ampere tension

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444 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-20, NO. 2, MARCH 19

Ampere Tension in E lectric ConductorsPETER GRANEAU

A bstract-It is shown that the applicationof the Ampere and Lorentzforce laws to a closed current in a metallic circuit results in two differentmechanical orcedistributionsaround hecircuit. naddition to thetransverse forces, which both laws predict, the A mpere electrodynamicsrequires aset of longitudinal forces that subject the conductor to tension.These longitudinal forces explain electromagnetic jet propulsion and therecoil mechanism n a railgun. Pulse current experiments are describedinwhichAmpere ensionshatteredsolidaluminum wires. Electronsmoving through the metal lattice are the basic current-elements of theLorentz force theory. But Ampere assumed his current-elements to beinf ini tely divisible. With the help of computer-aided analysis and experi-ment, t is demonstrated that the amperian current-element must alsobe of finite size andnvolve at least one l attice i on in additionothe con-duction electron. Calculations with Amperes formula have been foundto give reasonable results when the atom, or uni t atomic cell, is takento be the smallest possible current-element. Some technological conse-quences of A mpere tension are discussed briefly with regard to pulsecurrents in normal conductors and steady currents in superconductors.The use of large macroscopic current-elements of unit ength-to-widthratio gives rough approximations to the A mpere tension. The accuracyof the calculations can be mproved by resolving the conductor into anumber of parallel fi laments, each fi lament being subdivided into cubiccurrent-elements.

INTRODUCTIONA. HistoricalI THE 160 Y EARS which have elapsed since Oersteds dis-covery of electromagnetism [ l ] , Amperes orce law [2]reigned supremely during the first 80years. Maxwell [3] said

It is perfect in form, and unassailable in accuracy, and itis summed up in a ormula from which all the phenomenamay be deduced, and which must always remain the cardi-nal formula of electrodynamics.This statement ofMaxwells has lost li ttleof its validity duringthe past one hundred years provided the term phenomena isrestricted to mean electri c-current phenomena in metalli c con-ductors. L orentz [4] found it necessary to substitute thetrans-verse force known by his name for amperian repulsions andattractions, primarily to explain themotion of electrons invacuum. He recognized that thismight cause confl ict wi thNewtons third law of motion. By now generations of physi-

cists have unhesitatingly assumed that themagnetic force actingon a moving electron in vacuum remains unchanged when thiselectron travels through the metal lattice.Amperes force law formed thebasis of F. E. Neumanns math-ematical theory of electromagnetic nduction [SI . The result-Manuscript rcceived May 16, 1983; revised Septembcr 28, 1983. Thiswork was supportcd by the National Scicncc Foundation under GrantThe author is with the Massachusetts Insti tuteof Technology, NW16-ECS-8012995.158, Cambridge, M A 02139.

ing Ampere-Neumann electrodynamics, as taught in the ninteenth century, embraced much indirect experimental evidencof the existence of mechanical forces of electromagneticorigwhich acted parallel to the current streamlines. The iteratuof the ti me also contains two direct demonstrations of longitdinal forces [6], 7]. The authorsdiscovery of electromagnetjet propulsion [8] and Herings [9] liquid metal pumps reprsent further demonstrations of ongitudinal A mpere forces.B. The Force Laws

In the years from 1820 to 1825 Ampere carried out manexperiments concerning the mechanical forces exerted betwecurrent-carrying metall ic conductors. Eventually he singled oufour null experiments [2], that s, experiments in which thforces balanced each other, f rom which he deduced his fundmental force law for the interactionAF,,, of two current-ements i , dmand in dn. Of the many forms in which his fomula may be written, perhaps the most seful is

AF,,, =-i , in(dm.dn/v&,,) (2 cose- 3cos a! cos0 (1where dmanddn are the lengths of the two elements and,,nis the distance between their center points. Fig. 1 depicts ththree angles of Amperes force law. The incl ination of the urent-element vectors is defined by e, and a! and are the inclnations of the current elements to the distance vector rm,n.Aall three angles appear in cosines, and since the cosine is thsame for positive and negative angles, it does not matter n whidirection the vectors are turned to bring them intocoincidenceFurthermore, of the two possible orientations of the distancvector, either wil l give the same interaction force. t helps imagine that the directional properti es of the current-elemenare vested in the element length nd not in the current.Equation (1) has been treated as the defining equation ofundamental electromagnetic units. If the currents are insertein absolute ampere (1 ab-amp=10 A), the force is given idyne.The force law for two current-elements which has been useduring the past eighty years was first proposed in 1845by Grasmann [101 who also invented the vector calculus. t is an usymmetrical law and thereforehas to be statedby two equatioone for the force AF , on the element dm, and the other fothe orce AF , on the element dn. In vector form, with thepreviously employed notation, these twoequations maybewritten

AF , =(i , in/rh,n) X (d; X T7)AF , =(i, in/rh,n) ; X (d& X x)+

--f

0018-9464/84/0300-0444$01.0001984IEEE


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GRANEAU: AMPERE TENSION I N ELECTRIC CONDUCTORS 44

idmFig. 1. Angles of Amperesforcelaw.

where the directi on of unit distance vector 1, is along the lineconnecting the elements and pointing toward the element atwhich the force s being determined.In terms of modern field theory, the total force which anelectric charge eexperiences in thepresence, at its location, ofan electricE-field and a magnetic B-field is&=e(2t;x& (3)

where u is the relative veloci ty between the harge and the ourceof the B-field. The second term of (3) .is generally called theL orentz force and, with the help of the Biot-Savart law, it caneasily be shown to be identical to the Grassmann interactionforces of (2).The original papers and books on the mpere-Neumann elec-trodynamics do not appear o have been translated intoEngiish.More than anything else, this explains the uncertainty whichhas arisen as to themeasure of agreement between the Ampereand Lorentz forces on complete circuits and circuit portions.Many textbooks on electromagnetism wri tten in the last thirtyyears do not mention the old electrodynamics. Some wronglydescribe (2) as Amperes force law. Most books today refer toan Ampere law which relates electric current to ts magneticfield. It has nothing to do wi th Amperes theory whc h wasbased on action-at;a-distance and not f ields.In this confusion it i s sometimes held that the two aws givemathematically identical esults when applied to closed circuits.This is only a half -truth. They do agree on the ector sum reac-tion force between closed circuits, but disagree on the forcedistribution around the circuits. According to both laws theforce on an isolated current-element due to a separate closedcurrent is perpendicular to the element. The sharpest disagree-ment arises with regard to the orce distribution around an iso-lated arbitrarily shaped circuit. The L orentz orces are then stilleverywhere perpendicular to the circuit while the Ampere lawnow permits and requires the existence of ongitudinal forcecomponents which give rise to the experimental phenomena fAmpere tension, electromagnetic jets[8],and the ecoil of rail-gun accelerators [111. Because of integration singularities, thequantitative evaluation of longitudinal mpere forces s entirelydependent- on computer-aided finite current-element analysis.Therefore these forces could not have been fully appreciatedprior to, say, 1,960.This paper presents a quantitative analysis of electromagnet-ically generated ension n straight conductors according toAmperes empirical law. It goes on todiscuss the technologicalconsequences which should result from this tension. L ater sec-

+ tions describe an experimental demonstration of the action ofAmpere tension in aluminumwires.11. AM PERETENSION

When two current-elements ie on the ame straight streamlinwe have in (1) cos E =cos 01 =cos /3 = 1 and the mutual forcebecomesAF,,. = in dm.dn/r&J . (4

Modeling histheory on newtonian ravitation, A mpere assumthe force between two current-elements to act along the inconnecting hem.Thenhe arrangedhis formula (1) so thapositive interaction forces would stand for repulsion and negat.ive forces for attraction. Equation(4)cannot becomenegativand therefore nvariably describes repulsion. In this way Amperes electrodynamics leads to the remarkable prediction thalong straight current-carrying metallic conductors should finthemselves in tension unless the return circuit somehow cancethe tensile action.Ampere deduced his law from a host of experimental data the same way Newton arrived at the law of gravitation from atronomical observations. Amperes summarizing paper [2] entitled Mathematical theory of electrodynamics uniquely deduced from experiments was republished as recently as 195and goes to great length to demonstrate the method of dedution.Therefore, Ampere claimed his fournullexperimentsproved the existence of the longitudinalorces revealed by (4For a subsequent confirmation of this laim he devised with dLaRive [6]anadditionalexperimentwhichdirectlydemon-strates the repulsion between .conductor portions ying on thsame straight line.Accepting this position, there emains much tobe done in thway of calculating the tensile forces in practical conductor arangements and explaining why they emained hidden to generations of physicists who were not famil iar wi th the AmpereNeumann electrodynamics. Let us begin by considering a singfilament of conductor elements. Since the currents n Amperelaw are of finite magnitude, this filament has to have a finitcross-sectional area and not be merely a ine. Near the end othe twentieth century t would be absurd to pretend, as onmight have at Amperes time, that current-elements may binfinitely thin and infinitely short. The atomicity of conducmetals is likely toset defi nite l imits to the ubdivision of whmust be considered the fundamental particles of Ampereelectrodynamics. In the absence of any other proposal we wiassume that the atom, or unit lattice cell containing one atom


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446 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-20, NO. 2, MARCHThe specific tension contribution by the m - n element cobination is

X 2T 1 i = {l/(n m) }m = l n = x + l

X this will be aaximumhen x =2/2.Next we consider the interactions of current-elements nwi th other elements in sides BC and A D. The nteractionsquestion are all repulsions. This is due to the fact that thenfunction (2 cos E - 3 cos a cos p) is negative for all relevelement combinations because cos E =0 and cos a cos /3 =cosin a with 0


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GRANEAU: AM PERE TENSION N ELECTRIC CONDUCTORS

T , / i 2/'/--- \\\/ \

/' \\

T3/i2r l I I I I I_ _ ------_--500 I

Element number ( x )Fig. 3. Specific tension in free side of squarecircuit.

T3/ i2= (-l/r&,,) (- 2 t3 cos2 CY, )os01,x zm=l V = X + I

z xt (- l /r;,,) (-2 t 3cos2CY, )osa,

n=x+1 u=1

whererk,, = (D - m)2 t z 2 (14)T i , , =(n- u)2 tz2 (15)

cosCY =( n - u)/rn,,. (1 7)cos a, =( u - rn)/rm," (16)

The totalspecif ic tension in the ireAB may then be obtainedby adding(7), (8 ) , and (13):Tx/ i2=T 1i 2 t T 2i2 t T 3i 2. (18)

Fig. 3 is a plot of the three tension components and their sumfor z=1000. In the middle ofsideAB the tension s seen to belargely due to the repulsion of in-line current-elements. Nearthe ends of the ide it is mostly produced by actionsacross thecornersA and B . Side CDmakes only a small contribution tothe tension nA B.It can easily be shown that the computed tension increaseswith z. At first sight this appears to be an unsatisfactory out-come of the Ampere electrodynamics. However this dif ficultycan be overcome by making certainassumptionsabout helength-to-width ratio of the current-element. As a first step wecalculate the most important tension contribution iven by (7)

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TABLE ICOMPUTERVAI.UA.I.IONF (7) FO R z V A R Y I N GRO M 20 TO 200A N D X =212

z Ti i 220 3.18830.593 40.88050.103 60 4.28570 4.3968090 4.5734.691100 4.796

I 10 4.891120.978 130.058 140.133 150.202160 5.266170.327 180 5.384190.438 200 5.489

across the mid-plane at x =z/2. Table I lists the results for varying from 20 to 200. A regression analysis performed onthis data revealed a very close fit toT l / i 2=0.19 t n z. (1 9

It can be shown that the specific tension contributions T2/and T3/ i2 bey similar logarithmic laws. Hence Tx/ i2will alsbe a ogarithmic function of z. For z =1000, (19) gives thspecific tension of 7.098 as compared to 7.099 obtained byfinite element analysis. This extraordinarilygood agreemengives us confidence to extrapolate (19) to much larger valueof z which otherwise would have to be ascertained by a largcomputing expenditure.Equation (19) tends to infinity with z. Hence if the currenelement is assumed to be inf initely divisible, the Ampere electrodynamics,becomes absurd. The same is true for an electrodynamics based on the L orentz force law (2). We really havno choice but to accept fi nite size elements. Could the loweelement size limit be determined by the istance between neighboring atoms? In metal attices this would be of the order olo-' cm. It would amount to lo9 current-elements inAB oFig. 2, if the latter side is 100 cm ong. Equation (1 9) thengivea specif ic tension of 20.91 which is only three times the tenobtained for z =1000. It is not an unreasonably large numbeand therefore ends some support to the idea that the atomiccell is the extent of the asic current-element.It appears plausible thatany eduction in current-elemenlength from macroscopic to microscopic dimensions should baccompanied by a similar reduction in the cross-sectional dimesion of the element. n other words thepecific tension of 20.9probably applies to a conductor of cm in diameter and 1 mlong. For conductors of larger diameter, the atomic elementconcept requires the consideration of a bunch of parallel filaments, each being essentially a string of atoms.


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448 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-20, NO. 2, M A RCH 1

F, / i211_.-F+/i2b d

1 1 1 ;

(112) rnI T (i/2)! I 1 ;

z / 2 I z /2

/ I / /1 1 I , I I I

Fig. 4 Tension across perpendicular midplane of two parallel straightfilaments.What will be the tension n two adjacent trings of atoms which

share the current of one absolute ampere? Toobtain an answerconsider the two square-section fi laments of Fig. 4. They havebeen subdivided into four portions a,b,c, andd. Each portionconsists of 2/2 cubiccurrent-elementswith heir vectors allpointing in the same direction. The shape of a cube has beenchosen for the convenience with which a solid conductor maybe subdivided into cubic cells, and not because this is the atomiccell shape. Let us nowdetermine he specific tension T I / ?across the midplane of the fi lament combination wheneachfilament carries half the total current, or i/2. The tensile forcedue to the interactionf portionsa andbcan be derived directlyfrom (7) and (19). An equal component will arise from the in-teraction of portions c and d. Let these two components beTa,band Tc,d,then

Ta,~=Tc,d=(1/4)(0.19+lnZ)i 2. (20)For the calculation of componentsa,d nd Tc,d which, becauseof symmetry, are equal to each other, we find from Fig. 4 that

rk ,n=(m+n- ) + 1 (21)cos E =122)

cos a =cosp =( m+n - l)/rm,w. (23)Applying Amperes force law to portions aand dof the filamentpair of Fig. 4 and resolving the elemental interaction force in

the direction of the current,we obtain

(2 cos E - 3cosa cos p) cosa. (Solving the simultaneous equations21)-(24) by computer, applying regression analysis to the results, revealed the logarmic relationship

Hence the tension T1across the midplane of the filament cobination is

which is smaller than the force given by (19). This result deonstrates that themperian tension will be reduced if the curdivides between two adjacent fi laments. t is an importantfect which will henceforth be called longitudinal force dilution. Adding two more fi laments to the arrangement of F4, to make up a square-section conductor, would result ispecific tensile force of (- 1.23+In z ) which is distributed othe four filamentsnd thereby dilutes the tensioneven furtheIn summary it has been shown that, for the ircuit of Fig.Amperes force law predicts the existence of tension n a currcarrying conductor, and that this tension is not predicted bthe Lorentz force law (2). In the example of the quare circa current of, say, 6000 A =600 ab-amp would create a tenforce of 2385 g, if 2 is taken to be 1000. For an elementengto-diameter ratio of one, this would be the tension in a 1mdiameter wire of 100 cm straight length. The calculated tenswould be sufficient to break an aluminumwire heated to 5OOand it can therefore be verified or denied by experiment. This good reason to believe that the same kind of finite elemanalysis would also reveal tension in rectangular and other cuit geometries. In reducing thecurrent-element size frmacroscopic to microscopic dimensions, the tension in an invidual current filament appears to ncrease by as much as a ftor of three. However when the current is distributed ovenumber of closely bunchedadjacent ilaments, longitudiforce dilution akes place and his counteracts heapparentension increase wi th element-size reduction. Elsewhere [1i t has been proved by measurement that macroscopic curreelements of the shape of cubes give results which are n gooagreement with experiment.

111. MACROSCOPICCURRENT-ELEMENTNALYSISDuring the eighty years from 1820 to900,when the Amplaw was in wide use, current-elements were treated as beingfinitely divisible. The result was a continuum theorywhich to singularities in the integration of tensile forces because rmacross the nterface of conductor portions approached zero.This probably explains why so little was written n the niteenth century about amperian tensionn electric conductoLarge current-elements of a cross-section equal to the whconductor section have often been successfully employedcalculate the reaction orces between wocomplete circu


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GRANEAU: AM PERE TENSION IN ELE CTRI C CONDUCTORS

gm25 -20 -

p 15-

T I O -5 -

,,025 i nc h d io .c-

0 00 50 100 150 200 250x103A2- *Fig. 5. L if t force measurementsonrectangular circuit.

which were separated by at least ten current-element lengths.It therefore seemed worthwhile to nvestigate if single filamentrepresentations of practical conductors can be helpful in esti-mating the magnitude of mpere tension.To dohis a 100 X 40cm rectangular circuit, madeUP of 0.25-in diameter copper rod and two liquid mercury links, was setup in a vertical plane. As shown in Fig. 5, the uppermost sideand three centimeterof each vertical leg were cut off and recon-nected with liquid mercury containedn dielectriccups attachedto the bottom portion of theircuit. The liquid metal gaps wereless than one mill imeter long. The electromagnetic lif t force lon the upper portion of the circuit was measured with a beambalance. As the results plotted on Fig. 5 indicate, the reactionforce was found tobe proportional to the quare of the current,giving a .specif ic force of Fz/i2=10.30. According to acceptedpinch force theory [12] , the liquid mercury is responsible foraspecificupward thrust of 1.00.Hence Amperes force lawshould account for 9.30 of the pecific force.In the macroscopic current-element analysis of the lift force,the circuit was modeled as a single filament of one centimeterlong elements, making perfect right-angled joints at eachorner.Apart from computing the Ampere lift force, consisting partlyof ongitudinal components in the vertical legs and partly oftransverse forces on the horizontal branch, the L orentz forceon the horizontal conductorwas also computed byapplying (2)to the one centimeter long current-elements. The results wereas follows.

Ampere: Fl/ i2=11.2047ercentongitudinal)Lorentz: Fz/i2=11.24 (all transverse)ExDeriment: Fl l i2=9.30.

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Hence the single fi lament representation wi thcurrent-elemeof a length even longer than the conductor diameterave a rougguide to the magnitude of the reaction orces between parts othe same circuit. In the chosen example the Ampere force appeared to overestimate the force by 20 percent. Part of thisdiscrepancy may have been due to experimental errors.The other interestingevelation was that, for theame elemensize, both A mperes and the L orentz force theory predicted almost identical tension forces. However in Amperes electrodynamics 47 percent of the tensionas due to ongitudinal forceswhile theL orentz orce was of course entirely a ransverseforce which produced tension indirectly. This coincidence anthe prominence given to rectangular circuits in thepast has ndoubt contributed to the belief that Amperes force law is noneeded.How shouldone proceed fromhere?The oundersof heAmpere-Neumann electrodynamics were masters of analysisbut obviously did not succeed in finding an analytical solutiofor the directly nduced. conductor tension. Computer-aidedfinite element analysis was not available to scientists of the niteenth and the first half of the twentieth century. I t thereforis deemed desirable o pursue the latter techniquelittle furtheThe finite current-element has to e of a definite shape and thcube lies close at hand.Consider an a X a square-section conductor of straight lengt1. If I >>a, the important midplane tension s then largely independent of any further ncrease in length. In the case of Fig3, where l /a =z = 1000, over eighty percent of the midplanetension is being contributed by theepulsion of in-line elementTherefore, when dealing with very long straight conductorswe may ignore the return circuit and remember that this wiunderestimate the A mpere tension.For z =10000 the specific midplane tension from 1 9) cometo 9.40. This would apply, for example, to a 100 m ong conductorofonesquare-centimeter cross-section. Assuming thcurrent to be 1000 A =100ab-amp, the single filament modepredicts a tension of only 96g, or 0.096 kg/cm2 tensile stressI t would produce a negligible amount of strain in spite of thehigh continuous current density of 1000 A /cm2. t is therefore not surprising that Ampere tension has gone unnoticed iordinary wires and cables used for the transmission and distrbution of electrical energy.The tensile stresswil l become more severe in ful ly loadedryogenically cooled conductors. For nstance, an aluminum con-ductor of thebove dimensionsheld at the temperature ofnitrogen could possibly carry 10 000 A continuously, when iwould be subject to 9.6 kg/cm2 tensile stress. Even then thestrain is quite modest.Ampere tension is likely to be of critical importance inupeconductors. Type I1 superconducting filaments embedded in copper matrix and cooled with superfluid l iquid helium casupport current densities of the order of 100000A /cm2 ovethe combined copper and superconductor area. In the 100 mlong one square-centimeter conductor this would give rise t960 kg/cm2 tensile stress which not only produces noticeablestrain but would very likely change the superconducting proerties of the rod, known toe strain sensitive.


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450 IEEE TRANSACTIONSON MAGNETICS. VOL. MAG-20, NO. 2, M A R C H 1

I I1.' I 1 '9

I I I1 1I I I I I I1 I I I I I(a) (b) (c)

Fig. 6. Cubic element subdivisionof linear conductor.The single fi lament representation of the linear conductor isthe crudest model one can use. Finer subdivision of the con-ducting matter nto smaller cubes should result n better ap-proximations to the specific tension. Therefore, et every ele-ment of Fig. 6(a) be subdivided into eight smaller cubes, asshown by (b). This multiplies the computational work by atleast a factor of 64. Angles a and /3 are then no longer zero for

all relevant current-element combinations and (24) has to beused in addition to (7). To obtain a quantitative indication ofthe force dil ution resulting from filamentubdivision weanalyze a relatively short conductor of 2 m length and 1 cm2cross-section. The return circuit would make a significant con-tribution to the maximum tension in this short conductor butwe wil l not compute this. The midplane tension due only tothe straight portion was found to besingle filament, Fig. 6(a): T I i2=5.49four fi laments, Fig. 6(b): TI iz=4.71nine fi laments, Fig. 6(c): Tl/ i2=4.55.

This example demonstrates that the computedpecific Amperetension converges quite rapidly as the number of parallel fi la-ments increases. Hence onlyamodest degree of subdivisionwill give good approximations. A lthough the ultimate current-element of the Ampere-Neumann electrodynamics is likely tobe of atomic ize, the usefulness of calculationsnvolving macro-scopic current-elements has nowbeen demonstrated.

IV. PRACT ICALONSEQUENCESOFAM PERIAN TENSIONA . Normal Conductors

It has been shown that the specific A mpere tension can ap-proach a value of ten. For most practical purposes it wouldseem appropriate to assume that this tensionwill be at leastT Si2 (dyn) (27)with i expressed in absolute ampere. Using MKS units with thecurrent I in ampere, this approximationbecomes

Tz12/(20 981000) (kg). (28)Approximation (28) has been plotted in Fig. 7. For currentsbelow 10 kA the tension s seen to be less than 5kg. Few practi-cal circuits carry more than 10kA continuously. Furthermore,for steady currents of this order the size of the cross-sectionalareas of copper and aluminum conductors are ikely to vary

600

5oo]400

/+,E300100

CurreniFig. 7. Plot of approximation (28).

between 50 and 100cm2, resulting in stresses of less than 1gm/cm2. They are negligible compared to thermally nducstresses created by J oule heating and other stresses caused conductor supports.Substantially higher currents may flow for a short timewhpower circuits are accidentally short-circuited. Fault currentof this nature may e as high as50kA(RM S) and under excetional circumstances they may reach 100kA. This means thwi th the conservative approximations(27) and(28), power coductors could experience tensile impulses of the order of 1to 500kg lasting for a few cycles of the power frequency, uthe fault current is interrupted by circuit breakers. I t will realized that ac short-circuit currents produce tension corre-sponding to theRMS current ampli tude pulsating unidirectioally at twice the power frequency. The impulses appear stroenough to damage weak links suchas conductor joints.Many pulse current events should be dominated by ongitudinal forces. Examples are the rupturing of fuses and the eplosion of wires. Of particular nterest, in ths respect, is observationmade during many exploding wire experimen[13]. If capacitors are discharged through thin wires of a feinches length, the current will at first rise to a maximum anthen decay to zero without discharging all the energy stored the capacitors. A fter this interruption and dwell lasting for seeral microseconds, the currentwill start to flowagain and copletely discharge the capacitors. Photographic evidence provethat during the dwell period the wire is broken up in a nuof pieces or li quid drops. Reignition of the explosion isbrougabout by an arc forming in the surrounding gas or vapor whienvelops the wire pieces and usually evaporates all the metaCurrents inexploding wire experiments may each 50kA whshould result in tensile forces up to100kg.I t seems ikely that the unexpected current pause observin exploding wire experiments-which has never been satisfatorily explained-is the result of Ampere tension. The followisequence of events may be visualized. First the wire rupturat some weak point as the temperature approaches themeltinpoint. A gas arc immediately bridges the gap and restores tesion. Further breaksdevelopwhichare al bridged by arcEvery arc increases the voltage drop along the wire until th


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GRANEAU: I N ELECTRICONDUCTORS 45

VIfn4000

I O 0 0

0

I ///

/ 10 cm2

C ur ren t dens i t yFig. 8. Conductor tensile stress according to (29).

equals the applied voltage and extinguishes the current. Re-ignition of thedischarge is another process not connected withAmpere tension. t couldbecaused by arc plasmadiffusionwhich ultimately pBrmits the striking of an all enveloping arcof low voitage drop.B. Superconductors

L et A (cm2) be the cross-sectionalarea of a conductorcarryinga uniformlydistributedcurrent density j (A /cm2).It then follows from (28) that the tensile stress over the cqn-ductor section is given byu = T/A j2A /(20 X 981000) (kg/cm2). (29 )

This stress has been plotted in Fig. 8forA =1, 5 , and 10 cm2and current densities up to 100kA /cm2. Magnetic flux pene-trates type I1 superconductors and, at least on a macroscopicscale, thecurrentdistributes itself approximatelyuniformlyover the conductor cross-section. Hence Fig. 8should applytovery long type 1 superconductors. If conductors of tenquare-centimeter cross-section could be loaded to 100kA /cm2, theamperian tensile stress would be formidable.Since the advent of the CBS quantum theory of superconduc-tivity it appears that quenching of the superconducting state atthe critical current level is due to charge transport. However,for manyyears itwas believed that thecritical current ,, whicha straight type I superconductor could support,was the currentwhich established the critical fieldHeat thesurface of the con-ductor. This was known as Silsbees rule, and it was largelyborne out by experiment. If r is the radius of the straight con-ductor, then the Silsbee rule states, in EMU

H, =2ie/r. (30)Let the Ampere stress or this ritical condition be u e ,then from(27)and (30) t follows that

ue =T/A =5i2 / (m2)N 1.25rH:. (31)This ast condition implies that the superconductor quenchesat some critical tensile stress and the,cri tical current as well asthe cri tical field are measures of this stress. I t is well-known

that mechanical strain, resulting from stress, changes he criticalcurrentand he critical field strength.Thequestionnow arisesto what extent the critical parameters are affected by Amperetension?Experience has revealed that superconductors of one squarecentimeter or more in ross-section will not sustain the currendensities achievable wi th thin wires. The engineering solutionof this problem haseen to disperse large numbers of very thinsuperconducting filaments in a normal metal matrix andso decrease the effective current density over the combined crosssection ofnormalandsuperconductingmetal. This practicealso decreases the longitudinal stress in the composite conductor which is likely tocontribute tothe improved performance.In the first place the Ampere tension is felt by the superconducting filaments and from them it is transferred through themetallic bonding to the normal metal matrix.This mechanismshould set up shear stress at the metal interfaces. When thisstressexceedscertain imits, dissipative slip could result andgenerate heat which may lso contribute to thequenching process. Frictional slip between parts of superconductive magnetwindings is known tocause partial or complete quenching.To further elucidate the shear action, let us take an exampwhere 10 000 superconducting wires of 0.0254 cm diameteare embedded in a copper matrix to make up composite conductor of 10 cm2 cross-sectional area. When each wire carries10 A, the total current is 500 kA , resulting in an average current density of 50 kA /cm2. For this example Fig. 8 gives atensilestress of 1250 kg/cm2 anda total tension of 12 500kg. This would set up a shear force of 15.7kg/cm of peripherof the superconductingwires. I t does not seem impossible thatforces of this magnitude will break the bond between the twmetals, at least at some of the eaker spots.C Liquid Metal ConductorsThe simple experiment of Fig. 9 is capable o f showing an effect of ongitudinal conductor forces n liquid mercury. Theconductor dimension and the current magnitude re not criticalfor observing the effect. T he experimentwill work equally welwith direct and alternating current.The authorused 0.5 in2 copper ars set in a rectangular roovein a plastic board, part of the groove forming the iquid mercurytrough. When about 300A of current were switched on, an ir-regular wave pattern became apparent on the liquid mercurysurface close to the interfaces wi th the copper rod. The wavesdisappeared almost instantaneously as the current was switchedoff. The disturbances werestrongestright at the solid-liquidinterfaces and they died out wi thin a few centimeters of theinterfaces.Apart from a slight concave surface curvature on the mercurdue to surface tension of the liquid, the conductorwas of uni-form cross-section throughout. Transverse Lorentz forcesshould have pinched the liquid conductor equally all along itslength and not create anrregular situation at thenterfaces. Nodeformation of the mercury section due to pinching could beseen at the relatively small current o f 300A. This leads to theconclusion that the disturbances esponsible f or the observablesurface wave pattern were caused by ongi tudinalA mpereforces.


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452 IEEE TRANSACTIONSON MAGNETICS, VOL. MAG-20,NO. 2, M A R C H

/Y patterni V21nch quareopperrough Sectlon 1/21nchsquareopperi II with Liquid MercuryI I4 1 - 3 0 0 A II I1 I1 I

I

Remoteeturnonductorlthurrentource

Fig. 9. Generation of surface wave patternat solid-liquid conductorinterfaces.

Even when these facts have been recognized, it remains diffi-cult to see how longitudinal forces can produce relative motionbetween some liquid metal atoms and others, as demonstratedby the wave pattern. Starting wi th the assumption that someatoms of the liquid metal are propelled away from the solid in-terfaces, it fol lows that other must return to take their places,or the iquid level would continue to all at the interfaces, whichit does not. For some of the atoms to return to the interfaces,not all can experience the same repulsion. In any case we shouldexpect the tensile stress to be strongest in the center ofthe con-ductor and weakest along the corners. This will be better under-stood in conjunction with Fig. 6(c). There the center fi lamenthas eight close neighbors which all contribute to the ension inthe center. Each of the corner filaments has only three closeneighbors and should feel a correspondingly lower tension. Itwould therefore not be unreasonable for liquid to flow awayfrom the center of the nterfaces and return to the peripheryof them.We still have to explain why, in the experiment of Fig. 9, thewave motion and flow is strongest at the interfaces and disap-pea,s in the middle section of the liquid mercury trough. Thefollowing qualitative argument deals wi th this point. Each cur-rent-element is subject to twosets of forces. One set is generatedby the far-actions of Ampere's force aw. The other is due tothe hydrodynamic push from neighboring elements caused bycontact actions taking place throughout the liquid. In a solidconductor the contact action is taken up by the crystal latticeand it cannot be diminished by the conversion of some forceinto relative acceleration of atoms or current-elements. How-ever liquid conductor-elements may be accelerated relative tothe body of the fluid. This reduces the local contact action bythe product of mass times acceleration. The fi rst liquid elementat the solid interface will be backed up less strongly by contactaction than it would be in a solid and accelerates into the bodyof the liquid. This effect tends to zero halfway between thetwo interfaces. t provides the explanation why no wave mo-tion can be seen somedistanceaway from he solid-liquidinterfaces.A nother longitudinal force effect in a liquid metal conductorwas discovered by Northrup [12]. His primary objective wasto demonstrate the pinch effect on a horizontal iquid conduc-tor. To make theeffect more obvious he used avery light metal,that is a odium-potassium alloy which remained liquid at roomtemperature and had about the density of water. With quitesmall currents hewas able to produce deepand steep V-shapeddepressions in a horizontal conductor of rectangular section.Moreover the pinch was completely stable and it was of course

attributed to the transverse Lorentz force. But Northrup observed that liquid metal flowed rapidly uphill on the ssides of the stable depression. This meant of course that liqhad to flow elsewhere to the bottom of the depression. uphill flow was parallel to the tangential surface current inmetal and in every way compatible with the action of longdinal Ampere forces. These two experimentalobservations gest that ongi tudinal Ampere orces are likely to have a jor nfluence on magneto-hydrodynamic phenomena in liqmetals.

V. EXPERIMENTALDEMONSTRATIONF THEACTIONOF AM PERE TENSIONA. Exploding Wire Circuits

The circuit used at MIT for the experimental demonstraof current-induced tensile wire breaks is shown inFig.It is typical of exploding wire circuits except for the largeductance and the relatively high voltage. The purpose of inductance was to prolong the discharge. Higher voltages tnormal had to be employed to overcome the inductive impance and strike arcs to the test wire. It has been assumed telectric arcs in air have neither tensile nor longitudinal compsive strength and no forces can be transmitted through themthe exploding wire.Most of the experiments were carried out with 1000 pHcuit inductance and 8 pF capacitance. This resulted n a suimpedance of 11.2 s2 and required at least 56 kV to establiscurrent peak of 5 kA . Additional voltage was being dropacross the switching arc and the two connecting arcs to thewire.The circuit parameters caused the discharge current to oslate at 1830 Hz in the underdamped mode. When the capacbank was charged to 60 kV , the current would decay apprmately exponentially, as shown in the oscillogram of Fig. 1lwithout breaking the wire. It has been estimated that the 6discharge was accompanied by awire temperature rise of sevhundred degrees o f centigrade which produced a thermal expansion of the orderof one percent.By subsequent increases of the ischarge voltage in 2 kV a pulse current level was reached at which the wire broke in or more places. The hot fragments would fall to the laborafloor and be distorted on impact. When repeating the expment with a new wire and 2 kV additional voltage, the wwould break into a greater number of pieces. Finally, at70the wire would show clear signs of melting which obli teraany evidence of tensile breaks. The oscillogram of Fig. 1indicates discharge current limiting and quenching due to tvoltage drop along many series-connected arcs in air.B. Straight Wires

The most conclusive evidence for the existence of Amptension was obtained with straight wires mounted as shownFig. 12(a). In this arrangement all the L orentz forces act ppendicularly to the wire axis and could not possibly prodtensile stress n the direction of the wiring axis. Y et the 1.19 diameter aluminum wire was shattered by bri ttle fracture imany pieces, as shown in the photographs of Fig. 12. Electmicrographs, obtainedwitha scanning electron microsco


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GRANEAU: A MPERE TENSION IN ELECTRIC CONDUCTORS 45 3IOOltV INDUCTORL = O t o 2 0 0 0 pH

d100k V CAPACITORSC = I 6 x0 .5 p F

1111

r-- - -,LN-ZLJTORJ/ / / / / / / / / I / MECHANICAL

SWITCH800 k R

V=Oto + - 2 0 0 k V

ILy CAMERA

OSCILLOSCOPEWITH CAMERA-

INTEGRATORFig. 10. Exploding wire circuit.

(b) . .Fig. 11. Capacitor discharge current oscil lograms. (a) Without current-limiting arcs. (b) W ith current-limiting arcs.

showed surface melting on the.fracture.faces to depth ofsev-eral micrometers which is consistent with electric arcing acrossthe fracture gap.Longitudinal Ampere forces explain the brittle tensionbreaksin thesolid wire which was weakened by J ouleheating. Can wethink of any other mechanism which could explain this surpris-ing phenomenon? Three come to mind. They are 1) separa-tion due to pinchorces, and longitudinal fractureaused eitherby 2) acoustic vibrations or 3) thermal shock.According to classical pinch force theory [12] the maximumpressure in a circular conductor occurs on the xisand is givenby

P =iz/(?raz), dyn/cm2 (32)where i is the total conductor current in ab-amp and a is theconductor radius in centimeter. The 99percent pure aluminumwires of 1.19mm diameter were subjected to peak pulse cur-rents of at most6000A =600ab-amp. The maximumpressure

1 crn -

25 c m RAD. *

Fig. 12. (a), (b) Straight and semicircular aluminum wires which arearc-connected to the discharge circuit. (c) Coll ection of wire fragmentsproduced by discharge current. Distortion of fragments is due to hotpieces alling on aboratory loor. (d) Fourshortwirepieces spotwelded together by arcs. (e) Appearance of brittle fracture face.

on the wireaxiscalculated from (32) comes to 33 kg/cm2.Given sufficient time, pressures of the magnitudecould extrudea hot wire. . A careful micrometer survey of the fractured wirepieces revealed no evidence of any decrease in wire diameter.The extrusion forces existed only for a few mil liseconds. Thiswas too shorta time interval for producing plastic deformation,let alone wire breaks.The pinch force mustave oscillated at3660Hz which is twicethe discharge current requency. This representsanacousticvibration which in the long run could possibly lead to fatiguecracking and bri ttle fracture. However the compressive stressin the outer regions of the wire is much less than along the axisand thevibrations could at mosthave lasted for40cycles. Pinchforce-induced vibrations are therefore an unlikely cause of the


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454 IEEE TRANSACTIONSON MAGNETICS,VOL. MAG-20,NO. 2, MARCH 19wire breaks. Furthermore, any standing acoustic wave patternshould have resulted in wire pieces of equal length, which wasnot the case.Nasilowski 1141 was the first to observe how tensile breaksarrested a wire explosion. In one of his experiments he con-nected a ongitudinal vibration detector to the wire to be ex-ploded. He used a long dc current pulse which lasted for about20 ms. Longitudinal vibrations and arcing started 10ms afterdischarge initiation. It seems likely that the cause of the vibra-tions was tension relaxation of an initially taut wire when thefirst break occurred.The electricarcs struckin theMIT experiments were of courseaccompanied by oudreports. Acoustic waves in air are notknown obreak wires even when they are due o lightningstrokes. Hence acoustic vibrations maybe ruled out as thecause of tensile wire fractures.Finally we consider thermal shock. Natural cooling of wiresfrom red heat toambient temperature,whiletheendsare free,is not known to have caused wires to break in ension. Convec-tion and radiation cooling is a slow process and takes secondsif not minutes to be completed. Significant tensile stres couldbe set up by dif ferential thermal expansion. Pronounced skin-effect heating might achieve this. However the skin depth inaluminum at the ringing frequency of 1830 Hz is 2 mm andmore at elevated temperatures. This makes the current distri-bution over the 1.9 mm diameter wire almost uniform, givingno cause for thermal cracking.The 100cm long aluminum wire weighed three grams. Ac-cording to Fig. 3 the maximum specific tension was at least7.0.For a peak current of 6000 A this translates to a tension of2.57 kg and a corresponding tensile stress of 231 kg/cm2. It isequal to the ultimate strength f the material at around 300C.The mpact strength of the metal wil l be less. Therefore thefirst break in the wire could occur qui te early in the dischargecycleprovided the racture pieces manage to separate in theavailable time. The repulsion between the two wire portions isequal to the tension just before the break. If the break occurshalf way along the wire, it results in an acceleration of 2570/1.5 = 17 13 times that of gravity. This produces a wire separa-tion of 8X cm in 0.1ms which seems adequate or a cleanbreak. Themechanical isolation between theiecesmust owerthe tension in either portion and the current also decays wi thtime. However the reduction n strength wi th ncreasing temper-ature appears to permit further rupturing of thewire sections.C. Wire Semicircle

The typicalamperianmechanism of producing conductortension is not only active in straight wires but also in curvedsections. This will now be demonstrated wi th a wire semicirclewhich is arc-connected to the emainder of thedischarge circuit.First we examine themathematicalsituation in conjunctionwith Fig. 13. Any contribution o he semicircle tensionbythe interaction of i ts elements with the emainder of thecircuitwill be gnored.This is likely tounderestimate he Amperetension, but the error over the middle portion of the wire willbe qui te small.The semicircle A XB of Fig. 13 s divided into z equal elements

Fig. 13. Construction for calculating Ampere tension in semici rcle

of arc, each subtending an angle ofA0 =njz. (3

The elements along X A are labelled 1, 2, . . . ,m, . ., x ; anthose along XB are labelled 1, 2, . . . ,n, . .. , z - x. The ditance between the two general elements mand n is denoted bvm,,, and the arc mXn subtends the angle em,,. The angles the Ampere force law obey the relationships

a=P (3E =ern,,=201=20. (3

If R is the radius of the semicircle, thendm=drt =(n/z)R (3

andT A , , =~ ( 1 -ose,,,). (3It can easily be shown that for any element combination onsemicircle the angle function of (1) is negative and therefore aAmpere interactions to be considered are repulsions.With(33(37) Amperes force law may be writtenAFm,,/i2 =- ( T / z ) ~ 2 cos E - 3 cos2 (~/2)}/(22 cos e )

(3where E =(n/z) m+ I - 1) .

The tangential component of this repulsion produces tensioin a wire which is again assumed to behave like an ideal strinwi th no bending strength. The transverse component of (38tends to retain the shape of the semicircle and possibly acceleates the wire or ts fragments away from the center 0. Notinthat cos 01 =C O S ( E / ~ ) ,the elemental tension contribution matherefore be written

ATm,,/i2 =(AFm,,/i2)cos (~/2).39)Therefore the wire tension across the perpendicular plane containing OX is given by


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GRANEAU: AMPERE TENSION IN ELECTRICCONDUCTORS

I I I I I I

- I I

;J 201 I I I 1 , I I 10 IO 0 200 300 400 500 600 660

E l e m e n t n u m b e r ( x ) = zFig. 14. Computed specifi c Ampere tension n semicircle.

For the semicircle of Fig. 12(b) z comes to 660 elements.Using this in the computer evaluation of (40) gave the resultsplotted on Fig. 14.They show that the internally generatedtension is relatively constant from end to end of the emicircle.Comparing this with Fig. 3 we see that the tension in thesemi-circle is expected to be ofa similar magnitude as in the straightwire. Hence the same current pulses which fractured thestraightwire of Fig. 12(a) should also break thesemicircle. Experimentfully confirmed this prediction of the mpere force law.VI . CONCLUSION

With the particular circuit of Fig. 2 it has been proved that,under certain circumstances, the Ampere force law wil l predicttension in an electric conductor where it is not expected fromthe L orentz force law. The disagreement has been put tothetest by the experiment of Fig. 12(a) and thereby resolved infavor of Amperes,law.The fact that a component of the current-generated pondero-motive force does act in the direction of the current indicatesthat the metall ic current-element is not merely a moving con-duction electron, as hitherto assumed. A longitudinal force onthe electron would accelerate or retard it without exerting anyappreciable reaction on the metall ic body. From this t maybe concluded hat he amperian current-element involves atleast one lattice or li quid metal ion in addition to the conduc-tion electron. The redefinition of the amperian current-elementin terms of an electron-ion combination explains why the Am-pere force law was found incapable of dealing with isolatedcharges in vacuum or gasses. I t leaves the Lorentz force nchal-lenged in such mportant areas as particleaccelerators, massspectrometers, and electronoptics.While the element size is unimportant for the evaluation ofinteractions between separatecircuits, it becomes the dominantfactor when computing the force distribution around ansolatedcircuit. Ampere and his fol lowers believed the current-elementto be inf ini tely divisible. This created singularities in the inte-

4 55

gration of mechanical forces around individual circuits and soprecluded the quantitative assessment. of Ampere tension byanalytical methods. The computer-aided finite current-elementanalysis outlined in this paper furnishes the required solutions.It also revealed that the amperian current-element must be offinite size or the Ampere-Neumann electrodynamics becomesabsurd. Furthermore, it has been shown that the ultimate ele-ment may be of the size of an atom or unit atomicell.The resolution of conductors nto quite large macroscopiccurrent-elements gives approximate values of the Ampere ten-sion which agree reasonably well with measurements. Howeverthis agreementbecomes unsatisfactory unless the ength-to-widthratioof heelement is about one.Macroscopic cubiccurrent-elements have been found convenient and acceptablefor computer analysis. Using cubic elements it has been shownthat, at constant current; the mpere tension will decrease wi thincreasing conductor cross-section. This effect has been calledlongitudinal force dilution.The Ampere tension turns out to e quite small in air-cooledelectric power conductors carrying slowly varying ac and dccurrents. The much arger transient currentsarising accidentallyfrom lightning strokes and short-circuits of lines apply signifi-cant tensilestress to hecopperandaluminumconductorswidely used in power distribution. The practical consequencesof Ampere tension are marked in pulse current devices, as forexample railguns, exploding wires, and fuses.This tension isalso expected to influence the design of large superconductingmagnets for fusion and MHD generators.

ACKNOWLEDGMENTNeal Graneau was responsible for a number of computationaland experimental tasks.

REFERENCES[11 H . C . Oersted, E xperiment on the effects f a current on themag-netic needle,Annalsof Philosophy,vol. 16, 1820.[2] A. M . Ampere, Theorie Mathematique des Phenomenes Electro-Dynamques Uniquement Deduite de I Experience. Paris: Blan-chard, 1958.[3] J . C. Maxwell,A TreatiseonElectricity andMagnetism. Oxford:University, 1873, vol. 2, p. 175.[4] H. A. L orentz,The Theoryof Electrons. Leipzig: Teubner, 1909.[ 5 I; . E. Neumann, Die mathematischen Gesetze der inducirten elektrischen Stroeme, Akademe der Wissenschaften. Berlin, 1845.[6 A. M . Ampere, Memoirs SUT 1 Electrodynamc. Paris: Gauthier-V ill ars, 1885, vol. 1, XXV , pp. 329-331.[7 F. E. Neumann, Vorlesungen Ueber ElektrischeStroeme. Leipzig:

Teubner, 1884, p. 102.[SI P. Graneau,Electromagnetic et-propulsion n hedirectionofcurrent flow,Nature, vol. 295, pp. 311-312, 1982.[9] C. Hering,Electrodynamic orces in electric urnaces, Trans.Amer. Electrochem. Soc., vol. 39, pp. 313-330, 1921.[ I O] H. H. Grassmann, A new theory of electrodynamics, Poggen-dorfsAnnalen der Physik und Cheme,vol. 64, pp. 1-18, 1845.[ l l ] P. Graneau, A pplication of Amperes force law to railgun accel-erators, J . Appl. Phys., vol. 53, pp. 6648-6654, 1982.[12] E. F. Northrup, Some newly observed manifestations of forcesin the nterior of an electric conductor, Physical Rev., vol. 24,[13] W. G. Chace and H. K . Moore, Ed., Exploding Wires. New Y ork:[14] J . Nasilowski, Unduloidsandstriateddisintegration of wires,

PP. 474-497,1907.Plenum, 1959-1965, vols. 1-4.Exploding Wires. New Y ork: P lenum, 1964,vol. 3, pp.295-313.

ampere tension

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