The T.E.M. Field Equations

There are also the T.E.M. degrees of freedom. For them the Maxwell four-vector source

$$( S_{t}, S_{z}, S_{x}, S_{y})=\left( \frac{\partial I}{\part... ...c{\partial J}{\partial x},\frac{\partial J}{\partial y}\right)$$ (14)

is derived from two functions $I(t,z,x,y)$ and, $J(t,z,x,y)$, scalars on the 2-D Lorentz plane and the 2-D-Euclidean plane respectively. They are, however, not independent. Charge conservation demands the relation

$$\left( {\partial^2 \over \partial t^2} -{\partial^2 \over \... ...\over \partial x^2}+ {\partial^2 \over \partial y^2} \right)J$$

The T.E.M. four-vector potential

$$( A_{t}, A_{z}, A_{x}, A_{y})=\left( \frac{\partial \phi}{\p... ...tial \psi}{\partial x},\frac{\partial \psi}{\partial y}\right)$$ (15)

has the same form, but only the difference $\phi-\psi$ is determined by the field equations. Indeed, the T.E.M. field components are derived from this difference:

$E_{long.}:$	$$F_{zt}= 0$$
$E_x:$	$$F_{xt}=\partial_x A_t-\partial_t A_x= \frac{\partial }{\partial x}\frac{\partial (\phi-\psi)}{\partial t}$$
$E_y:$	$$F_{yt}=\partial_y A_t-\partial_t A_y= \frac{\partial }{\partial y}\frac{\partial (\phi-\psi)}{\partial t}$$
$B_{long.}:$	$$F_{xy}=0$$
$B_x:$	$$F_{yz}=\partial_y A_z-\partial_z A_y= \frac{\partial }{\partial y} \frac{\partial (\phi-\psi) }{\partial z}$$
$B_y:$	$$F_{zx}=\partial_z A_x-\partial_x A_z= -\frac{\partial }{\partial x} \frac{\partial (\phi-\psi)}{\partial z}$$

This e.m. field satisfies the Maxwell field equations if any two of the following three scalar equations,

$$-\left( {\partial^2 \over \partial x^2}+ {\partial^2 \over \partial y^2} \right)(\phi-\psi) = 4\pi I$$ (16)

$$\left( -{\partial^2 \over \partial t^2}+ {\partial^2 \over \partial z^2} \right)(\phi-\psi) = 4\pi J$$ (17)

$$\left( {\partial^2 \over \partial t^2} -{\partial^2 \over \partial z^2} \right)I = \left( {\partial^2 \over \partial x^2}+ {\partial^2 \over \partial y^2} \right)J$$ (18)

are satisfied. The last equation is, of course, simply the conservation of charge equation. Furthermore, it is evident that the T.E.M. field propagates strictly along the $z$-axis, the direction of the Pointing vector.