The TE Field

The result of deriving the Maxwell TE electromagnetic field components Eqs.(6.43)-(6.44) from the $TE$ potential Eq.(6.81), arising from the corresponding $TE$ source - all relative to the o.n. cylindrical coordinate basis - have been consolidated into Table 6.4.

Table 6.4: The $TE$ system: All components of any $TE$ e.m. field $(\vec E,\vec B)$ , as well as those of any four-vector $TE$ potential $(\vec A, \phi)$ , are derived from a single master scalar function $\Phi ^{TE}$ . Its source scalar $S^{TE}$ determines the vectorial charge flux vector field, which is purely transverse, i.e. it is tangent to the set of nested cylinders.

$TE$ Potential
$\hat A_r$	$\hat A_\theta$	$\hat A_z$	$\phi$
$\frac{1}{r}\frac{\partial \Phi^{TE}}{\partial \theta}$	$-\frac{\partial \Phi^{TE}}{\partial r}$	0	0
$TE$ Electric Field
$\hat E_r$	$\hat E_\theta$	$\hat E_z$
$-\frac{1}{r}\frac{\partial }{\partial \theta}\frac{\partial \Phi^{TE}}{\partial t }$	$\frac{\partial }{\partial r}\frac{\partial \Phi^{TE}}{\partial t}$	0
$TE$ Magnetic Field
$\hat B_r$	$\hat B_\theta$	$\hat B_z$
$\frac{\partial }{\partial z} \frac{\partial \Phi^{TE} }{\partial r}$	$\frac{1}{r}\frac{\partial }{\partial \theta} \frac{\partial \Phi^{TE}}{\partial z}$	$-\left( \frac{1}{r}\frac{\partial}{\partial r} r\frac{\partial}{\partial r} + \frac{1}{r^2}\frac{\partial^2}{\partial\theta^2} \right) \Phi^{TE}$
$TE$ Source
$\hat J_r$	$\hat J_\theta$	$\hat J_z$	$\rho$
$\frac{1}{r}\frac{\partial S^{TE}}{\partial \theta}$	$-\frac{\partial S^{TE}}{\partial r}$	0	0

The TE master scalar $\Phi ^{TE}$ from which this result is obtained satisfies the wave equation

$$\boxed{ \left( \frac{1}{r}\frac{\partial}{\partial r}r\frac{\part... ...artial z^2} - \frac{\partial^2}{\partial t^2} \right)\Phi^{TE}=-4\pi S^{TE} }~.$$ (686)