Test2d07, 7th 'benchmark test' data file for CPO2D
Capacitance of coaxial cylinders
A cylinder of radius 1mm is coaxial with another of radius 2mm. The potential difference is 1V. The total charge is found to be 8.02524E-14, compared with the correct result 8.02607E-14 ( = 2*pi*epsilon0*0.001/ln(2)). The error is therefore 0.01%, but the total number of segments is only 49.
(Also available on disk, as part of the CPO2D package.)
The number of segments used in the present example is small enough for the example to be run with the ‘demo’ version of CPO2D. Higher accuracy could of course be obtained with more segments, using the standard or full versions of CPO2D.
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.
Here 'planar' symmetry is used, so that the system extends an infinite distance in the +/- y directions. The electrodes are specified as arcs of circles that subtend 90 degrees. After reflection in the x=0 and z=0 planes these become full circles. With the planar symmetry they becomes cylinders of infinite length.
A cylinder of radius 1mm (divided into 18 segments) is coaxial with another of radius 2mm (divided into 31 segments). The potential difference is 1V. The z = 0 and x = 0 planes are used as planes of positive voltage reflection symmetry.
The printing level is put at 'a', for 'all', so that the charges are put in the output file tmp7a.dat. (These charges are entered only when they have newly calculated, and not when previously calculated charges are re-used.)
It can be seen that the total charge per unit length on the segments of the inner cylinder is 2.00630E-14 Coulomb/mm when the inner cylinder is at 1V and the outer is at 0V (see the cumulative charge for 18 segments) and is the same when the inner cylinder is at 0V and the outer at 1V. Multiplying by 4, for 2 reflection planes, the charge is therefore 8.02524E-14 Coulomb/mm. This is to be compared with the correct result 8.02607E-11 Coulomb/m ( = 2*pi*epsilon0/ln(2)). The error is therefore 0.01%, but the total number of segments is only 49.