Revista Brasileira de Ensino de Fsica, vol. 21, no. 4, Dezembro, Surface Charges and Electric Field in a TwoWire


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1 Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, Surfae Charges and Eletri Field in a TwoWire Resistive Transmission Line A. K. T.Assis and A. J. Mania Instituto de Fsia Gleb Wataghin' Universidade Estadual de Campinas  Uniamp Campinas, S~ao Paulo, Brasil Reebido em de Setembro, 998 We onsider a twowire resistive transmission line arrying a onstant urrent. We alulate the potential and eletri eld outside the wires showing that they are dierent from zero even for stationary wires arrying d urrents. We also alulate the surfae harges giving rise to these elds and ompare the magneti fore between the wires with the eletri fore between them. Finally we ompare our alulations with Jemenko's experiment. I Introdution One of the most important eletrial systems is that of atwowire transmission line, usually alled twinleads. We onsider here homogeneous resistive wires xed in the laboratory and arrying d urrents. Our goal is to alulate the eletri eld outside the wires. To this end we follow essentially the important works of Heald and Jakson, [] and []. They all attention to the surfae harges in a stationary resistive wire arrying a onstant urrent. These authors have shown that the distribution of these net harges is onstant in time if we have an stationary resistive wire with a d urrent produed by a battery. These harges reate not only the eletri eld inside the wire whih opposes the resistive frition, but also an external eletri eld in the surrounding medium (air, for instane). This fat is not realized by most authors who onsider only the magneti eld reated by these urrents. Heald, in partiular, onsidered the ase of a (twodimensional) urrent loop and Jakson that of a oaxial able of nite length with a return ondutor of zero resistivity. The ase of twinleads was rst onsidered by Stratton, [3, p. 6]. Although he alled attention to the eletri eld outside the transmission line, this has been forgotten by most authors as an be seen from the following quotation taken from Griths's book ([4, p. 96], our emphasys in boldfae): \Two wires hang from the eiling, a few inhes apart. When I turn on a urrent, so that it passes up one wire and bak down the other, the wires jump apart  they plainly repel one another. How do you explain this? Well, you might suppose that the battery (or whatever drives the urrent) is atually harging up the wire, so naturally the dierent setions repel. But this \explanation" is inorret. I ould hold up a test harge near these wires and there would be no fore on it, indiating that the wires are in fat eletrially neutral. (It's true that eletrons are owing down the line  that's what a urrent is  but there are still just as many plus as minus harges on any given segment.) Moreover, I ould hook up my demonstration so as to make the urrent owupboth wires; in this ase the wires are found to attrat!" In this work we will see that the wire is not eletrially neutral on any given segment as there are surfae harges distributed along its length. What reates the eletri eld anywhere along the transmission line are these surfae harges and not the battery, although the battery is essential to maintain these surfae harges in the ase of onstant urrent. As these surfae harges reate also an external eletri eld, a test harge plaed near it will experiene a fore, ontrary to Grith's statement. The existene of this fore has been on rmed by Jemenko's experiments, [5] and [6]. Despite this fat we show here that the eletrostati fore between two segments of the twin leads is many orders of magnitude smaller than the magneti fore between them. Our main goal is to all attention to the existene of the external eletri eld and to present analytial alulations whih were not performed by Jemenko. II TwoWire Transmission Line The geometry of the system is given in Fig.. We have two equal straight wires of irular rosssetions Web site: Also Collaborating Professor at the Department of Applied Mathematis, IMECC, State University of Campinas, Campinas, SP, Brazil.
2 470 A.K.T. Assis e A.J. Mania of radii a and length, surrounded by air. Their axes are separated by a distane R and are parallel to the z axis, symmetrially loated relative tothez and x axes. That is, the enters of the left and right wires are loated at (x; y; z) = (,R=; 0; 0) and (+R=; 0; 0), respetively. The ondutivityofthe wires is g and their extremities are loated at z =,= and z = +=. p Here we alulate the eletri potential and the eletri eld E ~ at a point (x; y; z) suh that r = x + y + z. Moreover, we also assume that R= >a, so that we an neglet border eets. Wewant to nd the potential and eletri eld when a urrent I ows uniformly over the left wire in the diretion +^z and returns uniformly over the right wire in the diretion,^z. The urrent densities in both wires are then given by ~ J =(I=a )^z and ~ J =,(I=a )^z, respetively. As we are onsidering homogeneous wires with a onstant resistivity g, Ohm's law yields the internal eletri eld in the wires as ~ E = (I=ga )^z. We don't need to onsider in ~ E the inuene of the time variation of the vetor potential as we are dealing with a d urrent in stationary wires, so ~ = 0 everywhere. We an then write ~ E =,r. Aswehave a onstant eletri eld in eah wire, this implies that the potential is onstant over eah ross setion and a linear funtion of z. In this work we onsider a symmetrial situation for the potentials so that in the left wire the urrent ows from the potential B at z =,= to A at z = = and returns in the right wire from, A at z = = to, B at z =,=, Figure We an then write L (z) = A, B = I ga ; () d R (z) =, L (z) : () In these equations L (z) and R (z) are the potentials as a funtion of z over the rosssetion of the left and right ondutors, respetively. In this work we are negleting the small Hall eet due to the poloidal magneti eld generated by these urrents. This eet reates a redistribution of the urrent density within the wires, and modies the surfae harges also. As these are usually small eets, they will not be onsidered here. Figure. Two parallel wires of radii a separated by a distane R. The left wire arries a onstant urrent I along the positive z diretion while the right one arries the return urrent I along the negative z diretion. We now nd the potential in spae supposing air outside the ondutors. As the ondutors are straight and the boundary onditions (the potentials over the surfae of the ondutors) are linear funtions of z, the same must be valid everywhere, [7]. That is, =(Az + B)f(x; y), where A and B are onstants and f(x; y) is a funtion of x and y. This funtion an be found by the method of images imposing a onstant potential o over the left wire and, o over the right one, [8, Setion.]. The nal solution for and E ~ satisfying the given boundary onditions, valid for the region outside the wires, is given by (x; y; z) =, A, B ~E =, A ln R,p R,4a a ln (x, p R, 4a =) + y (x + p R, 4a =) + y ; (3) p R, 4a, B ln R+p R,4a a
3 Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, (x, y + a, R =4)^x +xy^y x 4 + y 4 + R 4 =6 + a 4 +x y, R x =+a x + R y =, a y, R a = + A, B ln R,p R,4a a ln (x, p R, 4a =) + y (x + p ^z : (4) R, 4a =) + y The equipotentials at z = 0 are plotted in Fig.. It is also relevant to express these results in ylindrial oordinates (; '; z) entered on the left and right wires, see Fig. 3. For the left wire this an be aomplished replaing x by L os ' L, R=, y by L sin ' L,^x by ^ L os ' L, ^' L sin ' L and ^y =^ L sin ' L +^' L os ' L, yielding ( L ; ' L ; z)=, A, B ln R,p R,4a a lns p L, L os ' L (R + R, 4a )+R =, a + p R R, 4a = L, L os ' L (R, R, 4a )+R =, a, p R R, 4a = ; (5) p R, 4a ~E =, A, B ln R+p R,4a a ( L os ' L, L R + a os ' L )^ L + sin ' L ( L, a )^' L 4 L, 3 L R os ' L + L R + a 4 + L a (os ' L, sin ' L ), L Ra os ' L + A, B ln R,p R,4a a ln L, L os ' L (R + p R, 4a )+R =, a + R p R, 4a = L, L os ' L (R, p R, 4a )+R =, a, R p R, 4a = : (6) The density of surfae harges over the left and right wires, L and R, an then be found by " o =8:85 0, C N, m, times the radial omponent of the eletri eld over the surfae of eah ylinder, yielding (" o is the vauum permittivity): A, B L = R =, A, B "o "o p R, 4a ^z a ln R+p R,4a a p R, 4a a ln R+p R,4a a R=, a os ' L ; (7) R=+a os ' R ; (8) In order to hek our results we alulated the potential inside eah wire and in spae beginning with these surfae harges densities and utilizing (x; y; z) = 4" o + Z = Z L (' 0L )ad'0 L dz0 z 0 =,= ' 0 L =0 j~r, ~r 0 j Z = Z R! (' 0R )ad'0 R dz0 z 0 =,= ' 0 R =0 j~r, ~r 0 j : (9)
4 47 A.K.T. Assis e A.J. Mania Here we integrate over the surfaes of the left and right ylinders, S L and S R, respetively. We ould then hek our results assuming the orretness of the method of images for the eletrostati problem and utilizing the approximations j~rj and R= > a. The magneti eld of eah wire surrounded by air an be easily obtained by the iruital law HC B ~ d ~ = o I C, where I C is the urrent owing through the losed iruit C and o =4 0,7 kgmc, is the vauum permeability. For a long straight wire of radius a arrying a total urrent I we obtain: B( <a)= o I=a and B( >a)= o I=, both in the poloidal diretion. Adding the magneti eld of both wires taking into aount that they arry urrents in opposite diretions yields the magneti eld anywhere in spae (in this approximation that r). will assume A = 0 in order to simplify the analysis. The distribution of surfae harges for a given z is similar to the distribution of harges in the eletrostati problem given the potentials o and, o at the left and right wires, without urrent. That is, L (' L ) > 0 for any ' L and its maximum value is at ' L = 0. The density of surfae harges at the right wire, R, has the same behaviour of L with an overall hange of sign, with its maximum magnitude happening at ' R =. A qualitative plot of the surfae harges at z = 0 is given in Fig. 4. A quantitative plot of L is given in Fig. 5 supposing R=a = 0=3 and normalizing the surfae harge density by the value of L at ' L =. It should also be remarked that for a xed ' L the surfae density dereases linearly from z =,= toz = =, the opposite happening with R for a xed ' R. Figure 4. Qualitative distribution of surfae harges for the two parallel wires at z =0. Figure. Equipotentials in the plane z = 0 given by Eq. (3). Figure 3. Left (L) and right (R) ylindrial oordinates for the left and right wires, respetively. III Disussion and Conlusions Figure 5. Surfae harge density at the left wire in z = 0 for R=a =0=3 as a funtion of 'L, normalized by its value at 'L = : L('L)=L() 'L. We an integrate the surfae harges over the periphery The rst aspet to be disussed here is the qualitative of eah wire obtaining the integrated harge interpretation of these results. In all this Setion we per unit length (z) as: Z L (z) = a L (' L )d' L 'L=0 " o =, ln ((R, p A, B : (0) R, 4a )=a) R (z) =,Z 'R=0 a R (' R )d' R =, L (z) : ()
5 Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, One important aspet to disuss is the experimental relevane of these surfae harges in terms of fores. That is, as the wires have a net harge in eah setion, there will be an eletrostati fore ating on them. We an then ompare this fore with the magneti one. This last one is given essentially by (fore per unit length) d ~ F E dz = Z 'L=0 a L (' L ) ~ E( L = a; ' L ;z =0)d' L " o B ln R=a df M dz = oi R ; () where we are supposing R= a. Wenow alulate the eletri fore per unit length on the left wire integrating the fore over its periphery. We onsider a typial region in the middle of the wire, around z =0, and one more suppose R= a: ^x R + ^z : (3) d From Eqs. () and (3) the ratio of the magneti to the radial eletri fore is given by (with Ohm's law B =I = R o =(=ga ), R o being the resistane of eah wire): F M F E o=" o R o ln R a : (4) As o =" o =:4 0 5 this ratio will be usually many orders of magnitude greater than. This would be of the order of when R o 370 (supposing ln R=a ). This is a very large resistane for homogeneous wires. In order to ompare this fore with the magneti one we suppose typial opper wires of ondutivities g =5:7 0 7 m,,, lengths =m, separated by a distane R =6mm and diameters a =mm. This means that by Ohm's law B =I = R o 5 0,4. With these values the ratio of the longitudinal eletri fore to the magneti one is of the order of 7 0,, while the ratio of the radial eletri fore to the magneti one is of the order of 0,8. That is, the eletri fore between the wires due to these surfae harges is typially 0,8 times smaller than the magneti one. This shows that we an usually neglet these eletri fores. Despite this fat it should be remarked that while the magneti fore is repulsive in this situation (parallel wires arrying urrents in opposite diretions), the radial eletri fore is attrative, as we an see from the harges of Fig. 4. It must be stressed that the surfae harges are essential for understanding the origins of the eletri eld driving the urrent. The role of the battery is to separate the harges and keep this distribution of harges xed in time for d urrents. But what reates the eletri eld inside and outside the wires is not the battery but these surfae harges. Moreover, this external eletri eld an also be seen and measured if we have a dieletri material whih an be polarized by the eletri eld, but whih is not inuened by the magneti eld. This was the tehnique employed by Jemenko, [5] and [6, Setion 96 and Plate 6]. In his experiment heob tained the lines of eletri eld utilizing grass seeds, in a similar way that we obtain the lines of magneti eld utilizing iron llings. The situation desribed in this paper is very similar to the experiment performed by Jemenko whose results are presented in Fig. 5 of [5] or in Plate 6 and Fig. 9.3 of [6]. We an ompare his experiment with our theoretial alulations by plotting the equipotentials obtained here. Jemenko did not give the dimensions of his experiment but from Fig. 5 of [5] or from Plate 6 of [6] we an estimate the ratio of the important distanes as R=a 0=3, =R 5= and =a 50=6. With these values and ' A = 0 and ' B =V we obtain the equipotentials given by Eq. (3) at y = 0, Fig. 6. Figure 6. Equipotentials in the plane y = 0 given by Eqs. (), () and (3) with the dimensions orresponding to Jemenko's experiment, from z =,= to =. These lines an also be interpreted as lines of Poynting eld ~ S = ~ E ~ B= o, where ~ B is the magneti eld. That is, they may also represent the energy ow from the battery (at z =,=) to the wires given by Poynt
6 474 A.K.T. Assis e A.J. Mania ing vetor throughout the spae. This has been pointed out in general by Heald in his important work, []. The lines of eletri eld orthogonal to the equipotentials an be obtained by the proedure desribed in Sommerfeld's book, [9, p. 8]. We are looking for a funtion (x; y =0;z) suh that r(x; 0; z) r(x; 0; z) =0: (5) The equipotential lines an be written as z (x) = K, where K is a onstant (for eah onstant wehave a dierent equipotential line). Analogously, the lines of eletri fore will be given by z (x) = K, where K is another onstant (for eah K wehave a different line of eletri fore). From Eq. (5) we get dz =dx =,=(dz =dx) Integrating this equation we obtain (x; 0; z). This yields the following solutions in the plane y = 0 outside the wires: out (x; 0;z)=,Bz + A x(x, 3x o) ln (x, x o) 6x o (x + x o ) + x o 3 ln[(x, x o) (x + x o ) ], x 3 =3, z ; (6) where A = ( A, B )=, B = ( A + B )= and x o = p R, 4a =. The lines of eletri eld inside the left and right wires an be written as, respetively: L (x; 0;z)=,Ax ; (7) R (x; 0;z)=Ax ; (8) The lines of eletri eld are then plotted imposing (x; 0; z) = onstant. With Jemenko's dimensions for R, a and we obtain the lines of fore by these equations as given in Fig. 7. This numerial plot is extremely similar to Jemenko's experiment as presented in Fig. 5 of [5] or in Plate 6 of [6]. Although our alulation is stritly valid only for r, our numerial plot goes from z =,= to=. As the result is in very good agreement with Jemenko's experiment, we onlude that the exat boundary onditions at z = = are not very important in this partiular onguration. Our work might be onsidered as a omplementation of Jemenko's one, as he realized the experiment but made no theoretial alulations for the transmission line onsidering straight ylindrial wires. The only aluations he presented in [6, Setion 96] were restrited to the urrent owing over one surfae of a resistive apaitor plate and returning through the other. He didn't onsider twinleads nor ylindrial ondutors. Figure 7. Lines of eletri eld in the plane y = 0 given by Eqs. (6), (7) and (8) with the dimensions orresponding to Jemenko's experiment, from z =,= to =. We an also estimate the ratio of the radial omponent of the eletri eld to the axial one just outside the wire. We onsider the left wire at three dierent heights: z =,=, z = 0 and z = =. The axial omponent E z is onstant over the ross setion and does not depend on z. On the other hand the radial omponent E x is a linear funtion of z and also depends on ' L. In this omparison we onsider ' L = 0. With these values and Jemenko's data in Eq. (4) we obtain E x =E z at z =,=, 6 at z =0and0at z = =. That is, the radial omponent of the eletri eld just outside the wire is typially one order of magnitude larger than the axial eletri eld responsible for the urrent. Jemenko's experiment gives a lear onrmation of this fat. Aknowledgements: The authors wish to thank FAPESP for nanial support, Prof. Mark A. Heald for many important suggestions related to the rst version of this paper and J. A. Hernandes for helping with the omputational alulations. Referenes [] M. A. Heald. Eletri elds and harges in elementary iruits. Amerian Journal of Physis, 5:5{56, 984. [] J. D. Jakson. Surfae harges on iruit wires and resistors play three roles. Amerian Journal of Physis, 64:855{870, 996. [3] J. A. Stratton. Eletromagneti Theory. MGrawHill, New York, 94. [4] D. J. Griths. Introdution to Eletrodynamis. Prentie Hall, Englewood Clis, seond edition, 989.
7 Revista Brasileira de Ensino de Fsia, vol., no. 4, Dezembro, [5] O. Jemenko. Demonstration of the eletri elds of urrentarrying ondutors. Amerian Journal of Physis, 30:9{, 96. [6] O. D. Jemenko. Eletriity and Magnetism. Eletret Sienti Company, Star City, nd edition, 989. [7] B. R. Russell. Surfae harges on ondutors arrying steady urrents. Amerian Journal of Physis, 36:57{ 59, 968. [8] J. D. Jakson. Classial Eletrodynamis. John Wiley, New York, seond edition, 975. [9] A. Sommerfeld. Eletrodynamis. Aademi Press, New York, 964.
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