The TEM Field

The Maxwell TEM electromagnetic field components relative to the o.n. cylindrical coordinate basis, the corresponding vector potential and its source have been consolidated into Table .

Table 6.6: The $TEM$ system: All components of any $TEM$ e.m. field $(\vec E,\vec B)$ are derived from a single master scalar function, the difference $\Phi -\Psi$ between the two scalar functions. Even though both, separately, are necessary for the definition of the $TEM$ vector potential $(\vec A, \phi)$ , it is only their difference which is determined by an inhomogeneous Poisson equation and an inhomogeneous wave equation, Eqs.(6.88) and (6.89).

$TEM$ Potential
$\hat A_r$	$\hat A_\theta$	$\hat A_z$	$\phi$
$\frac{\partial\Psi}{\partial r}$	$\frac{1}{r}\frac{\partial\Psi}{\partial\theta}$	$\frac{\partial \Phi}{\partial z}$	$- \frac{\partial \Phi}{\partial t}$
$TEM$ Electric Field
$\hat E_r$	$\hat E_\theta$	$\hat E_z$
$\frac{\partial }{\partial r} \frac{\partial (\Phi-\Psi) }{\partial t}$	$\frac{1}{r}\frac{\partial }{\partial \theta} \frac{\partial (\Phi-\Psi)}{\partial t}$	0
$TEM$ Magnetic Field
$\hat B_r$	$\hat B_\theta$	$\hat B_z$
$\frac{1}{r}\frac{\partial }{\partial \theta}\frac{\partial (\Phi-\Psi)}{\partial z }$	$-\frac{\partial }{\partial \theta} \frac{\partial (\Phi-\Psi)}{\partial z}$	0
$TEM$ Source
$\hat J_r$	$\hat J_\theta$	$\hat J_z$	$\rho$
$\frac{\partial I^{~}}{\partial r}$	$\frac{1}{r}\frac{\partial I^{~}}{\partial \theta}$	$\frac{\partial J^{~}}{\partial z}$	$- \frac{\partial J^{~}}{\partial t}$

The underlying master scalar is the difference function $\Phi -\Psi$ . It satisfies the two separate equations. The first is an equation for the two-dimensional amplitude profile in the transverse plane,

$$\boxed{ \left( \frac{1}{r}\frac{\partial}{\partial r}r\frac{\part... ...1}{r^2}\frac{\partial^2}{\partial \theta^2} \right) (\Phi-\Psi)=-4\pi J^{~}~. }$$ (688)

The second is the equation for the propagation of this profile along the $z$ -direction,

$$\boxed{ \left( \frac{\partial^2}{\partial z^2} -\frac{\partial^2}{\partial t^2} \right) (\Phi-\Psi)=~~4\pi I^{~}~. }$$ (689)

these two equations are consistent because the source satisfies the charge conservation law

$$\left( \frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial}{\pa... ...frac{\partial^2}{\partial z^2} -\frac{\partial^2}{\partial t^2} \right)J=0~,$$

which is the polar coordinate version of Eq.(6.67).