言 0.5 0.2 Fig. 6. Surface charge density (radial electric field at surface in units of VIL) versus z/L for three different resistivity ratios, r=0.5, 5, and 50, for a centered battery and resistor Column and outer cylinder parameters are the same as in Fig. 5, with r/L=. 2. The r=0.5 example can be thought of two resistors (lead) with a wire(iron) between. The r=5 (50) example might be resistor/wires of tin/gold or iron/gold (steel/copper or nichrome/aluminum an be generated by reflection through the point (x=0.5, tial drop across the resistor (SV/V-0. 85 for r=50, 8V/V y=0). There is very little change in the surface charge den- 0.36 for r=5). Away from the end of the resistor, the sur- sity right at the end of the resistor(and not much change face charge density is negative, the more so the smaller the further away) for R/L>0.4. For smaller R values, the prox- resistivity ratio, because the potential along the bottom wire imity of the cage and its particular variation of voltage with determined by the resistive properties of the column) is de- z begins to influence the surface charge. For R/L=0.1, the creasing in z more rapidly while the cage potential at the intuitive positive spike at the end of the resistor is still same z is still at its peak value. Larger radial electric fields present, but otherwise the charge density is of opposite sign, occur for smaller resistivity ratios and are reflected in the even on most of the bottom half of the resistor. This counter- surface charge density along the wire. The example of r=0.5 intuitive behavior along the resistor can be traced to the cir- should be compared with those for r>1. It can be thought of cumstance that the potential on the nearby cage is a step as two symmetric resistors of length b(perhaps made of function at z/L=0.5, while the potential drop across the re- lead) separated by a(iron)wire of length d sistor is linear from z/L=0.45 to z/L=0.55. The reader more comfortable with field lines is invited to draw sketches of those for large and small R/a ratios in order to understand the peculiarities of the surface charge density as a function of II. COMPARISON OF SURFACE CHARGE DISTRIBUTIONS FOR CLOSED AND OPEN D Different conductivity ratios CIRCUITS Figure 6 illustrates the effect of different conductivity ra We now turn to the comparison of the surface charge dis- tios(and so different voltage drops along the wires and re- tributions for the closed circuit of Fig. 1 and the previous sistor)on the surface charge distributions for a resistor and section with those of the electrostatic system of conductors battery both centered in z. The parameters are the same as in (called open circuit, for brevity)obtained by removing the Fig. 5, except that R/L=0. 2 is fixed and the resistivity ratios high resistivity segment of the central column(shaded part in are r=0.5, 5, and 50. The intuitive spike is smaller, the Fig. 1). For simplicity and to have a finer mesh in the relax aller the resistivity ratio, in accord with the small ler poten- ation calculations, we consider only resistors and batteries Am J Phys., VoL 64, No. 7, July 1996 J D. Jackson