The TM Field

The result of deriving the TM electromagnetic field components Eqs.(6.43)-(6.44) from the $TM$ potential Eq.(6.83), arising from the corresponding $TM$ source - all relative to the o.n. cylindrical coordinate basis - have been consolidated into Table 6.5.

Table 6.5: The $TM$ system: All components of any $TM$ e.m. field $(\vec E,\vec B)$ , as well as those of any four-vector $TM$ potential $(\vec A, \phi)$ , are derived from a single master scalar function $\Phi ^{TM}$ . Its source scalar $S^{TM}$ determines the vectorial charge flux vector field, which is purely longitudinal.

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

The TM master scalar $\Phi ^{TM}$ for these results satisfies the wave equation

$$\boxed{ \left( \frac{1}{r}\frac{\partial}{\partial r}r\frac{\part... ...artial z^2} - \frac{\partial^2}{\partial t^2} \right)\Phi^{TM}=-4\pi S^{TM} }~.$$ (687)