A comparison of two approximations for the capacitance of a circle concentric with a cross

The maximum and minimum capacitances on circles concentric with an internal cross are determined for a four-lobed and an eight-lobed equipotential distribution. It is found that the relative difference between the maximum and minimum effective capacitances of the eight-lobed curves is about an order of magnitude lower than the same difference in the four-lobed curves, for comparable geometries. It is shown that the average and the geometric means of the maximum and minimum effective capacitances on the multilobed curves are excellent approximations of the exact values. In fact, the error in the average of the maximum and minimum effective capacitances decreases exponentially as their relative difference decreases so that the average value for the eight-lobed case is a better approximation than the average for the four-lobed case by at least an order of magnitude, in the cases of most interest. The increased accuracy obtained from the eight-lobed equipotential distribution is presented in graphical form