Test2d25, the ‘743’ test.
This is the test described by A Asi, Proc SPIE Vol 4510, 138-147 (2001), which is called the '743' test in that publication.
This is one of the tests offered by Integrated Engineering Software (IES).
Here 'planar' symmetry is used (in the second line), so that the system extends an infinite distance in the +/- y directions.
As can be seen by running the present file, the system consists of two flat plates in the xz plane, one at -1V and at z < 0 while the other is at +1V and at z > 0. The distance between them is very small (1.E-6). Both plates are infinite in the +/- y direction and extend very far (+/- 320000) in the z direction.
For plates that are infinite in the +/- z direction, the potential distribution is
V(x,z) = u*theta, where theta = arctan(z/x) and u = 2/pi.
The components of the field are therefore
Ex = u*sin(theta)/r, Ey = -u*cos(theta)/r, where r = sqrt(x**2 + z**2).
The magnitude of this field is therefore proportional to 1/r and it has no radial component. The equipotentials are purely radial.
In the '743 test' an electron starts at (x,z) = (0,-1) with zero velocity. This electron eventually hits the right-hand plate at z = zf = 743.345446.
A program for integrating the trajectory to find this value of zf is given in the footnoted of test2d25.
In the present simulation (and also that of IES) the two plates do not touch but are separated by 1.E-6. Tests show that the value of zf is insensitive to the size of this gap, provided that it is <= 1.E-4.
In the present simulation the available accuracy parameters have been optimised to give the best resolution in a short computing time (32 seconds to find the surface charges and 10 seconds to trace the rays, using a 2GHz PC). The 'zero kinetic energy' option has been used with a 'cathode'
that consists of one segment, to simulate the required initial condition.
IES obtain zf = 741, an error of 0.32%
The present simulation gives zf = 743.342(5), an error of 0.0007%
The dependence of zf on the distribution and lengths of the segments, and on all the other available parameters can easily be studied. The error that we give here results from such a study (see the footnotes of test2d25).
The program to integrate trajectory in exact (theoretical) field is given in the footnotes to test2d25.