Xmpl2d19, 19th 'example' data file for CPO2D
Field and potential penetration through a cylindrical mesh of round wires.
Planar symmetry is used to simulate an infinitely long cylindrical mesh composed of thin parallel wires. The mesh has a transparency of 97.5 percent and is surrounded by a complete cylinder at a different potential. It is found that the external field penetrates only a short distance through the mesh, but that there is a finite change in the potential throughout the mesh.
One of a set of files:
test2d18.dat: Field penetration through a flat mesh of flat strips.
xmpl2d19.dat: Field penetration through a cylindrical mesh of round wires.
xmpl2d20.dat: Field penetration through a flat mesh of round wires.
xmpl3d23.dat: Field and potential penetration through a flat mesh of crossed round wires.
xmpl3d29.dat: Field and potential penetration through a flat mesh of crossed flat wires.
For further details see the papers:
Short and long range penetration of fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward Nucl. Instr. Meth. A427, 363367 (1999).
Penetration of electrostatic fields and potentials through meshes, grids or gauzes, by F H Read, N J Bowring, P D Bullivant and R R A Ward, Rev. Sci. Instrum. 69, 20002006 (1998).
The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.
The mesh is composed of parallel wires (in planar symmetry). The mesh has a radius of R=1.0 and the 40 wires have radii of r0=0.002. The mesh is surrounded by a cylinder of radius R2=2.0, at a potential of .69315, to create a field of E=1.0 at the position of the mesh. The spacing of the wires is s=0.15708 (measured along the arc).
In the limit of a vanishingly small ratio r0/s, Read and Bowring (to be published) show that the potential along the axis if the cylinder is raised by
phi0 = (0.5*E*s/pi)*ln(s/(2*pi*r0)) + 0.5*E*r0,
which for the present example is 0.06414.
The program gives the value 0.05794.
The computed value can (and should) be corrected in three different ways:
(1) The number N of segments should be extrapolated to infinity. It is found that phi0 is linear in (1/N)**2, and that the extrapolated value is 0.05791.
(2) The outer potential 0.693147 should corrected by adding phi0, which gives the new value 0.6275.
(3) The radius R0 of the outer cylinder should be extrapolated to infinity. It is found that the appropriate dependence is (1/R2)**1.5, giving the value 0.06324.
The fully extrapolated value differs from the theoretical value (which applies exactly only when r0=0) by 1.4 percent.
The program has also been used to find the fully extrapolated value of phi0 for two other values of r0:
r0 
limiting 
computed value 
difference 

theoretical phi0 
(fully extrapolated) 
(percent) 
0.004 
0.04782 
0.04601 
3.48 
0.002 
0.06414 
0.06324 
1.42 
0.001 
0.08097 
0.08059 
0.47 
The theoretical and computed values seem to converge satisfactorily as r0 is decreased.
To estimate the potential penetration for a cylindrical mesh of square holes, we can use the results obtained for plane meshes composed of parallel or crossed wires. Then the penetration for a cylindrical mesh of square holes is half the penetration of a mesh of parallel wires of the same transparency. The transparency is (12r0/s) for parallel wires, and approximately (14r0/s) for crossed wires.
For example a mesh of crossed wires of radius 0.002 has the same opacity as a mesh of parallel wires of radius 0.004, and so the value of phi0 would be approximately 0.5*0.4601=0.2301.
The above data file can also be modified to simulate a thin wire outside a cylindrical mesh. This can be done by using the (x,z) reflection symmetries (0,1) (on the 6th line of data), by placing the external wire at nonzero x, zero z, and by adding a second copy of the mesh wires with negative x.
The following results are then obtained (not extrapolated) for a mesh of radius R=1.0, constructed of 40 parallel wires of radius r0=0.002, outside which is a second cylinder of radius R2 and potential V2 centred at (x,z)=(X,0):
X 
R2 
V2 
phi0 
1.5 
0.05 
1.0 
1.88078E02 
2.0 
0.05 
1.0 
1.50837E02 
3.0 
0.05 
1.0 
1.22758E02 
5.0 
0.05 
1.0 
1.01328E02 
10.0 
0.05 
1.0 
8.26179E03 
2.0 
0.02 
1.0 
1.23766E02 
2.0 
0.05 
1.0 
1.50837E02 
2.0 
0.10 
1.0 
1.80870E02 
The field inside the mesh is nonzero (but small).
The above values of phi0 presumably scale approximately as
phi0 = V2*s*ln(s/(2*pi*r0)).