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


 Margery Hopkins
 1 years ago
 Views:
Transcription
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.
SHAFTS: TORSION LOADING AND DEFORMATION
ECURE hird Edition SHAFS: ORSION OADING AND DEFORMAION A. J. Clark Shool of Engineering Department of Civil and Environmental Engineering 6 Chapter 3.13.5 by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220
More information10.1 The Lorentz force law
Sott Hughes 10 Marh 2005 Massahusetts Institute of Tehnology Department of Physis 8.022 Spring 2004 Leture 10: Magneti fore; Magneti fields; Ampere s law 10.1 The Lorentz fore law Until now, we have been
More informationAnother Look at Gaussian CGS Units
Another Look at Gaussian CGS Units or, Why CGS Units Make You Cool Prashanth S. Venkataram February 24, 202 Abstrat In this paper, I ompare the merits of Gaussian CGS and SI units in a variety of different
More informationFrom (2) follows, if z0 = 0, then z = vt, thus a2 =?va (2.3) Then 2:3 beomes z0 = z (z? vt) (2.4) t0 = bt + b2z Consider the onsequenes of (3). A ligh
Chapter 2 Lorentz Transformations 2. Elementary Considerations We assume we have two oordinate systems S and S0 with oordinates x; y; z; t and x0; y0; z0; t0, respetively. Physial events an be measured
More information) ( )( ) ( ) ( )( ) ( ) ( ) (1)
OPEN CHANNEL FLOW Open hannel flow is haraterized by a surfae in ontat with a gas phase, allowing the fluid to take on shapes and undergo behavior that is impossible in a pipe or other filled onduit. Examples
More informationUser s Guide VISFIT: a computer tool for the measurement of intrinsic viscosities
File:UserVisfit_2.do User s Guide VISFIT: a omputer tool for the measurement of intrinsi visosities Version 2.a, September 2003 From: Multiple Linear LeastSquares Fits with a Common Interept: Determination
More informationClassical Electromagnetic Doppler Effect Redefined. Copyright 2014 Joseph A. Rybczyk
Classial Eletromagneti Doppler Effet Redefined Copyright 04 Joseph A. Rybzyk Abstrat The lassial Doppler Effet formula for eletromagneti waves is redefined to agree with the fundamental sientifi priniples
More informationComay s Paradox: Do Magnetic Charges Conserve Energy?
Comay s Paradox: Do Magneti Charges Conserve Energy? 1 Problem Kirk T. MDonald Joseph Henry Laboratories, Prineton University, Prineton, NJ 08544 (June 1, 2015; updated July 16, 2015) The interation energy
More informationarxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 USA
LBNL52402 Marh 2003 On the Speed of Gravity and the v/ Corretions to the Shapiro Time Delay Stuart Samuel 1 arxiv:astroph/0304006v2 10 Jun 2003 Theory Group, MS 50A5101 Lawrene Berkeley National Laboratory
More informationElectrician'sMathand BasicElectricalFormulas
Eletriian'sMathand BasiEletrialFormulas MikeHoltEnterprises,In. 1.888.NEC.CODE www.mikeholt.om Introdution Introdution This PDF is a free resoure from Mike Holt Enterprises, In. It s Unit 1 from the Eletrial
More informationElectromagnetic Waves. Physics 6C
Eletromagneti Waves Physis 6C Eletromagneti (EM) waves are produed y an alternating urrent in a wire. As the harges in the wire osillate ak and forth, the eletri field around them osillates as well, in
More informationMathematics 1c: Solutions, Homework Set 6 Due: Monday, May 17 at 10am.
Mathematis : Solutions, Homework Set 6 Due: Monday, May 7 at am.. ( Points) Setion 6., Eerise 6 Let D be the parallelogram with verties (, 3), (, ), (, ) and (, ) and D be the retangle D [, ] [, ]. Find
More informationWaveguides. 8.14 Problems. 8.14. Problems 361
8.4. Problems 36 improving liquid rystal displays, and other produts, suh as various optoeletroni omponents, osmetis, and hot and old mirrors for arhitetural and automotive windows. 8.4 Problems 9 Waveguides
More informationImpedance Method for Leak Detection in Zigzag Pipelines
10.478/v1004801000360 MEASUREMENT SCIENCE REVIEW, Volume 10, No. 6, 010 Impedane Method for Leak Detetion in igzag Pipelines A. LayEkuakille 1, P. Vergallo 1, A. Trotta 1 Dipartimento d Ingegneria
More informationMagnetic Materials and Magnetic Circuit Analysis
Chapter 7. Magneti Materials and Magneti Ciruit Analysis Topis to over: 1) Core Losses 2) Ciruit Model of Magneti Cores 3) A Simple Magneti Ciruit 4) Magneti Ciruital Laws 5) Ciruit Model of Permanent
More informationTHE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES
Proeedings of FEDSM 98 998 ASME Fluids Engineering Division Summer Meeting June 225, 998 Washington DC FEDSM98529 THE PERFORMANCE OF TRANSIT TIME FLOWMETERS IN HEATED GAS MIXTURES John D. Wright Proess
More informationREDUCTION FACTOR OF FEEDING LINES THAT HAVE A CABLE AND AN OVERHEAD SECTION
C I E 17 th International Conferene on Eletriity istriution Barelona, 115 May 003 EUCTION FACTO OF FEEING LINES THAT HAVE A CABLE AN AN OVEHEA SECTION Ljuivoje opovi J.. Elektrodistriuija  Belgrade 
More informationRole of the Reference Frame in Angular Photon Distribution at ElectronPositron Annihilation
Journal of Modern Physis,, 5, 353358 Published Online April in SiRes. http://www.sirp.org/journal/jmp http://dx.doi.org/.36/jmp..565 Role of the Referene Frame in Angular Photon Distribution at EletronPositron
More informationSinglePhase Transformers
SinglePhase Transformers SinglePhase Transformers Introdution Figure shows the transformer hemati symbol and the orresponding ommonly used Steinmetz model is shown in Figure. Therein, the model voltage
More informationHEAT CONDUCTION. q A q T
HEAT CONDUCTION When a temperature gradient eist in a material, heat flows from the high temperature region to the low temperature region. The heat transfer mehanism is referred to as ondution and the
More informationPropagation in Lossy Rectangular Waveguides
10 Propagation in Lossy Retangular Waveguides Kim Ho Yeap 1, Choy Yoong Tham, Ghassan Yassin 3 and Kee Choon Yeong 1 1 Tunku Abdul Rahman University Wawasan Open University 3 University of Oxford 1, Malaysia
More informationTHE UNIVERSITY OF THE STATE OF NEW YORK THE STATE EDUCATION DEPARTMENT ALBANY, NY
P THE UNIVERSITY OF THE STATE OF NEW YORK THE STATE EDUCATION DEPARTMENT ALBANY, NY 4 Referene Tables for Physial Setting/PHYSICS 006 Edition List of Physial Constants Name Symbol Value Universal gravitational
More informationON THE ELECTRODYNAMICS OF MOVING BODIES
ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries
More informationON THE ELECTRODYNAMICS OF MOVING BODIES
ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 905 It is known that Maxwell s eletrodynamis as usually understood at the present time when applied to moing bodies, leads to asymmetries
More informationSpecial Relativity and Linear Algebra
peial Relativity and Linear Algebra Corey Adams May 7, Introdution Before Einstein s publiation in 95 of his theory of speial relativity, the mathematial manipulations that were a produt of his theory
More informationChapter 1 Microeconomics of Consumer Theory
Chapter 1 Miroeonomis of Consumer Theory The two broad ategories of deisionmakers in an eonomy are onsumers and firms. Eah individual in eah of these groups makes its deisions in order to ahieve some
More informationPhys 232 Lab 8 Ch 21 Interactions with Magnetic Fields 1
Phys 3 Lab 8 Ch 1 Interations with Magneti Fields 1 Equipment: omputer with VPython, single e/m apparatus for qualitative experimenting: Fore on dipole: blak power supply, TeahSpin Magneti Fore apparatus,
More informationEE 201 ELECTRIC CIRCUITS LECTURE 25. Natural, Forced and Complete Responses of First Order Circuits
EE 201 EECTRIC CIUITS ECTURE 25 The material overed in this leture will be as follows: Natural, Fored and Complete Responses of First Order Ciruits Step Response of First Order Ciruits Step Response of
More informationTheory of linear elasticity. I assume you have learned the elements of linear
Physial Metallurgy Frature mehanis leture 1 In the next two letures (Ot.16, Ot.18), we will disuss some basis of frature mehanis using ontinuum theories. The method of ontinuum mehanis is to view a solid
More informationConversion of short optical pulses to terahertz radiation in a nonlinear medium: Experiment and theory
PHYSICAL REVIEW B 76, 35114 007 Conversion of short optial pulses to terahertz radiation in a nonlinear medium: Experiment and theory N. N. Zinov ev* Department of Physis, University of Durham, Durham
More informationVOLTAGE CONTROL WITH SHUNT CAPACITANCE ON RADIAL DISTRIBUTION LINE WITH HIGH R/X FACTOR. A Thesis by. HongTuan Nguyen Vu
VOLTAGE CONTROL WITH SHUNT CAPACITANCE ON RADIAL DISTRIBUTION LINE WITH HIGH R/X FACTOR A Thesis by HongTuan Nguyen Vu Eletrial Engineer, Polytehni University of HCMC, 1993 Submitted to the College of
More informationpss Comparison of AlGaN/GaN MISHEMT powerbar designs solidi status physica
Phys. Status Solidi C 11, No. 3 4, 906 910 (2014) / DOI 10.1002/pss.201300490 Comparison of AlGaN/GaN MISHEMT powerbar designs physia pss www.pss.om urrent topis in solid state physis Steve Stoffels *,
More informationPY1002: Special Relativity
PY100: Speial Relativity Notes by Chris Blair These notes over the Junior Freshman ourse given by Dr. Barklie in Mihaelmas Term 006. Contents 1 Galilean Transformations 1.1 Referene Frames....................................
More informationSebastián Bravo López
Transfinite Turing mahines Sebastián Bravo López 1 Introdution With the rise of omputers with high omputational power the idea of developing more powerful models of omputation has appeared. Suppose that
More informationIntelligent Measurement Processes in 3D Optical Metrology: Producing More Accurate Point Clouds
Intelligent Measurement Proesses in 3D Optial Metrology: Produing More Aurate Point Clouds Charles Mony, Ph.D. 1 President Creaform in. mony@reaform3d.om Daniel Brown, Eng. 1 Produt Manager Creaform in.
More informationChapter 5 Single Phase Systems
Chapter 5 Single Phase Systems Chemial engineering alulations rely heavily on the availability of physial properties of materials. There are three ommon methods used to find these properties. These inlude
More informationSPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S
SPECIAL RELATIVITY. MATH2410 KOMISSAROV S.S 2012 2 Contents Contents 2 1 Spae and Time in Newtonian Physis 9 1.1 Spae............................................ 9 1.1.1 Einstein summation rule..............................
More informationChapter 38A  Relativity. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chapter 38A  Relativity A PowerPoint Presentation by Paul E. Tippens, Professor of Physis Southern Polytehni State University 27 Objetives: After ompleting this module, you should be able to: State and
More informationGravitational Forces Explained as the Result of Anisotropic Energy Exchange between Baryonic Matter and Quantum Vacuum
Journal of odern Physis, 05, 6, 548 Published Online July 05 in Sies. http://www.sirp.org/journal/jmp http://dx.doi.org/0.46/jmp.05.687 Gravitational Fores Explained as the esult of Anisotropi Energy
More information5.2 The Master Theorem
170 CHAPTER 5. RECURSION AND RECURRENCES 5.2 The Master Theorem Master Theorem In the last setion, we saw three different kinds of behavior for reurrenes of the form at (n/2) + n These behaviors depended
More informationRelativity in the Global Positioning System
Relativity in the Global Positioning System Neil Ashby Department of Physis,UCB 390 University of Colorado, Boulder, CO 8030900390 NIST Affiliate Email: ashby@boulder.nist.gov July 0, 006 AAPT workshop
More informationChannel Assignment Strategies for Cellular Phone Systems
Channel Assignment Strategies for Cellular Phone Systems Wei Liu Yiping Han Hang Yu Zhejiang University Hangzhou, P. R. China Contat: wliu5@ie.uhk.edu.hk 000 Mathematial Contest in Modeling (MCM) Meritorious
More informationEffects of InterCoaching Spacing on Aerodynamic Noise Generation Inside Highspeed Trains
Effets of InterCoahing Spaing on Aerodynami Noise Generation Inside Highspeed Trains 1 J. Ryu, 1 J. Park*, 2 C. Choi, 1 S. Song Hanyang University, Seoul, South Korea 1 ; Korea Railroad Researh Institute,
More informationCombining refractive and topographic data in corneal refractive surgery for astigmatism
Combining refrative and topographi data in orneal refrative surgery for astigmatism A new method based on polar value analysis and mathematial optimization Kristian Næser Department of Ophthalmology, Randers
More informationImpact Simulation of Extreme Wind Generated Missiles on Radioactive Waste Storage Facilities
Impat Simulation of Extreme Wind Generated issiles on Radioative Waste Storage Failities G. Barbella Sogin S.p.A. Via Torino 6 00184 Rome (Italy), barbella@sogin.it Abstrat: The strutural design of temporary
More informationchapter > Make the Connection Factoring CHAPTER 4 OUTLINE Chapter 4 :: Pretest 374
CHAPTER hapter 4 > Make the Connetion 4 INTRODUCTION Developing seret odes is big business beause of the widespread use of omputers and the Internet. Corporations all over the world sell enryption systems
More information10 UNSTEADY FLOW IN OPEN CHANNELS
0 UNTEY FLOW IN OEN CHNNEL 0. Introdution Unsteady flow in open hannels differs from that in losed onduits in that the eistene of a free surfae allows the flow rosssetion to freely hange, a fator whih
More informationTECH LETTER #3 MEASUREMENT OF LINE AND LOAD REGULATION OF DC POWER SUPPLIES HARRISON LABORATORIES DIVISION OF HEWLETTPACKARD COMPANY
TECH LETTER #3 MEASUREMENT OF LNE AND LOAD REGULATON OF DC POWER SUPPLES (F HARRSON LABORATORES DVSON OF HEWLETTPACKARD COMPANY 100 Loust Avenue Berkeley Heights, New Jersey 07922 .    _q_ TECH
More information1.3 Complex Numbers; Quadratic Equations in the Complex Number System*
04 CHAPTER Equations and Inequalities Explaining Conepts: Disussion and Writing 7. Whih of the following pairs of equations are equivalent? Explain. x 2 9; x 3 (b) x 29; x 3 () x  2x  22 x  2 2 ; x
More informationChemical Equilibrium. Chemical Equilibrium. Chemical Equilibrium. Chemical Equilibriu m. Chapter 14
Chapter 14 Chemial Equilibrium Chemial Equilibriu m Muh like water in a Ushaped tube, there is onstant mixing bak and forth through the lower portion of the tube. reatants produts It s as if the forward
More informationIn order to be able to design beams, we need both moments and shears. 1. Moment a) From direct design method or equivalent frame method
BEAM DESIGN In order to be able to design beams, we need both moments and shears. 1. Moment a) From diret design method or equivalent frame method b) From loads applied diretly to beams inluding beam weight
More informationChapter 1: Introduction
Chapter 1: Introdution 1.1 Pratial olumn base details in steel strutures 1.1.1 Pratial olumn base details Every struture must transfer vertial and lateral loads to the supports. In some ases, beams or
More informationCapacity at Unsignalized TwoStage Priority Intersections
Capaity at Unsignalized TwoStage Priority Intersetions by Werner Brilon and Ning Wu Abstrat The subjet of this paper is the apaity of minorstreet traffi movements aross major divided fourlane roadways
More informationTEACHING FUNDAMENTALS OF ELECTRICAL ENGINEERING: NODAL ANALYSIS
TACHING FUNDAMNTALS OF LCTICAL NGINING: NODAL ANALYSIS Đorđević, A..; Božilović, G.N.; Olćan, D.I.; Sh. of letr. ng., Univ. of Belgrade, Belgrade, Serbia 00 I. Personal use of this material is permitted.
More informationComputational Analysis of Two Arrangements of a Central GroundSource Heat Pump System for Residential Buildings
Computational Analysis of Two Arrangements of a Central GroundSoure Heat Pump System for Residential Buildings Abstrat Ehab Foda, Ala Hasan, Kai Sirén Helsinki University of Tehnology, HVAC Tehnology,
More informationChapter 6 A N ovel Solution Of Linear Congruenes Proeedings NCUR IX. (1995), Vol. II, pp. 708{712 Jerey F. Gold Department of Mathematis, Department of Physis University of Utah Salt Lake City, Utah 84112
More informationA Holistic Method for Selecting Web Services in Design of Composite Applications
A Holisti Method for Seleting Web Servies in Design of Composite Appliations Mārtiņš Bonders, Jānis Grabis Institute of Information Tehnology, Riga Tehnial University, 1 Kalu Street, Riga, LV 1658, Latvia,
More informationMirror plane (of molecule) 2. the Coulomb integrals for all the carbon atoms are assumed to be identical. . = 0 : if atoms i and j are nonbonded.
6 Hükel Theory This theory was originally introdued to permit qualitative study of the πeletron systems in planar, onjugated hydroarbon moleules (i.e. in "flat" hydroarbon moleules whih possess a mirror
More informationcos t sin t sin t cos t
Exerise 7 Suppose that t 0 0andthat os t sin t At sin t os t Compute Bt t As ds,andshowthata and B ommute 0 Exerise 8 Suppose A is the oeffiient matrix of the ompanion equation Y AY assoiated with the
More informationCHAPTER J DESIGN OF CONNECTIONS
J1 CHAPTER J DESIGN OF CONNECTIONS INTRODUCTION Chapter J of the addresses the design and heking of onnetions. The hapter s primary fous is the design of welded and bolted onnetions. Design requirements
More informationTrigonometry & Pythagoras Theorem
Trigonometry & Pythagoras Theorem Mathematis Skills Guide This is one of a series of guides designed to help you inrease your onfidene in handling Mathematis. This guide ontains oth theory and exerises
More informationWeyl s Theory under Chiral Approach
February 06 Volume 7 Issue pp. 360366 Weyl s Theory under Chiral Approah 360 Artile H. TorresSilva, J. LópezBonilla * & R. LópezVázquez Esuela de Ingeniería Elétria y Eletrónia, Universidad de Tarapaá,
More informationImpact of High Voltage Shunt Capacitor Banks on General Purpose Circuit Breakers
a b a b Impat of High Voltage Shunt Capaitor Banks on General Purpose Ciruit Breakers M. Alawie *, Y. Filion, A. Coutu Abstrat It is well known that during a fault on a bus bar with the presene of a shunt
More informationHeat Generation and Removal in Solid State Lasers
Chapter 1 Heat Generation and Removal in Solid State Lasers V. Ashoori, M. Shayganmanesh and S. Radmard Additional information is available at the end of the hapter http://dx.doi.org/10.577/63 1. Introdution
More informationMechanics of the tapered interference fit in dental implants
ARTICLE IN PRESS Journal of Biomehanis 36 (2003) 1649 1658 Mehanis of the tapered interferene fit in dental implants Din@er Bozkaya, Sinan M.uft.u* Department of Mehanial Engineering, Northeastern University,
More informationSupply chain coordination; A Game Theory approach
aepted for publiation in the journal "Engineering Appliations of Artifiial Intelligene" 2008 upply hain oordination; A Game Theory approah JeanClaude Hennet x and Yasemin Arda xx x LI CNRUMR 668 Université
More informationThe Basics of International Trade: A Classroom Experiment
The Basis of International Trade: A Classroom Experiment Alberto Isgut, Ganesan Ravishanker, and Tanya Rosenblat * Wesleyan University Abstrat We introdue a simple webbased lassroom experiment in whih
More informationA novel active mass damper for vibration control of bridges
IABMAS 08, International Conferene on Bridge Maintenane, Safety and Management, 37 July 008, Seoul, Korea A novel ative mass damper for vibration ontrol of bridges U. Starossek & J. Sheller Strutural
More informationMICROWAVE COMPONENT ANALYSIS USING A NUMERICAL ELECTROMAGNETIC FIELD SOLVER
LLN R. JLON, CRIG R. MOORE, and JOHN E. PENN MICROWVE COMPONENT NLYSIS USING NUMERICL ELECTROMGNETIC FIELD SOLVER With the ret developmt of numerial eletromagneti field solvers, mirowave gineers an aurately
More information17. Shaft Design. Introduction. Torsion of circular shafts. Torsion of circular shafts. Standard diameters of shafts
Objetives 17. Shaft Design Compute fores ating on shafts from gears, pulleys, and sprokets. ind bending moments from gears, pulleys, or sprokets that are transmitting loads to or from other devies. Determine
More informationMeasurement of Powder Flow Properties that relate to Gravity Flow Behaviour through Industrial Processing Lines
Measurement of Powder Flow Properties that relate to Gravity Flow ehaviour through Industrial Proessing Lines A typial industrial powder proessing line will inlude several storage vessels (e.g. bins, bunkers,
More informationVapor Pressure of a Solid by Knudsen Effusion
Vapor Pressure of a Solid by Knudsen Effusion Readings: Atkins (8 th ed) pages 1718 and 746757; Garland, Nibler, and Shoemaker (GNS) (8 th ed) pages 11917, 587594, and 597. Introdution Methods of measuring
More informationVOLUME 13, ARTICLE 5, PAGES 117142 PUBLISHED 05 OCTOBER 2005 DOI: 10.4054/DemRes.2005.13.
Demographi Researh a free, expedited, online journal of peerreviewed researh and ommentary in the population sienes published by the Max Plank Institute for Demographi Researh KonradZuse Str. 1, D157
More informationImproved Vehicle Classification in Long Traffic Video by Cooperating Tracker and Classifier Modules
Improved Vehile Classifiation in Long Traffi Video by Cooperating Traker and Classifier Modules Brendan Morris and Mohan Trivedi University of California, San Diego San Diego, CA 92093 {b1morris, trivedi}@usd.edu
More informationDerivation of Einstein s Equation, E = mc 2, from the Classical Force Laws
Apeiron, Vol. 14, No. 4, Otober 7 435 Derivation of Einstein s Equation, E = m, from the Classial Fore Laws N. Hamdan, A.K. Hariri Department of Physis, University of Aleppo, Syria nhamdan59@hotmail.om,
More informationAsimple analytic method for transistor
A Simple Analyti Method for ransistor Osillator Design his straightforward mathematial tehnique helps optimize osillator designs By Andrei Grennikov Institute of Miroeletronis, Singapore Asimple analyti
More information4.15 USING METEOSAT SECOND GENERATION HIGH RESOLUTION VISIBLE DATA FOR THE IMPOVEMENT OF THE RAPID DEVELOPPING THUNDERSTORM PRODUCT
4.15 USNG METEOSAT SECOND GENEATON HGH ESOLUTON VSBLE DATA FO THE MPOVEMENT OF THE APD DEVELOPPNG THUNDESTOM PODUCT Oleksiy Kryvobok * Ukrainian HydroMeteorologial nstitute Kyiv, Ukraine Stephane Senesi
More informationExplanatory Examples on Indian Seismic Code IS 1893 (Part I)
Doument No. :: IITKGSDMAEQ1V.0 inal Report :: A  Earthquake Codes IITKGSDMA Projet on Building Codes Explanatory Examples on Indian Seismi Code IS 1893 (Part I) by Dr. Sudhir K Jain Department of
More informationFundamentals of Chemical Reactor Theory
UNIVERSITY OF CALIFORNIA, LOS ANGELES Civil & Environmental Engineering Department Fundamentals of Chemial Reator Theory Mihael K. Stenstrom Professor Diego Rosso Teahing Assistant Los Angeles, 3 Introdution
More informationJournal of Colloid and Interface Science
Journal of Colloid and Interfae Siene 338 (9) 93 Contents lists available at SieneDiret Journal of Colloid and Interfae Siene www.elsevier.om/loate/jis Contat angle of a emisperial bubble: An analytial
More informationHierarchical Clustering and Sampling Techniques for Network Monitoring
S. Sindhuja Hierarhial Clustering and Sampling Tehniques for etwork Monitoring S. Sindhuja ME ABSTRACT: etwork monitoring appliations are used to monitor network traffi flows. Clustering tehniques are
More informationWeighting Methods in Survey Sampling
Setion on Survey Researh Methods JSM 01 Weighting Methods in Survey Sampling Chiaohih Chang Ferry Butar Butar Abstrat It is said that a welldesigned survey an best prevent nonresponse. However, no matter
More informationNeural networkbased Load Balancing and Reactive Power Control by Static VAR Compensator
nternational Journal of Computer and Eletrial Engineering, Vol. 1, No. 1, April 2009 Neural networkbased Load Balaning and Reative Power Control by Stati VAR Compensator smail K. Said and Marouf Pirouti
More informationStatic Fairness Criteria in Telecommunications
Teknillinen Korkeakoulu ERIKOISTYÖ Teknillisen fysiikan koulutusohjelma 92002 Mat208 Sovelletun matematiikan erikoistyöt Stati Fairness Criteria in Teleommuniations Vesa Timonen, email: vesatimonen@hutfi
More informationHOW TO CALCULATE PRESSURE ANYWHERE IN A PUMP SYSTEM? Jacques Chaurette p. eng. www.lightmypump.com April 2003
HOW TO CALCULATE PRESSURE ANYWHERE IN A PUMP SYSTEM? Jaques Chaurette p. en. www.lihtmypump.om April 003 Synopsis Calulatin the total head of the pump is not the only task of the pump system desiner. Often
More informationCIS570 Lecture 4 Introduction to Dataflow Analysis 3
Introdution to Dataflow Analysis Last Time Control flow analysis BT disussion Today Introdue iterative dataflow analysis Liveness analysis Introdue other useful onepts CIS570 Leture 4 Introdution to
More informationF220 Series. Installation Instructions. Photoelectric Smoke/Heat Detectors
F0 Series EN Installation Instrutions Photoeletri Smoke/Heat Detetors F0 Series Installation Instrutions.0 General Information EN.0 General Information. F0B6 Series Bases Use with the F0 Series Heat and
More informationThe analysis of brushing tool characteristics
ARCHIVES OF CIVIL AND MECHANICAL ENGINEERING Vol. IV 004 No. 4 The analysis of rushing tool harateristis Kiele University of Tehnology, al. Tysiąleia P. P. 7, 534 Kiele In this paper, an analytial proedure
More informationWATER CLOSET SUPPORTS TECHNICAL DATA
WATER CLOSET SUPPORTS TECHNICAL DATA Smith engineers have developed an unusually omplete line of fixture supports for mounting all types of "off the floor" fixtures. Supports have been designed for water
More informationBEARING CAPACITY OF SOIL
BEARING CAPACITY OF SOIL Dr. S. K. Prasad Professor of Civil Engineering S. J. College of Engineering, Mysore 7.0 Syllabus. Definition of ultimate, net and safe bearing apaities, Allowable bearing pressure
More informationDifferential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation
Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of
More informationJournal of Engineering Science and Technology Review 6 (5) (2013) 143148. Research Article
Jestr Journal o Engineering Siene and Tehnology Review 6 (5) (13) 143148 Researh Artile JOURNAL OF Engineering Siene and Tehnology Review www.jestr.org Numerial Analyses on Seismi Behaviour o Conreteilled
More informationTHE EFFECT OF WATER VAPOR ON COUNTERFLOW DIFFUSION FLAMES
THE EFFECT OF WATER VAPOR ON COUNTERFLOW DIFFUSION FLAMES by Jaeil Suh and Arvind Atreya Combustion and Heat Tkansfer Laboratory Department of Mehanial Engineering and Applied Mehanis The University of
More informationA Comparison of Default and Reduced Bandwidth MR Imaging of the Spine at 1.5 T
9 A Comparison of efault and Redued Bandwidth MR Imaging of the Spine at 1.5 T L. Ketonen 1 S. Totterman 1 J. H. Simon 1 T. H. Foster 2. K. Kido 1 J. Szumowski 1 S. E. Joy1 The value of a redued bandwidth
More informationTo the Graduate Council:
o the Graduate Counil: I am submitting herewith a dissertation written by Yan Xu entitled A Generalized Instantaneous Nonative Power heory for Parallel Nonative Power Compensation. I have examined the
More informationCOMPUTATION OF THE PROPAGATION CHARACTERISTICS OF TE AND TM MODES IN WAVEGUIDES WITH THE USE OF THE GENERALIZED DIFFERENTIAL QUADRATURE METHOD
Figure 4 Input impedane behavior for an ETNUTL versus the length of the line when the tapering fator is varying resonane peak that is shifted when the length of the line is varying. Furthermore, this peak
More informationEquivalence between the formulas for inductance calculation
357 Equivalence between the formulas for inductance calculation Marcelo Bueno and A.K.T. Assis 1. Introduction Abstract: We demonstrate the equivalence for the selfinductance of closed circuits, with
More informationA ContextAware Preference Database System
J. PERVASIVE COMPUT. & COMM. (), MARCH 005. TROUBADOR PUBLISHING LTD) A ContextAware Preferene Database System Kostas Stefanidis Department of Computer Siene, University of Ioannina,, kstef@s.uoi.gr Evaggelia
More informationImproved SOMBased HighDimensional Data Visualization Algorithm
Computer and Information Siene; Vol. 5, No. 4; 2012 ISSN 19138989 EISSN 19138997 Published by Canadian Center of Siene and Eduation Improved SOMBased HighDimensional Data Visualization Algorithm Wang
More informationSet Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos)
A.1 Set Theory and Logi: Fundamental Conepts (Notes by Dr. J. Santos) A.1. Primitive Conepts. In mathematis, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existene
More information