The charge is concentrated close to the ends of the resistor, ith charge at the end where the negative charge where it leaves Intuition would demand this behavior-there must be a strong electric field across the resistor to maintain the current flow in a medium of high resistivity. Care must be exercised with intuition, however since the continuity of current flow and ohm's law dictates that there is a discontinuity in the internal longitudinal elec tric field at the interface between wire and resistor. Thus there are intermal surface charge densities at each end of the the free surface of the resistor do no necessarily relate to the current flow. In some situations, il lustrated below, the sign of surface charge(and normal elec- tric field) along the side of the wire seems to oppose the current flow, and in any event are unrelated to the small internal longitudinal electric field that drives the current in the highly conducting wire. harge density is illustrated by comparison of the upper and lower surface charge densities in Fig. 3, corresponding to the two locations of the battery shown in Fig. 2. The resistor, of length d/L=0. 2, is located near the bottom plate(b/L=0. 1) resistor When the battery is near z=0( Fig. 2, top), the potential drop is concentrated in the region of small z at all radial distances Above the top of the resistor (Z/L >0.3), the potential within ig. 4. Energy fiow in the circuits 2 and 3. The arrows represent the column is less than 8% of its peak value, in rough cor- relative values of the radial coordinate times the Poynting vector. The base respondence with the other parts of the circuit. The surface the surface of the column (p=a), the points (pz)are displaced upward charge density(Fig. 3, top)is localized to the resistor and the the midpoint of the resistor. In contrast, when the battery is placed near the top of the cage(Fig. 2, bottom), the potential changes from being at its peak value for almost all z values all the potential drop occurs across the resistor, formerly the at the cage(p=R=0.5L)to being near zero on the top wire gap. The charge distribution and the electric field configura- (psa, 0.3<z/L<1). Only a short distance away, the poten- tion at the resistor are changed in detail (there is now surface tial has appreciable positive values; the closeness of the con- charge on the resistor itself, and at the intemal interfaces tours implies a large radial electric field at the wire. In con- between the wires and the ends of the resistor), but not as sequence, the surface charge density becomes skewed(Fig much as one might think away from the resistor terminals, 3, bottom), with an extensive negative surface charge density the surface charges are much the same as in the absence of along most of the top wire the resistor because the voltages around the circuit are largely the same. Depending on the configuration of the vari- ous parts of the circuit, the surface charge density on the B. Energy flow from battery to resistor near(but not at)the resistor may be of the same sign The second role of the surface charge densities, the proy opposite to that in the immediate neighborhood of the end of sion of the electric field throughout the space between the the resistor. Except in the most extreme situations, the sur- circuit elements, is important for the pattern of energy flow face charge distribution along the resistor itself is the intui- described by the Poynting vector, SoEXB. The magnetic tive one-positive at the end where the current enters and field vanishes outside(z<0, z>L, or p>R), and in the inte- negative where it exits. What follows are explicit demonstra- rior region is purely azimuthal and given by Ampere's inte tions of these remarks with the circuit of Fig. 1 gral law, Bop/a- for 0<p<a and bo1/p for a<p< This(perhaps initially surprising)result follows from the ob I l. EXAMPLES OF SURFACE CHARGE DENSITIES servation that all the current flows(in the column, top and ON THE WIRES AND RESISTOR AND bottom plates, and outer cage) give rise to only azimuthal ENERGY FLOW magnetic fields that are functions of p alone--just apply the right-hand rule! The remarks on p. 170 of Ref. 15 not with- general features described in the introduction are now standing, we may examine the contributions of the current illustrated in the next several figures. Unless stated other- flow in the different segments of the circuit. The flow in the wise, the ratio of resistivities is pl/po=50. Figure 2 presents z direction within the column and in the axially symmetric the equipotentials for circuits with two different locations of return path of the outer cage clearly lead to only an azi- the localized battery, while Fig. 3 shows the corresponding muthal component of B with no dependence. The surface surface charge densities. Before noting the differences occa- current density on the top and bottom plates is radially out sioned by the different positions of the battery, we comment ward and independent of azimuth, decreasing inversely with on the grossest feature of the surface charge distributions. radius for p>a. Application of the right-hand rule to succes- Am J. Phys., Vol 64, No. 7, July 1996 859