Transient Behavior of Simple RC Circuits

Transient behavior of simple RC circuitsNorris W. Preyera)

Department of Physics and Astronomy, College of Charleston, Charleston, South Carolina 29424

~Received 25 October 2001; accepted 26 July 2002!

I report on simulations of the evolution of surface charges, the retarded electric potential, and theelectric field in a simple resistor–capacitor~RC! circuit and in an RC circuit with a switch. Thecircuits have high resistance, so radiation and inductive effects are not significant. The simulationsillustrate nonquasistatic effects due to the finite speed of light, and should be useful in teachingstudents how circuits respond to changes. ©2002 American Association of Physics Teachers.

@DOI: 10.1119/1.1508444#

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I. INTRODUCTION

The wires in a circuit with steady currents are electricaneutral in their interiors,1 but have small amounts of chargon their surfaces. These surface charges are critical tobehavior we expect from circuits, for they make the elecfield parallel to the wires and ensure that the same amouncurrent flows in different parts of the circuit.2 If the electricfield and surface charges do not create the conditionssteady-state currents, rapid changes occur in the surcharge distribution until these conditions are satisfied. This feedback because the electric fields change the surcharges and the surface charges create the electric field.feedback between charges and the electric field is not instaneous, because changes in the field propagate at the sof light c.

Without these three ideas in mind~surface charges modifythe electric field in circuits, electric fields modify the surfacharges, and propagation delays!, even simple questionabout circuits are difficult to answer. However, with theconcepts, questions such as, ‘‘I know the charges in a wmove very slowly, so why does the light come on immeately when I flip the switch?’’ can now be answered cleaand correctly.3–6

The concepts of surface charge and feedback are brmentioned in many introductory and intermediate phystexts, but were not central to the pedagogical developmenelectrical circuits until the textbook by Chabay anSherwood.3 These authors link the usually disparate topicselectrostatics and circuits by means of the surface chaand give students a qualitative understanding of circuithavior. They also discuss the transient behavior of circwhen, for example, a switch is closed.4,7,8

In a previous paper9 I presented the results of calculationof surface charges and fields in simple resistor–capac~RC! circuits. These calculations showed, for the first timthe actual distribution of surface charges in circuits chofor their pedagogical value rather than for their analytitractability. The importance of feedback between the surfcharges and the fields in creating equilibrium in the circwas emphasized. However, that work was incomplete,cause it did not show the true time dependence of the surcharges and fields. Rather, it treated the time constant Rbeing so large that propagation delays were unimportThese calculations were sufficient to discern the feedbmechanisms at work, but could not indicate how the circwould respond to sudden changes. This work addressesimportant area.

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II. PRIOR WORK

Much study has been given to equilibrium solutions fthe surface charges and fields in circuits. There are sevcircuit geometries that are amenable to analytical solutithe infinite straight wire and coaxial wires,10–13finite coaxialwires,2,14 circular loops,15 a spherical battery,16 and a‘‘squared coil’’ 17 or ring18 in a magnetic field. Time-independent solutions are found by solving Laplace’s eqtion. However, these solutions cannot show the transienthavior of the circuit before the establishment of the steastate.

Rosser19 has given a short calculation showing the smnumber of charges needed to guide current through a bSherwood and Chabay4 have given an extensive review othis literature. Some lecture demonstrations have been plished that show the electric fields20 and surface charges~de-tected with an electroscope!21,22around a circuit. These demonstrations allow arbitrary one-dimensional and twdimensional circuits with resistors, capacitors, and batterbut cannot show transient responses, which are only alight-crossing times~the time light takes to cross the circuit!.

Jefimenko’s textbook23 was perhaps the first to recognizand discuss surface charges in an introductory text~see alsohis answer to a student question about current flow24!.Hartel25–27 discussed the pedagogical importance of surfcharges in circuits. Swartz,28 Swartz and Miner,29 andGriffiths30 are among the texts that discuss the role of surfcharges and feedback in circuits. Chabay and Sherwoo3,31

have produced an excellent introductory textbook using sface charges to link electrostatics and circuit concepts, rathan the typical text which treats these as disjoint topics.

Computer calculations have been done to illustrate theculiarities of the 1/r 2 force law32 and to augment the analytical calculations.2 White, Frederiksen, and Spoehr33 used acomputer simulation of a transport model of charges incircuit to study the effectiveness of various conceptual mels in teaching electric circuits. Little computational work33

has been done on the transient behavior of surface chargcircuits, but there is much qualitative description.3–5,27,28

Moreau has derived Kirchhoff’s laws from the transient bhavior of circuits.34

III. ANALYSIS OF THE PROBLEM

The system is a seriesLRC circuit with C50.68 pF andL520 nH. The resistance of the circuit was made relativlarge (R53.5 kV, corresponding to a resistivityr50.80V m) for reasons discussed below. The mathemat

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solution of this circuit is well known if we ignore retardatioeffects. If we write the capacitor chargeq5q0 ert , the solu-tions for r are

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where g5R/2L58.8531010 s21 and v251/LC57.3931019 s22. Becauseg2@v2, the roots are all real and thcharge does not oscillate, but decays to zero. If we usebinomial expansion and keep the positive root, we obtain

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We see that the charge will behave as it would in a simRC circuit,

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This result would be incorrect if an assumption of tmodel, uniform current motion, is violated. For the circustudied here, the surface charges and electric fields caimmediately adjust themselves to produce a uniform currRather, the time scale of the feedback process is relatethe shortest time that information can be exchanged incircuit, which is the time light takes to cross the circuit.

There are two time scales to compare in this problem:light-crossing timetc5,/c, where, is a characteristic sizeof the system, and the resistor–capacitor time constanttRC

5RC. Depending on the relative sizes of these two timthe system is in one of three different regimes:

~1! If tRC,tc , the relaxation of the circuit is limited by thspeed of light, that is, the excess charges decay asidly as possible. In this case we expect that radiationinductive effects will be important. This case is ntreated in this work because it involves solving Maxwequations with retardation effects. Complex numericodes are available for this problem,35 but the hope ofhaving students understand both the simulation tenique and the results would be lost.

~2! If tRC@tc , the relaxation of the circuit is limited by thproductRC. The speed of light is effectively infinite anpropagation delays can be ignored. This situationtreated in previous work in which it was assumed thatthe charges in the circuit instantaneously interact weach other and retardation effects can be ignored.9

~3! If tRC.tc , the dynamics of the circuit will display features due to both the finite speed of light and the finrelaxation time of the circuit. For the parameters choshere, tRC52.4 ns, tc'2.5 cm/c'80 ps, so tRC

'30 tc , which is large enough that inductive and radtive effects are not important, but sufficiently small thpropagation delays are quite visible in the simulation

A third time scale in the system is the free charge detime,30 tr5re0'7.1 ps. This is the time constant for thdissipation of excess free charge in the circuit and setstime scale for the transition of the switched circuit from opto closed~this circuit is discussed below!. This dissipation is

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a local effect, in contrast to the global nature of the RC timconstant and the light-crossing time. Note thattr andtRC areproportional to each other because each depends on thsistivity.

IV. DETAILS OF THE CALCULATIONS

The resistor–capacitor circuits are similar to those meled previously,9 following two examples in Chabay anSherwood.31 All have capacitor plates 1734.5 mm on a side,separated by 1.0 mm. The resistive connecting wires hasquare cross-section 4.5 mm on each side. Equal and osite charges are placed on the inner faces of the capaplates, and then the evolution of the circuit’s charges afields is calculated.

The circuit is assumed to obey a simple Drude model:wires are filled with equal densities of positive and negatcharges of magnitudee, and the local current density is related to the electric field and the conductivitys :

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The circuit is divided into cubic computational cells 0.5mm on a side and contains approximately 50350310523104 cells. Any excess charge is assumed to reside incenter of each cell. The electric field is calculated atcenter of each face of a cell. Charges are then moved acthe face from one cell to another, conserving charge, accing to Eq.~4! multiplied by s2Dt,

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whereEn is the normal component ofE at this surface,s2 isthe area of the face (@0.50 mm#2), and Dt50.75 ps. Thistime step is about one-half the light-travel time acrosscell.

We cannot use Coulomb’s law to find the electric fiebecause the circuit is not in quasistatic equilibriuJefimenko23,30,36 has generalized Coulomb’s law for timedependent charges and currents to the form

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wherer is the distance between the field point and the soupoint, and @r# represents the retarded charge density,@r#5r(t2r /c). The second and third terms correspond todiation from the system.37 As described in Sec. V A 1, thecircuits have an initial ‘‘free-expansion’’ phase with largcurrents and rapidly changing charge distributions. Becaof the high resistance of the connecting wires, however, eduring this phase the second and third terms in Eq.~6! werenever more than 10 % and 2 %, respectively, of the first teHence, the electric field at the face of a cell was calculasolely from the retarded charge density, consistent withassumed absence of radiative effects,

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The calculation is started by placing643105 e/mm2 onthe inner faces of the capacitor and the rest of the circumade neutral. The program makes a loop over all the cellthe circuit, calculating fields using Eq.~7!, updating charges

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using Eq.~5!, and then repeating using the updated chadistribution. Enough previous steps are held in memoryenable the retarded charges to be retrieved~or interpolated,as required!.

This calculation is done by direct summation, rather thby using a tree code38 or other sophisticated technique, bcause the number of cells is relatively small and becathese codes do not include retardation effects. This simsummation technique also means that freshman physicsdents can understand how the calculations were performThere is a substantial time penalty, however; each integrastep takes about 5 min on six 400 MHz Pentium II computthat were run in parallel.

Plots of the charge density, electric field vectors, and slar potential were made. The scalar potentialV can be ex-actly calculated from the retarded charge density,30,36

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The range of magnitudes of the electric field vectors islarge to easily plot, so the square root of the magnitude ofvectors is shown in the figures. The very large and uninesting electric fields between the capacitor plates areplotted.

V. RESULTS

Two resistor–capacitor circuits were simulated that dfered only in the arrangement of the connecting wire:

Uniform resistive wire: a single fat wire of high resistancconnects the capacitor plates. At timet50, equal and oppo-site charges are placed on the capacitor plates and the syis allowed to evolve in time~see Fig. 1!.

Fig. 1. The simple resistor–capacitor circuit 30 ps after equal and oppcharges have been placed on the capacitor plates. The diagram is asection through the midplane of the circuit. The various gray shades resent the excess charge density, ranging between65000e/mm3 ~the chargedensity on the capacitor exceeds this range, which was chosen to illusthe surface charges on the wires!. The arrows plot the square root of thelectric field. The large field vectors between the plates have been omAlso shown are the equipotentials from15.0 V to25.0 V in steps of 0.4 V.Positive~negative! equipotentials have a solid~dashed! line.


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Electrical switch: the single fat resistive wire now hasthin slice that forms a switch. If the conductivity of the slicis zero, the switch is open and there is an open circuit. Ifslice is conducting~switch closed!, then the circuit is identi-cal to the one described above. This circuit, with the swiopen, was prepared by placing equal and opposite chargethe capacitor plates. The circuit was simulated for severaltime constants, which allowed the charges to distribute theselves around the circuit and the circuit to nearly reach eqlibrium. At time t50, the insulating segment becomes coducting, simulating the closing of a switch, and the systagain evolves in time~see Fig. 2!.

A. Uniform resistive wire

Figure 1 shows the circuit after 30 ps~40 computationalsteps! have elapsed since charges were placed on the cator plates. Figure 3 shows six steps~0, 15, 30, 45, 90, and150 ps! in the evolution of the system.

1. Retardation effects

Retardation effects are visible in the left-hand upper wof the 30 ps picture~Fig. 1, or the upper-right panel in Fig3!. The steep potential gradient and large electric fielddue solely to the left-hand positively charged plate. The fifields to reach this point will come from the charges closto this point in the circuit; because changes in the fiepropagate with speedc, these charges are on the light-coof this point. These strong fields give rise to an initial freexpansion phase that rapidly removes charges from thepacitor plates~see Fig. 4!.

The potential becomes much more uniform~and the elec-tric field much smaller! to the right of this point, because thelectric field in this region has contributions from boplates. We now have the much weaker fringe field of a pof oppositely charged plates. This same pattern is sthroughout the circuit.

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Fig. 2. The second circuit, a resistor–capacitor circuit with an initiallysulating region~a switch! in the center of the lower wire. Equal and opposicharges were placed on the capacitor plates and the circuit allowed to ebrate for several time constants. This diagram shows the circuit 15 psthe switch begins conducting. The diagram is a cross section throughmidplane of the circuit. The gray scale and other details are the sameFig. 1.


plates. Thetures mor

Fig. 3. The evolution of the simple RC circuit. The diagrams are at 0, 15, 30, 45, 80, and 160 ps after charges have been placed on the capacitorgray scale and other details are the same as in Fig. 1. The electric fields are drawn 5 times larger in the last three diagrams to display small feaeclearly.

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2. Approach to equilibrium

It takes approximately 40 ps for the combined electfield and potential of both plates to reach the first turn inupper wires. During this interval there are essentiallyother surface charges to modify the large electric field ofclosest plate of charge. After this time, however, we seeface charges starting to build up on all the wires andcircuit begins to reach steady state. It takes at least one licrossing time (tc'80 ps, lower-middle panel of Fig. 3! be-fore the electric field vectors in different parts of the circbegin to point correctly, and after 2tc ~lower-right panel! thefield vectors are still not uniform in magnitude.

Fig. 4. The solid line is the charge on one capacitor plate plotted afunction of time. The dashed line corresponds to an exponential decaya time constant of 2.67 ns.


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The feedback processes in circuits3,9 become clear afterabout two light-crossing times~compare the last two panelof Fig. 3!. As charges move into the upper corners of tcircuit, the surface charges there increase until they dcurrent around the bend and down. We see this effectflected in the changing direction ofE, from pointing horizon-tally into the corner to being tipped downward. We also ssubtle modifications of surface charge gradients betwthese two panels. These surface charges gradients arepart of a feedback loop that ensures that the same curflows throughout the circuit.3,10–12,39

3. When is the time constant equal to RC?

Figure 4 shows a graph of the charge on one capacplate as a function of time and shows that it takes aboutps ('4tc) before the decay becomes approximately exnential with a time constant of 2.67 ns~close to the estimatedtRC'2.4 ns). The very large slope in the first 40 or 50corresponds to the current driven by a single capacitor plas discussed above.

Haertel27 suggests the current in such an RC circuit shochange, in a step-wise fashion, every light-crossing timethe circuit ‘‘learns’’ what other parts are doing. The initiafall in Fig. 4 can be taken as one of Haertel’s steps,thereafter the decline is smooth. There are two reasonsthe discrepancy: his example assumes thattRC'4tc , so hehas a few large steps in one RC time. The simulated circhere havetRC'30tc , so there are many small steps in o

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Fig. 5. The evolution of the switch region of circuit No. 2. The diagrams are at 0, 7.5, 15, 30, 60, and 160 ps after the switch is closed. The grayother details are the same as in Fig. 1, except that the scale of the equipotentials is from13.0 V to 23.0 V in steps of 0.24 V.
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RC decay time. In addition, the circuit learns continuouabout other parts of the circuit, rather than a single, gloupdate, which also smears out the steps.

B. Switched circuit

The second circuit considered had an initially insulatiregion in the middle of one of the conducting wires. Charg


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were placed on the capacitor plates and the circuit equbrated for several RC time constants. The switch was mconducting and the circuit allowed to evolve in time.

Figure 2 shows the circuit after 15 ps~20 computationalsteps! have passed since the switch was closed. Figurshows an enlargement of the region around the switchvarious times after the switch was closed. Figure 6 showsentire circuit at various times after the switch was closed

ails are the
Fig. 6. The evolution of circuit No. 2. The diagrams are at 0, 15, 30, 60, 160, and 320 ps after the switch is closed. The gray scale and other detsame as in Fig. 1. The electric fields are drawn 5 times larger in the last three diagrams to display small features more clearly.

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1. Effects near the switch

The t50 picture of Fig. 5~the top-left panel! shows theelectric fields and potential of the~almost! equilibrated opencircuit in the region around the switch. As expected,~almost!all of the potential drop in the circuit occurs across the gand hence the electric field in this region is very large. Potive and negative surface charges, partially creating this fihave built up on either side of the switch region so itsembles another capacitor.

After 15 ps~top-right panel of Fig. 5!, the surface chargeon the ends of the switch region are beginning to neutraand the electric field between them decreases. This neuization occurs on the free-charge dissipation timetr

'7.1 ps and is almost completely finished after 30~bottom-left panel of Fig. 5!.

In addition, an unexpected effect appears: the electric filines are no longer parallel to the axis of the wires. Teffect is due to retardation and superposition: the previoualmost zero electric field in this region was the sum offields of the local switch charges~a finite capacitor! and dis-tant surface charges. The converging pattern of field lifrom distant charges originally matched the diverging pattof field lines from the finite capacitor.3 The electric field ofthe local charges has decreased, but the distant chargesnot yet learned of the switch closing so their electric field hnot changed. When distant surface charges learn ofswitch closing and begin changing, the electric field arouthe switch region decreases, straightens, and eventumatches that everywhere else in the circuit~last panel of Fig.5!. See Ref. 3, pp. 636–638 for a thorough discussion.

2. Global effects

Figure 6 shows the behavior of the entire circuit asswitch is closed. Perhaps the most interesting feature isbehavior of the equipotential surfaces. Initially, the voltaacross the switch was very large and almost all the equtentials were spaced across this gap. As news of the swclosing propagates outward with speedc, we see the surfacecharges, electric fields, and equipotentials responding inferent parts of the circuit. The feedback between curresurface charges, and electric field works to create unifocurrent flow.3,9 This requires an electric field of constamagnitude throughout the circuit, and hence evenly spaequipotentials. The last panel of Fig. 6 shows that after flight-crossing times, the electric field is approximately uform everywhere in the circuit, and the equipotential surfaare equally spaced around the circuit.

VI. DISCUSSION

The transient behavior of circuits is important for devoping a qualitative understanding of the feedback procein circuits.6 The simulations have shown how surfacharges modify the electric field and current, which in tuaffect the surface charges, leading to steady-state behaThe simulations are also helpful in demonstrating the tighcoupled nature of a circuit:27 we have to use both local andistant charges to understand the complex dynamics arothe closing switch.

We also see several distinct time scales: the light-crostime tc sets the scale for global equilibration, the excecharge timetr sets the scale for local dissipation of excecharges at switches, and the RC time constant sets thescale for the slow decay of charges and fields to zero.


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Animations and large color versions of the figures in thpaper and other figures are available athttp://galaxy.cofc.edu/rcircuits.html . The computercodes are also available upon request.

VII. SUGGESTIONS FOR FURTHER STUDY

These projects require the author’s code. The code issigned to run on multiple computers, but can be run mslowly on a single computer~depending on the number anspeed of your computers, you may need hours to weekcomputer time!. The software uses the message passingterface ~MPI!40 and the graphics are produced with thDISLIN41 library. The code is written inFORTRAN 90and willneed to be compiled.

It is necessary to keeptRC large compared totc , or theassumptions in the program will be violated.

~1! Investigate a circuit that has a resistor formed fromnarrow segment of wire or a region of increased resisity. Does the circuit come to equilibrium more quicklor less quickly, than the simulations shown here?

~2! Investigate a circuit with parallel branches. You coumake the branches have equal or different resistance

~3! Start with different charge distributions than the onshown in this paper. Does the circuit come to equilrium more quickly, or less quickly, than the simulationshown here?

ACKNOWLEDGMENT

I thank Laney Mills for his careful reading of the papera!Electronic mail: [email protected]; URL: http://galaxy.cofc.edu1But see M. A. Matzek and B. R. Russell, ‘‘On the transverse electric fiwithin a conductor carrying a steady current,’’ Am. J. Phys.36, 905–907~1968! and W. G. V. Rosser, ‘‘Magnitudes of surface charge distributioassociated with electric current flow,’’ibid. 38, 265–266~1970!, whoshow that the Hall effect produces a very small excess charge densithe interior of a wire. For a current of 1 A in a copper wire with crossectional area 1 mm2, the volume charge density is about 6000 ions/m3.

2J. D. Jackson, ‘‘Surface charges on circuit wires and resistors play troles,’’ Am. J. Phys.64, 855–870~1996!.

3R. W. Chabay and B. A. Sherwood,Matter and Interactions II: Electricand Magnetic Interactions~Wiley, New York, 2002!.

4B. A. Sherwood and R. W. Chabay, ‘‘A unified treatment of electrostatand circuits,’’ http://www4.ncsu.edu:8030/ ;rwchabay/mi/circuit.pdf .

5W. G. V. Rosser, ‘‘What makes an electric current ‘flow’,’’ Am. J. Phy31, 884–885~1963!.

6B. A. Thacker, U. Ganiel, and D. Boys, ‘‘Macroscopic phenomena amicroscopic processes: student understanding of transients in directrent electric circuits,’’ Am. J. Phys.67, ~Suppl.! S25–S31~1999!.

7B. A. Sherwood and R. W. Chabay, ‘‘Electrical interactions and the atostructure of matter: Adding qualitative reasoning to a calculus-based etricity and magnetism course,’’ pp. 23–35, in Ref. 8.

8Proceedings of the NATO Advanced Research Workshop on Learningtricity and Electronics with Advanced Educational Technology, edited byM. Caillot ~Springer-Verlag, Berlin, 1993!.

9N. W. Preyer, ‘‘Surface charges and fields of simple circuits,’’Am. J. Ph68, 1002–1006~2000!.

10A. Sommerfeld,Electrodynamics~Academic, New York, 1952!, pp. 125–130.

11A. Marcus, ‘‘The electric field associated with a steady current in a locylindrical conductor,’’ Am. J. Phys.9, 225–226~1941!.

12B. R. Russell, ‘‘Surface charge on conductors carrying steady currenAm. J. Phys.36, 527–529~1968!.

13See also A. K. T. Assis, W. A. Rodrigues, Jr., and A. J. Mania, ‘‘Telectric field outside a stationary resistive wire carrying a constantrent,’’ Found. Phys.29, 729–753~1999!; R. N. Varney and L. H. Fisher,‘‘Electric fields associated with stationary currents,’’ Am. J. Phys.52,1097–1099~1984! give a review and critique of early work on this problem.


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14A. K. T. Assis and J. I. Cisneros, ‘‘Surface charges and fields in a resiscoaxial cable carrying a constant current,’’ IEEE Trans. Circuits SystFundam. Theory Appl.47, 63–66~1999!.

15M. A. Heald, ‘‘Electric fields and charges in elementary circuits,’’ Am.Phys.52, 522–526~1984!.

16W. M. Saslow, ‘‘Consider a spherical battery...,’’Am. J. Phys.62, 495–501~1994!.

17J. M. Aguirregabiria, A. Hernańdez, and M. Rivas, ‘‘An example of surface charge distribution on conductors carrying steady currents,’’ AmPhys.60, 138–141~1992!.

18J. M. Aguirregabiria, A. Hernańdez, and M. Rivas, ‘‘Surface charges anenergy flow in a ring rotating in a magnetic field,’’ Am. J. Phys.64,892–895~1996!.

19W. G. V. Rosser, ‘‘Magnitudes of surface charge distributions associwith electric current flow,’’ Am. J. Phys.38, 265–266~1970!.

20O. Jefimenko, ‘‘Demonstration of the electric fields of current-carryconductors,’’ Am. J. Phys.30, 19–21~1962!.

21W. R. Moreau, S. G. Ryan, S. J. Beuzenberg, and R. W. G. Syme, ‘‘Chdensity in circuits,’’ Am. J. Phys.53, 552–553~1985!.

22S. Parker, ‘‘Electrostatics and current flow,’’ Am. J. Phys.38, 720–723~1970!.

23O. Jefimenko,Electricity and Magnetism~Appleton-Century-Crofts, NewYork, 1966!, pp. 295–304.

24O. Jefimenko, ‘‘Electric fields in conductors,’’ Phys. Teach.15, 52–53~1977!.

25H. Hartel, ‘‘The electric voltage,’’ pp. 353–362 in Ref. 26.26Aspects of Understanding Electricity: Proceedings of an Internatio

Conference, edited by R. Duit, W. Jung, and C. von Rhoneck~IPN/Schmidt & Klaunig, Kiel, Germany, 1985!.

27H. Haertel, ‘‘New approach to introduce basic concepts in electricity,’’5–21 in Ref. 8.


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28C. E. Swartz,Phenomenal Physics~Wiley, New York, 1981!.29C. E. Swartz and T. Miner,Teaching Introductory Physics: A Sourceboo

~AIP Press, Woodbury, NY, 1997!.30D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. ~Prentice Hall,

Upper Saddle River, NJ, 1999!.31Ref. 3, pp. 636–638.32E. Martın and R. Chicoń, ‘‘Computer assisted learning of basic concepts

electricity and electromagnetic wave propagation,’’ pp. 211–226, Ref.33B. Y. White, J. R. Frederiksen, and K. T. Spoehr, ‘‘Conceptual models

understanding the behavior of electrical circuits,’’ in Ref. 8, pp. 77–These models were designed for educational research, not physicalracy, and ignore the distinction between volume and surface chargesallow only nearest-neighbor interactions.

34W. R. Moreau, ‘‘Charge distribution on DC circuits and Kirchhoff’s lawsEur. J. Phys.10, 286–290~1989!.

35See, for example,http://www.emclab.umr.edu/emap.html orhttp://www.brunel.ac.uk/research/fdtd/emu.html

36D. J. Griffiths and M. A. Heald, ‘‘Time-dependent generalizations of tBiot–Savart and Coulomb laws,’’ Am. J. Phys.59, 111–117~1991!.

37K. T. McDonald, ‘‘The relation between expressions for time-dependelectromagnetic fields given by Jefimenko and by Panofsky and PhillipAm. J. Phys.65, 1074–1076~1997!.

38S. Pfalzner and P. Gibbon,Many-Body Tree Methods in Physics~Cam-bridge University Press, New York, 1996!.

39A. Walz, ‘‘Fields that accompany currents,’’ pp. 403–412 in Ref. 26.40The MPI software library is available for most varieties of UNIX/LINUX,

Macintosh OSX, and for WINDOWS NT/2000 from http://www-unix.mcs.anl.gov/mpi/ or http://www.lam-mpi.org/

41H. Michels, DISLIN software library, available for most varieties of UNIX/LINUX and WINDOWS NT/2000 fromhttp://www.linmpi.mpg.de/dislin/dislin.html

. This one,the extent

at the pistJr.,

Vacuum Pump. As the nineteenth century began to wind down, table-top vacuum pumps like this one were standard issue for college collectionsat Kenyon College in Gambier, Ohio, was made by James W. Queen of Philadelphia; in 1881 it cost $45, which included a syphon gauge to indicateof the vacuum. The apparatus has a density of air sphere screwed into the pump plate. One of the problems with a single-barrel vacuum pump is thonmust be raised against the pressure of the atmosphere, a hard job for a large-diameter pump piston.~Photograph and notes by Thomas B. Greenslade,Kenyon College!


Transient Behavior of Simple RC Circuits

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