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, 0<z<L. The plates are assume the components. Furthermore for a circuit in which the re- an么re图m0 are equipotentials. the retur same current flow, depending on the location of the rest of part of the circuit at p=R is such that the electros p(R z) ere is q(R,z)=Vfor0≤z≤b sistor has a very different resistance from the low-resistance =V[1-(z-b/d'] for b'<zsb+d' and (R, z)=0 for wires connecting it, there is localization of charge at each b'+d<zsL In the limit of d-0, the potential at p=R is end of the resistor that is similar to the charge present if that of a localized ring battery at z=b. In the limit b-0, current flow is stopped by removal of the resistor. This fact d'+L, there is a uniform potential drop from z=0 to z=L demonstrates that the surface charge densities play multiple akin to the outer cage of a cylindrical time projection cham roles in keeping the current confined and maintaining the ber, a particle physics detector. The cylindrical geometry potential and fields in and around the circuit and azimuthal symmetry permits solution of the boundary What do we offer beyond the previous literature The value problem for the potential in terms of modified Bessel treatments of the long straight wire, 9-2 or localized con- functions of order zero in p and trigonometric functions in z, figurations of such wires illustrate nicely the presence of with arbitrary choices of all parameters. The details of the surface charge densities on current-carrying circuit elements, solution are given in Appendix A but do not eliminate all the apparent confusion. Our contri- Another generalization is our consideration of the com- bution is a generalization of the long wire of uniform con- parison electrostatic problem that occurs when the resistor is ity connected to a remote batter discussion clos- removed. A relaxation technique is used to obtain numerical est to ours is that of Heald, who treats a heterogeneous solutions. For simplicity of computation, we restrict our circuit in two dimensions--an infinitely long circular cylin- comparisons to examples of centrally located (in z)resistors der of negligible wall thickness, whose wall has zero resis- and either a centrally located battery(d=0, b=L/)or a ivity, except over an angular range 0=+a(region of the uniform potential drop(b=0, d'=L). The relaxation grid resistor), and a battery across 6=±丌 spans one half of the circuit shown in Fig. 1, namely,Osz We consider the circuit shown in Fig.1. The azimuthally L≤0.5(0≤i≤M)and0≤p≤R(0≤j≤N), with M≤40 symmetric geometry is retained, as is the straight central <60. For the"open circuit''examples given, the charge conductor of uniform circular cross section of radius a densities for z>L/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,am<l, can be tested In(R/a2) Since the final expression is of order unity, the condition on the resistance B assures that aT<1 We show below that the capacitance of the circuit of Fig i is of the order C=O(4丌∈y with R2/4L, the first contribution from the surface charges on the column and the second from the charge on the annular plates. The crite- rion a72<1 implies that4丌60e多≤1. But the " electro statics criterion requires ael a72=O(4丌 T∈noa The joint conditions 0 4丌2 between b/L-0.1 and(b+d)/L=0.3, and a ring battery(d =0)at can surely be satisfied for long enough wavelengths. with b'/L=0.05(top)and b'/L=0.95(bottom). The contours(left to right )are a=1 cm and L=r=10 cm, the lower end of the inequality p/v=0.95(0.05)0.05. The other circuit parameters are column radius a/L is(v/vo)2 with vo0.8 GHz, while.851 v/ohms 0.05, outer cylinder radius R/L=0.5, resistivity ratio r=50. Dotted lines To complete the discussion, consider the requirement of mark the ends of legitimate neglect of Maxwell,s displacement current. The current density is of the order J=O(/Ta2)while the dis placement current is of the order of constant. Details of the two-dimensional relaxation method D/dt=O(wEo V/L)=O(we/ B/L). The static approxima for cylindrical coordinates (p, z), including the method of tion for the magnetic field is thus valid provided handling the region containing the z axis(p=), are given in ppendix b While our discussion is phrased in terms of steady-state 龙≤377 currents and voltages, the considerations are applicable to low-frequency alternating currents. The range of applicabil- e note in passing that when the resistance equals its upper ity is for frequencies a such that wT <1 and wt2 <l, where bound, oT2=0(4Laefa2), a number somewhat \areen two er than 71=9198 and T2=C. [We are forced into somewhat un- bounds e aT2, the resistance is bracketed bet T1, are the inductive and capacitive relaxation times conventional symbols for inductance and resistance by our L previous use of L and R as lengths! At frequencies satisfy 377≤≤377 with the voltage, without phase lag or lead. The steady-state It is easy to see that the bracketing criteria for aT and. 8 fail language may be interpreted as instantaneous in time, every at y the same frequency, to wit, when the wavelength here around the circuit is no longer very large compared with the dimensions of the The range of validity of the quasistatic approximation can circuit be explored as follows. First of all, electrostatics is predi For the reader with little interest in the details, we sum cated on VxE=0, but in fact VxE=-dB/at. In the quasi- marize the rather obvious conclusions For simplicity, con- static limit, a current I(C) in the central column produces an sider a circuit consisting of a resistor connected by wires to a azimuthal magnetic induction, Bo(n=uo/(t)p/2ma for battery (or low-frequency ac source), Assume that the resis- O<p<a, and Bo(t)=Ho (t)/2mp for p>a. 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, 0.3<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