charge density on end of wire vs. r/ charge density at wire sistor interface vs. r/L Z/L (or r/L, radial distance) Fig. 8. Surface charge densities along the side and end of the bottom wire for the open circuit(solid points)and along the side and interface between wire i he gap beyond the end of the Fig. 7, with resistivity ratio, /=50. The last three underlined points on the right are ues terface and end.The and resistor for the closed circuit(continuous curves). The line and points at the left, extending in r/L to a/L=0.05, are for the figuration and parameters are 0.250. The resistor's aspect ratio thus varies from that of tor of Table Il. For each value of a/L there is a monotonic oup can to a deep-dish apple pie. In units of Va, the total increase of the ratio with increasing r. The small differences charge in either situation varies by approximately 15%0-20% between r=500 and r=2000 show that the limiting values as a function of a/l, except for the smallest a/L value. The for r-o0 cannot be much greater(independent investigation ratio of the closed-circuit charge to the open-circuit charge confirms this belief). The ratio does not approach unity, at varies by only 2%-3% as a/L changes by a factor of 10, and least for the range of geometries shown. As indicated above, is eclosed/@onen=0.85-0.88 for both styles of battery. With a the differences in detail of the charge distributions for the centered battery, but a resistivity ratio of 5, closed and open circuits precludes a ratio of unity except in eclose/@open =0.253-0.263 for the same range of a/L. For a the extreme circumstances of the wires separated by a gap resistivity ratio of 500, &closed open=1.04-+1. o1 as all (resistor)that is very small compared to their diameter for 0.025-0.250. The approach of the ratio to unity as r-o which almost all the charge is found on each end surface is not universal, but a reflection of the large ratio of radius to (interface)]. Even at the largest a/L value in Table Ill, the nly in the limit of a/d>l (ar charges approach the naive parallel plate capacitor result aspect ratio is only 2a/d=5/4 with negligible fringing fields The sampling of results in Tables I-Ill indicates that, at least for practical resistors with large resistances compared to esistor(d/L=0.4)and a different set of aspect ratios, are the connecting leads, the total charge or equivalently the ca- sented in Table Il(for a centered zero-thickness battery). pacitance of the resistor -leads combination is approximately For a/L=0.025, the two numbers for gonen are for two dif- the same(at the 15%0-25% level)as is found for the same ferent Gauss's law surfaces; for larger alL, the two surfaces circuit configuration, but with the resistor removed. If one yielded the same results to within less than 0.3%. Here the speaks of the capacitance of the wires for the open circuit, different aspect ratio of the resistor leads to @closed/ @open one may equally speak of the capacitance of the wires and larger than unity by 5%0-20% for r=50 resistor. A resistor and its leads are one extreme of a lossy Table I addresses the question of behavior of the ratio of capacitor, with rather less capacitance for its resistance than charges as a function of resistivity ratio for the longer resis- one expects from a useful capacitor Am. J. Phys., Vol. 64, No. 7, July 1996 J D. Jackson