tween the two calculations is very satisfactory, except for the one point exactly at the end of the resistor, where the finite mesh size causes a rounding off of the distribution. Smaller radii show some differences, but even at a radius of 2 units the results are only slightly poorer than those shown in Fig. The potential contours for the bottom half (0<z<0.5)of the circuit with a centered battery and centered resistor are shown in Fig 7 for the open and closed circuits with a re- sistivity ratio of 50 for nearly identical, with only a slight broadening toward z=0 for the closed circuit-the o=0.95 v contour reaches the resistor at z/L-0.3, not at z/L>0.45. with the potentials around the circuit in the two situations so similar, it is not surprising that the surface charge densities(on both the sides of the column and the ends of the wires)shown in Fig 8 are o similar. In the open circuit, the fat termination of the en of the wire results in a singular charge density at the circular edge, varying as 5 where s is the limiting distance from the edge, either in z or in p. The discrete mesh of the relaxation method cannot exhibit such a singularity, but it integrable and so the total amount of charge can be estimated reliably. Note that, while the charge at the interface of the wire and resistor is less than the charge on the end of the wire with the resistor absent, there is charge on the side of the resistor (z/L>0. 45), not present in the open circuit. It is s if the installation of the resisto ment of the charge in the immediate neighborhood, without much change elsewhere--more on this subject in the next The quantitative changes that occur with changes in the ig. 9. with 00, the charge densities on the side of 0 cylinder axis-一 sentially identical, whether current is flowing or not. Even tb350 the end and interface densities approach each other, at least on axis. When r=5, the surface charge is much diminished Fig. 7. Equipotential rs for the bottom half(O<z/Ls0.5)of the for the closed circuit as compared to the open, but still peaks open circuit( top)and closed circuit(bottom) for centered battery and resis- at the end of the resistor. tor Resistivity ratio r=50; column radius a/L=0.05(4), outer cage radius R/L=0.75(60), half-resistor length d/2L=0.05 (4), length of wires b/L 0. 45(36). Numbers in parentheses are the mesh units for the relaxation IV. COMPARISON OF TOTAL CHARGES ON alculation of the open circuit. Potential contours (left to right) are o/ CENTRAL COLUMN FOR OPEN AND CLOSED =095(005)0.50 CIRCUITS A final aspect is the total charge or capacitance associated with a resistor and its leads, compared with the charge or that are centrally located in z. We can then use the lower half capacitance of the leads without the resistor. We consider the of the cylinders(0<z/L<0.5)and determine the behavior total charge on the bottom wire of our circuit(and the charge on the adjacent one half of the resistor, when present) for a in the upper half by symmetry arguments a word needs to be said about the accuracy of the relax- symmetrically placed resistor and"battery""(either battery ation calculations in the cylindrical geometry. A coordinate of zero thickness at z/L =0.5, or the linear voltage drop transformation is needed to convert the azimuthally symmet- dong the cage). The top wire(and half resistor)have equal and opposite charge. The charge on the flat top and ric Laplace equation in p and z into an equation with a Car- discs and on the cage at p-R are not includedIn tesian Laplacian( See Appendix B). If the boundary of the circuit calculations, the total charge on the wire two-dimensional region contains the z axis (p=0), special mined via Gauss's law, with a surface of integration removed methods are needed to avoid serious loss of precision Even from the conducting surfaces in order to assure an integrand if the z axis is excluded errors creep in if the smallest value that is as smooth as possible. For the closed circuit, the of p is only a few mesh points away from the axis. To es- charge densities are integrated analytically before summation tablish plausible limits, sample relaxation calculations for the of the series surface charge density were performed with the closed cir Comparisons between the total charges on the half cuit and compared with the"""Bessel-Fourier series column, with and without the resistor, are given in Tables I solution of Appendix A (An example is given in Fig. 10, for and Il for various aspect ratios of the resistor for a resistivity which the 40X60 lattice had a central column radius of 4, ratio of 50. In Table I, the resistor is short, of length d/l equivalent to column radius a/L=0.05. The agreement be- =0.1, while the column radius is varied from a/L=0.025 to 862 Am J. Phys., Vol. 64, No. 7, July 1996 D. Jackson