How can electric field representation be obtained from Enge representation using Maxwell's equations? Suppose we have a long electric capacitor. Let $L$ be its length ($z$ coordinate), $W$ its width ($y$ coordinate), and $D$ its full height (full aperture; $x$ coordinate). Let $L\gg W\gg D$.
The shape of it is similar to that of parallel plate capacitor. Actually, it is an electrostatic deflector and is slightly curved (general concept shown in figure), which can apparently be ignored to some extent.

The fringe field falloff is described by the Enge function
$$
F\left(z\right)=\frac{1}{1+\exp\left(a_{1}+a_{2}\left(\frac{z}{D}\right)^{2}+a_{3}\left(\frac{z}{D}\right)^{3}+a_{4}\left(\frac{z}{D}\right)^{4}+a_{5}\left(\frac{z}{D}\right)^{5}\right)}
$$

How can the field near the reference orbit, which runs through the centerline of the device and onwards, be found from the Enge function using Maxwell's equations?
The Enge function coefficients were obtained from analysis of the degenerate case of a parallel plate capacitor composed of infinite semi-planes.
 A: Let $E_{0}$ be the $x$ component of the electric field at $z\rightarrow-\infty$. Then the $x$ component of the electric field $E\left(0,0,z\right)
 $ on the reference orbit can be approximated by $E_{x}\left(0,0,z\right)
 =E_{0}F\left(z\right)$.
By Gauss's law $\nabla E\left(x,y,z\right)=0$ near the reference orbit. Assuming the absence of magnetic fields, the Maxwell-Faraday's equation gives $\nabla\times E\left(x,y,z\right)=0$. Also, due to midplane symmetry and $W\gg D$ we can take $\frac{\partial}{\partial y}E\left(x,y,z\right)=0$ near the reference orbit, as well as $E_{y}\left(x,y,z\right)=0$.
The relevant Maxwell's equations are:


*

*$\frac{\partial}{\partial x}E_{x}+\frac{\partial}{\partial z}E_{z}=0$; and

*$\frac{\partial}{\partial x}E_{z}+\frac{\partial}{\partial z}E_{x}=0$.


We apply the second equation at points $\left(0,0,z\right)$ to obtain the Taylor's expansion of $E_{z}$ by $x$ and $z$. Then we apply the first equation at points $\left(x,0,z\right)$ to obtain the Taylor's expansion of $E_{x}$ by $x$ and $z$.
