spontaneous-force optical trap,". Ketterle, K. B. Davis, M. A. BEC in many diff Joffe,. Martin, and D. E. Pritchard, Phys. Rev. Lett. 70, 2253(1993). A. Gallagher, and. Wieman, "Colli- trap,"Phys. Rev. Lett. 63, 961-964 C. Wieman, "Collective behavior of 64,408--411(1990);D Behavior of neutral atoms in a spon- Cornell, "Behavior of neto-optical trap, "J. Opt. Soc. Am. B 11, (North-Holl nd E. A. Cornell, "Stable ing of neutral atoms," U.P., Cambridge tom traps," in Pro- . Raab, M. Prentis Physics, edited neutral sodit cientific, Singapore, 2634(1987) C. Monroe, Einstein condensate in a phys.rev.a53,r1954-1957(1996 0. Mewes, N.. vanDruten, D. S. Durfe, D. M. Dyrn, of light-assisted collisic phys.rev.a50,3597-3600 atoms in Phys. Rev. Lett. 75, 3969(1995) Surface charges on circuit wires and resistors play three roles Unive. Jac niversity of Ca lifornia 94720-7300 (Received 11 Sept d 1 November 1995) The significanc s assciated with current-carrying circuits is often not apprec surface ctors of a current-carrying circuit ircuit must have nonuniforr orm ele in the potential around the circuit. (2) to provide the surface n vary greatly, depending on the location and orienta s with a circuit consisting of a resistor and a rmits solution with a four tor sizes and location of the batter res ow from the battery to the resistiv irtace charge densities around h the resistor removed. F lements, defined in terms of the surface the same as the capacitance of th oen circuit alone. The discussion is in terms of time-independent currents applies also to low-frequency ac circuits. 1996 American Association . INTRODUCTION A cursory inspection of some beginning undergraduate The ideas of electric charges and potentials of conducting texts-in the Berkeley Physics Library showed that only surtaces in electrostatics on the one hand and current flow in one (the new book by Chabay and Sherwood mentione d elementary physics textbook showing circuit has plus and minus signs next tricity and magnetism ther this is a hint charges, then cor charges presen face charge d treat a practic simple cir scribed as charges ments of the circuit, but able ex or ad into current densities c rrents obeying Ohm's law. In vanced texts15-18 are no better. My book does not even treat electrostatics, charges are always stationary; in circuits, circuits, except in a few pr charges are always in motion. ms associated with capaci- tance or inductance. It is very true that the amounts of charge 855 Am. J. Phys. 64(7), July 1996 1996 American Association of Physics Teachers 855
on the wires in a circuit are generally small-the capacitance per unit length of an ordinary lamp cord is measured in pi cofarads per meter-but they are significant, nonetheless. Z=L Over the years the pages of this journal and a few books have contained discussions of one or another aspect of the electric fields or stationary surface charges associated with current-carrying conductors or simple circuits. 9-30Alread mentioned are the books by Jefimenko and Chabay and resistor Sherwood. The correct analytic solution for the special case of a uniform straight cylindrical wire with a cylindrically symmetric return path and remote battery appeared 63 years ago in a book by Schaefer, and was published indepen 豐①=0 dently 50 or more years ago by Marcus" and by d' battery Sommerfeld. Notable at the qualitative level are the class- room demonstrations of the electric fields and charges ac- companying circuits by Jefimenko, Parker, and moreau et al.> Some of the discussion focuses on what makes a current fiow, especially what makes it turn a corner when wire is bent 23-25 That there are localized accumulations of surface charge to assure that the current does not escape from Fig. 1.Sketch A central circular the wire is made clear-"".. when the current is steady it is column of radius a and length L consists of two wires, one of length b and guided'along the conducting wire, 23" his linear varia- the other of length(L-b-d), with a cylindrical resistor of radius a and tion of the charge distribution [for a system of long straight length d between (shaded region). The wires have resistivity Ap, the resistor, wires」 does in deed produce uniform axial electric fields R and zero resistivity. The circuit is completed by a hollow cylindrical within the conductor surfaces. 24 battery cage such that the potential on the bottom plate and for a distance nfortunately, the discussions are too qualitative or in- z=bup the cage is V. The potential falls linearly to zero at z=b'+d',and equally important aspects of the surface charges, those of battery. When the resistor p plate. The region, z=bto z=b+d'is the complete or so specialized as to omit what I believe are is zero beyond and on the in place, current flows up the central column maintaining the potential around a circuit and providing the (and b and df. when the resistor is absent. the bottom plate and wire are as magnetics of a circuit is ultimately determined by the dispo- potential V, while the top plate and wire are at zero potential sition of all the charges--in the wires as current, on the surfaces as stationary charge densities, and within the battery but consists of wires (resistivity Po)on either side of a resis- or other source of emf. Nevertheless, statements such as tor(resistivity P1). The total length of the central column is (describing a simple circuit of switch, wires, resistor, and L; the resistor is of length d; the bottom wire is of length b battery),""Surface charges are set up immediately after the the top wire is of length(L-b-d). The circuit is completed tch is closed and the resulting electric fields drive current by circular plates of radius R at z=0 and z=L, and a cylin in the circuit, 'mislead in that the surface charge densities drical battery cage at p=R, 0L/2 are the negatives of those at z-L now of finite length. The conductor is not uniform, however, and the potentials possess an obvious symmet Am J. Phys., Vol 64, No. 7, July 1996 J. D. Jackson
B>c, where ic 4n=377 Here a is the free-space wavelength associated with the fre- quency o. The inductance of the circuit of Fig. 1 is approxi mately, -(u/4T)L In(R"1a), neglecting the contribution from the interior of the central column. The putative crite rion,ama. The time- tance of the resistor is large compared to the internal resis varying flux produces an additional axial electric field at tance of the battery and that of the wires. The circuit is p=a of magnitude AE2I=Hool/4, as can be seen from opened and closed by removing and inserting the resistor the integral form of Faraday's law with an appropriate path ( think of screwing in a light bulb), with the wires and battery in the p-z plane. The electrostatic axial electric field varies otherwise undisturbed. When the circuit is open, charge is along the central column, but its order of magnitude is distributed along the surfaces of the wires in such a manner E=O(VIL). Putting V=1 and requiring that the elec- that the potential on each wire is constant and the same as at trostatic electric field be very large compared to AEZI, we the corresponding terminal of the battery. At the end of each find the criterion of approximate validity of the electrostatic wire, where the resistor would be, there is a larger accumu- description of the electric fields to be lation of charge, opposite in sign, one from the other, to Am J. Phys., Vol. 64, No. 7, July 1996 J. D. Jackso
Left-hand scale 0 -----==---===7 s卫 0.6 Z/L Left-hand scale Right-hand scale Fig. 3, Surface charge densities (radial electrical field at surface in units of V/L)and voltage drop(in units of V) along the wires op along the column is surface charge density(top) is confined to the immediate neighborhood of the resistor. when the voltage distribution along the is very different from that along the column, the charge distribution along the wire(bottom)is large and negative for z/L>0.3 in 二 dial clectric field at the wire and maintain its potential near zero provide the electric field across the gap ne circuit is circuit determined by current conservation and Ohm's law closed by inserting the resistor, current d there inside the wires and resistor, regardless of the circuit's geo- changes in the surface charges and the of various metrical configuration. But because the resistance of the rest parts of the circuit, with the potential at any point around the of the circuit is small compared to that of the resistor, almost 1. D. Jackson 858
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, 0.3L, 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 0a. Application of the right-hand rule to succes- Am J. Phys., Vol 64, No. 7, July 1996 859
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
tween the two calculations is very satisfactory, except for the one point exactly at the end of the resistor, where the finite mesh size causes a rounding off of the distribution. Smaller radii show some differences, but even at a radius of 2 units the results are only slightly poorer than those shown in Fig. The potential contours for the bottom half (00.45. with the potentials around the circuit in the two situations so similar, it is not surprising that the surface charge densities(on both the sides of the column and the ends of the wires)shown in Fig 8 are o similar. In the open circuit, the fat termination of the en of the wire results in a singular charge density at the circular edge, varying as 5 where s is the limiting distance from the edge, either in z or in p. The discrete mesh of the relaxation method cannot exhibit such a singularity, but it integrable and so the total amount of charge can be estimated reliably. Note that, while the charge at the interface of the wire and resistor is less than the charge on the end of the wire with the resistor absent, there is charge on the side of the resistor (z/L>0. 45), not present in the open circuit. It is s if the installation of the resisto ment of the charge in the immediate neighborhood, without much change elsewhere--more on this subject in the next The quantitative changes that occur with changes in the ig. 9. with 00, the charge densities on the side of 0 cylinder axis-一 sentially identical, whether current is flowing or not. Even tb350 the end and interface densities approach each other, at least on axis. When r=5, the surface charge is much diminished Fig. 7. Equipotential rs for the bottom half(O<z/Ls0.5)of the for the closed circuit as compared to the open, but still peaks open circuit( top)and closed circuit(bottom) for centered battery and resis- at the end of the resistor. tor Resistivity ratio r=50; column radius a/L=0.05(4), outer cage radius R/L=0.75(60), half-resistor length d/2L=0.05 (4), length of wires b/L 0. 45(36). Numbers in parentheses are the mesh units for the relaxation IV. COMPARISON OF TOTAL CHARGES ON alculation of the open circuit. Potential contours (left to right) are o/ CENTRAL COLUMN FOR OPEN AND CLOSED =095(005)0.50 CIRCUITS A final aspect is the total charge or capacitance associated with a resistor and its leads, compared with the charge or that are centrally located in z. We can then use the lower half capacitance of the leads without the resistor. We consider the of the cylinders(0<z/L<0.5)and determine the behavior total charge on the bottom wire of our circuit(and the charge on the adjacent one half of the resistor, when present) for a in the upper half by symmetry arguments a word needs to be said about the accuracy of the relax- symmetrically placed resistor and"battery""(either battery ation calculations in the cylindrical geometry. A coordinate of zero thickness at z/L =0.5, or the linear voltage drop transformation is needed to convert the azimuthally symmet- dong the cage). The top wire(and half resistor)have equal and opposite charge. The charge on the flat top and ric Laplace equation in p and z into an equation with a Car- discs and on the cage at p-R are not includedIn tesian Laplacian( See Appendix B). If the boundary of the circuit calculations, the total charge on the wire two-dimensional region contains the z axis (p=0), special mined via Gauss's law, with a surface of integration removed methods are needed to avoid serious loss of precision Even from the conducting surfaces in order to assure an integrand if the z axis is excluded errors creep in if the smallest value that is as smooth as possible. For the closed circuit, the of p is only a few mesh points away from the axis. To es- charge densities are integrated analytically before summation tablish plausible limits, sample relaxation calculations for the of the series surface charge density were performed with the closed cir Comparisons between the total charges on the half cuit and compared with the"""Bessel-Fourier series column, with and without the resistor, are given in Tables I solution of Appendix A (An example is given in Fig. 10, for and Il for various aspect ratios of the resistor for a resistivity which the 40X60 lattice had a central column radius of 4, ratio of 50. In Table I, the resistor is short, of length d/l equivalent to column radius a/L=0.05. The agreement be- =0.1, while the column radius is varied from a/L=0.025 to 862 Am J. Phys., Vol. 64, No. 7, July 1996 D. Jackson
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
ooaog5 charge density on end of wire vs. r/L charge density at wire resistor interface vs. r/L 0 0.1 0.2 0.3 0.4 05 Z/L(or r/L, radial distance >15 charge density on end of wire vs. r/L 0LA··…···· charge density at wit resistor interface vs. r/L 0.0 0.2 03 4 0.5 Z/L (or r/L, radial distance Fig 9. The same as Fig. 8, except that the resistivity ratio between resistor and wire is r=500(top)or 5(bottom ). The lines and points at the left, extending in r/L to a/L=0.05, are for the interface and end Am J Phys., Vol 64, No. 7, July 1996 J. D. Jackson
