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