15 10 RL=0.5 55 nLnan U.0 0.1 0.2 0.3 0.4 0.5 Z/L Fig. 5. Surface charge density (radial electric field at surface in units of V/L)on the bottom wire and bottom half of the resistor for different values of battery ge radius R. Battery and resistor are centered in z: column radius a/L=0.05, length of resistor d/L=0. 1, length of wires b/L=0.45, battery thickness d/L=0, battery location b/L=0.5; resistivity ratio r=50. For the smallest R/L values, the proximity of the battery cage influences the surface charge istribution away from the end of the resistor, and even at its end sive infinitesimal pie-shaped segments of current fow on visible in the bottom part of Fig. 4. This flow is proportional each plate shows that the sum of these contributions will to Eo, that is, to the surface charge density. The choice of a result in only a o component of B, with the discontinuity in much smaller value of resistivity ratio (e.g, r=5)is neces Bo decreasing as 1/p for p>a. Once the magnetic field is sary to show clearly the small radially inward component of established to be azimuthal and independent of azimuth, it is s at the surface of the wires (proportional to e,x), although it safe to apply Ampere's integral law to a centered circular is very visible for the resistor. path of radius p at fixed z to determine its value(and depen- Despite these diagrams there may be a lingering belief that dence on z and p) For P<R and 0<z<L we find the stan- much of the energy flows within the wires from battery to dard result, as if the wire were infinitely long. If either or resistor. Quite the contrary! within the central column there both of p and z are outside those ranges, we find B=0. In is an azimuthal magnetic field proportional to p and only an fact, apart from the central column not being a thin conduct- axial electric field, largest in the resistor. The Poynting vec ing tube, our circuit is an ideal toroid, with its well-known tor points radially inward everywhere within the wires and magnetic field. resistor. It is proportional to p and corresponds to uniform The components of the Poynting vector are evidently only (heating) throughout a column segment of adial and axial: S oc-E2Boo-E2p, S2oEpBg Ep/p. In a given resistivity. Most of the heating is in the resistor, of Fig. 4 we display for the two battery positions of Fig. 2 the course, as the lengths and directions of the arrows in Fig. 4 relative values of components of 2 pS(p, z), the integral over just outside the column indicate azimuth of the Poynting vector, because that is the meaning The reader may wish to ponder the reason for the similari ful quantity in making a two-dimensional projection of the ties between Fig. 4(Poynting vector) and Fig. 2(potential), azimuthally symmetric three-dimensional circuit. The base special to some particular geometries and current fows of each vector is at the point(p, z) where S is evaluated, hile its length is proportional to 2pS. The flow of energy C. Influence of proximity of other circuit elements attern in space is governed by the magnetic and electric Figure 5 demonstrates another aspect of the influence of fields there, the latter determined by the locations and sizes the rest of the circuit, the proximity of the cage to the col- of the resistor and battery, as well as the resistivity ratio. umn. This time the battery and resistor are both centered at noteworthy is the significant axial flow of energy toward the z/L =0.5. The resistor is small and stubby(d/L=0. 1, a/L resistor outside but close to, the central column, especially =0.05). Only half of the range in z is shown. The other half 60 Am J Phys., Vol. 64, No. 7, July 1996 J D. Jackson
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