. Field

For the T.E. degrees of freedom the components of the charge-current four-vector are

( S_{t}, S_{z}, S_{x}, S_{y})=\left(0,0,
\frac{\partial S}{\partial y},
-\frac{\partial S}{\partial x}\right)~.
\end{displaymath} (9)

The components of the T.E. vector potential are
( A_{t}, A_{z}, A_{x}, A_{y})=\left(0,0,
\frac{\partial \psi}{\partial y},
-\frac{\partial \psi}{\partial x}\right)~,
\end{displaymath} (10)

and those of the e.m. field are
$ E_{long.}:$ $ F_{zt}=0 $
$ E_x:$ $ \displaystyle F_{xt}=
-\frac{\partial }{\partial y}\frac{\partial \psi}{\partial t } $
$ E_y:$ $\displaystyle F_{yt}=
\frac{\partial }{\partial x}\frac{\partial \psi}{\partial t} $
$ B_{long.}:$ $\displaystyle F_{xy}=
\frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2} \right) \psi $
$ B_x:$ $\displaystyle F_{yz}=
\frac{\partial }{\partial x}
\frac{\partial \psi }{\partial z} $
$ B_y:$ $\displaystyle F_{zx}=
\frac{\partial }{\partial y}
\frac{\partial \psi}{\partial z}$
These components are guaranteed to satisfy all the Maxwell field equations with T.E. source, Eq.(9), whenever $\psi$ satisfies the inhomogeneous scalar wave equation, Eq.(5).

Ulrich Gerlach 2001-10-09