Test2d19, 19th 'benchmark test' data file for CPO2D
Potential near the tip of a cone.
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.
See also the later 3D version test3d35.
According to J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon Press, Oxford)
V = Pn(cos(theta))*Rn
Here:
R is the distance from the tip,
theta is the angle between the axis of the cone and the vector from the tip to the point,
Pn is the associated Legendre function of order n,
n is a (nonintegral) number that satisfies:
Pn(cos(theta0)) = 0
where theta0 is the angle between the axis of the cone and the surface of the cone.
In the present example, theta0 = 150 degrees (that is, the internal halfangle of the cone is 30 degrees). Therefore n = 0.346184.
Using the program given in program15 some values of Pn(cos(theta)), together with the differential with respect to theta (calculated by numerical differentiation) are:
theta 
Pn 
d(Pn)/d(theta) 
0.0 
0.0000E+00 
0.0 
30.0 
9.6795E01 
0.1229 
60.0 
8.7022E01 
0.2521 
90.0 
7.0029E01 
0.4022 
120.0 
4.3748E01 
0.6211 
150.0 
1.0174E07 
1.1649 
The total charge on the cone between 0 and R is proportional to R**(n+1). The power p for subdiving the cone has been treated empirically, and the best value has been found to be approximately 3.
The cone extends from z = 0 to z = cos(30), and is at a potential of 1V.
A sphere of radius 1 has been used as an outer shield. The formula given above has been used, with the help of the program given below, to give the required potentials at 5 degree intervals around the circle. For simplicity, linear voltage gradients are used between these values.
Results obtained:
r 
z 
potential 

er 

ez 



prog 
exact 
prog 
exact 
prog 
exact 
0.000 
0.000 
0.99961 
1.00000 
0.000 
0.000 
1.5E6 
inf 
0.000 
0.002 
0.88371 
0.88368 
0.000 
0.000 
20.13 
20.13 
0.000 
0.004 
0.85217 
0.85213 
0.000 
0.000 
12.79 
12.80 
0.000 
0.006 
0.82989 
0.82985 
0.000 
0.000 
9.815 
9.817 
0.000 
0.008 
0.81208 
0.81203 
0.000 
0.000 
8.132 
8.133 
0.000 
0.010 
0.79699 
0.79694 
0.000 
0.000 
7.028 
7.030 
0.002 
0.000 
0.91856 
0.91854 
14.10 
14.10 
23.40 
23.39 
0.004 
0.000 
0.89648 
0.89645 
8.959 
8.962 
14.87 
14.87 
0.006 
0.000 
0.88088 
0.88085 
6.873 
6.875 
11.41 
11.41 
0.008 
0.000 
0.86840 
0.86837 
5.695 
5.696 
9.450 
9.450 
0.010 
0.000 
0.85784 
0.85780 
4.922 
4.923 
8.171 
8.167 
For the potentials, the differences between the computed and exact values are of the order of 0.05 percent. In fact these errors are probably due mostly to the approximate simulation of the potential distribution on the shielding electrode. The following table shows the percentage errors obtained for various potentials and fields as a function of the size of the intervals into which the outer shield is divided for the application of voltages.
interval percentage errors
in degrees potential at (r,z) = ez at (r,z) = er at (r,z) =
 
(0,0) 
(0,.01) 
(0.01,0) 
(0,0.01) 
(0.01,0) 
(0.01,0) 
30 
.038 
.258 
.170 
.930 
.932 
1.046 
5 
.039 
.0062 
.0039 
.029 
.049 
.030 
The errors (excluding those for the tip itself) seem to be approximately proportional to the square of the interval in degrees.
Higher accuracy can of course be obtained by using more segments than the 1100 used here.
See program15 for evaluation of the Legendre function.