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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,

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

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

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

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

$\displaystyle \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).


next up previous contents index
Next: Spherical Coordinates Up: Cylindrical Coordinates Previous: The TM Field   Contents   Index
Ulrich Gerlach 2010-12-09