TEAL Physics and Mathematics Documentation

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1 Vesin. 7/7/008 TAL Phsics and Mathematics cumentatin Jhn Belche, Stanislaw Olbet, and Nman eb IN PF FORMAT THIS OCUMNT HAS BOOKMARKS FOR NAVIGATION CLICK ON TH LFT BOOKMARK TAB IN TH PF RAR Vesin., Jul 7, 008 This dcument pvides desciptins f the phsics and mathematics f man f the applets and visualiatins ceated f the Technlg nabled Active Leaning (TAL/Studi Phsics Pject at MIT and f elated NSF and Fundatin gants that peceded fllwed that pject. Sample cde is included, as apppiate. All units ae SI unless thewise indicated. In μ the tanslating fmulas int cde, facts f and ae usuall dpped, unless 4π 4πε thewise indicated. Cmments and questins t jbelche@mit.edu

2 Vesin. 7/7/008 Table f Cntents (LIV LINKS Suppting Mathematics Special Functins The Cmplete lliptic Integal Cde f Cmputing The Cmplete lliptical Integal Gemet Vect Tansfmatins F An M Object With An Axis Of Smmet.7.. Pint-Nmal Fm f a Line in Tw imensins Flux Functins Geneal Cnsideatins f Axismmetic Cnfiguatins The Time ependence f Field Lines In Magnet-quasi-statics....4 Rtatinal namics (N. eb Chsing a Cdinate Sstem and Specifing the State Oientatin f the Bd Quatenin Algeba quatin f Mtin... 6 lectstatics Tw imensinal lectstatics (and Magnetstatics Geneal Cnsideatins The lectstatic ( Magnetstatic iple and Line f iples 9..3 Cnstant lectic Field Line f Chage Line f Cuent Thee imensinal lectstatics..... Pint Chage in a Unifm lectic Field..... Pint Chage Being Chaged B a Line Cuent Flux f a Pint Chage thugh a Lp (fllwing N. eb Magnetstatics Magnetic Field Lines f a Pint 3 iple The Vect Ptential in Clindical Cdinates Cmpnents f the Magnetic Field in Clindical Cdinates The Flux Functin in Clindical Cdinates Magnetic Field Lines f A Cicula Lp Of Cuent The Vect Ptential in Clindical Cdinates Cmpnents f the Magnetic Field and the lectic Field The Flux Functin in Clindical Cdinates Magnetic Field f a Line f 3 Magnetic iples The Ptential f a 3 iple The Ptential f a Line f 3 iples The Field f a Line f 3 iples alng the x-axis Tw Magnetic iples Inteacting Faada s Law The Falling Ring The quatin f Mtin... 39

3 Vesin. 7/7/ etemining an quatin f I fm Faada's Law imensinless Fm f the quatins Cnsevatin f neg Numeical Slutins The Tplg f The Field Magnetic iple Mving Nea a Cicula Cnducting Lp (N. eb Flux thugh the Lp Limits quatins f Mtin Cicuits The isplacement Cuent Radiatin lectic iple Radiatin The lectic and Magnetic Fields The Flux Functin f a iple Oiented Alng The Z-Axis An Oscillating lectic iple A Rtating lectic iple Linea Antenna (S. Olbet and N. eb Ntatins, efinitins, Basics Flux Functin f Linea Antenna Singula Pints f -Field f Linea Antenna Appendices What is new in Vesin. cmpaed t Vesin.0? Time vlutin f Field Lines Using Flux Functins Refeences Index... 73

4 Vesin. 7/7/008 4 Figue Captins Figue.-: Pint nmal fm f equatin f line... 8 Figue.3-: A magnet levitating abve a disk with e esistance.... Figue.3-: Iscntu levels f the flux functins f tw ings... 4 Figue.-: The ientatin f the cicle with espect t the pint chage... 4 Figue Figue.3-: A 3 magnetic diple... 8 Figue.4-: Lp f cuent... 9 Figue.4-: Lp flux functin f < a... 3 Figue.4-3: Lp flux functin f > a Figue.4-4: Field line f a lp with a flux functin value f Figue 3.-: Gemet f the falling ing Figue 3.-: iple flux thugh a pla cap Figue 3.-3: Numeical Slutins f the Falling Ring quatins Figue 3.-: The gemet f the ing and diple f the flux calculatin Figue 3.-: Gemetic intepetatin f R and R Figue 6.-: Field lines and flux functin f an scillating electic diple Figue 6.- t vs (a.00, π/4, π / Figue 6.-: Singula pints f the esnant antenna... 68

5 Vesin. 7/7/008 5 Suppting Mathematics. Special Functins.. The Cmplete lliptic Integal The geneal cmplete elliptic integal is cel( k c, p, a, b π / 0 (cs ( a cs β psin β bsin β cs β dβ β k c sin β (... We efe t this definitin in a numbe f places in what fllws, f example in Sectin 4... Sme useful elatins f cel ae: cel( k π / π / dβ,,, 0 cs β k sin β 0 dβ c k sin β c π / k kc cel( kc,,, kc k sin β dβ 0 ( k K( k (... (...3 (...4 whee K and ae Legende s standad fms f the cmplete elliptic integals f the fist and secnd kind. Sme assciated elliptic integals ae π / sin β dβ K cel( k sin β k 0 π / cs β dφ kc K B cel( k k sin β k 0 k c,,0, c,,,0 (...5 ( Cde f Cmputing The Cmplete lliptical Integal package teal.math; public class SpecialFunctins { /** * lliptic integals: * This algithm f the calculatin f the cmplete elliptic * integal (CI is pesented in papes b Rnald Bulisch,

6 Vesin. 7/7/008 6 * Numeical Calculatin f lliptic Integals and * lliptic Functins, Numeische Mathematik 7, * (965 and Rnald Bulisch: Numeical Calculatin * f lliptic Integals and lliptic Functins III, * Numeische Mathematik 3, (969. The definitin * f the cmplete elliptic integal is given in equatin (... * f the dcument " TAL Phsics and Mathematics cumentatin " */ public static duble ellipticintegal(duble kcc, duble pp, duble aa, duble bb, duble accuac { duble ca, kc, p, a, b, e, m, f, q, g; ca accuac; kc kcc; p pp; a aa; b bb; if ( kc! 0.0 { kc Math.abs(kc; e kc; m.0; if (p > 0. { p Math.sqt(p; b b/p; } else { f Math.pw(kc,.0; q.-f; g.-p; f f-p; q q*(b-a*p; p Math.sqt(f/g; a (a-b/g; b -q/(p*math.pw(g,.0 a*p; } f a; a b/p a; g e/p; b.0*(f*g b; p p g; g m; m m kc;

7 Vesin. 7/7/008 7 } while (Math.abs(g - kc > g*ca { kc.0*math.sqt(e; e kc*m; f a; a b/p a; g e/p; b.0*(f*g b; p p g; g m; m m kc; } etun (Math.PI /.*(a*m b/(m*(m p; } else { etun 0.0; } }. Gemet.. Vect Tansfmatins F An M Object With An Axis Of Smmet We fequentl want t find the cdinates f an bsevatin pint in a pimed cdinate sstem centeed n an electmagnetic bject that has an axis f smmet, with the pime axis alng the axis f smmet. This ccus, f example, when the expessin f the field f that bject takes n an especiall simple fm in this pimed cdinate sstem (e.g., the field f a pint magnetic diple. T get the cdinates f an abita bsevatin pint in this pimed cdinate sstem, we d the fllwing. Let M be the smmet axis f the electmagnetic bject (f example, the magnetic diple mment vect. Let X bject be the psitin f the bject, and X the psitin f the bsevatin pint. efine the vects Z ˆ M / M R X - X bject R R - (R Z ˆ Zˆ pep ˆ R / R (... pep pep Then if we cmpute ˆ R and Zˆ R, we have the cdinates f u bsevatin pint in a fame in which the electmagnetic bject is at the igin, the pime axis is alng the smmet axis f the bject, and the pime axis is in the plane defined b the

8 Vesin. 7/7/008 8 smmet axis M and the diectin t the bseve R, and pependicula t the smmet axis. We then use these cdinates t calculate b means f elativel simple fmulas the cmpnents f the field in this cdinate sstem, sa B and B. We then ecnstuct the field in u unpimed iginal cdinate sstem using B B Zˆ (... B ˆ.. Pint-Nmal Fm f a Line in Tw imensins... efining quatin Given tw endpints A and B (lines d nt have endpints, but this is a tpical applicatin, it is ften useful t have a fm f the line based n a pependicula vect called the nmal. The nmal specifies the "facing" diectin f the line the insideutside half spaces f the line ceated b the spaces n each side f the line. The nmal is said t be the "fnt facing" diectin f the line the diectin f the "inside" space. Hweve, the nmal can be facing eithe diectin (thee ae tw diectin pependicula t a line in tw dimensins s the "inside diectin" "fnt-facing" nmal is meel a cnventin t be detemined b the pgamme. The methd that fllws will be a cnsistent epesentatin f all lines and epesents the nmal as a 90 degee tatin f the line in a cunte-clckwise diectin. Figue.-: Pint nmal fm f equatin f line

9 Vesin. 7/7/008 9 The Pint-Nmal fm f a line is np ˆ, whee the vect p lcates an pint n the line. The nmal is simpl the diectin vect f the line fm pint A t pint B ( B A tated 90 degees in a cunte-clckwise diectin. The equatin np ˆ is the mathematical statement that the dt pduct f the nmal t the line and an pint n the line equals the distance t the line fm the igin alng the nmal vect (see Figue.-. T find, we can use the nmal n, the pint A, and the Pint-Nmal equatin t deive the fllwing expessin. nˆ A x x n A n A (... F clait, if we expand ut the Pint-Nmal fm f the line we will btain the me cmmn line equatin in the fm a x b n x np ˆ nxx n x n n ( Intesectin with a Line A a is defined paameticall b the equatin p( t s d t p(t whee p(t is a pint n the line at "time" t, s is the stating pint, and d is the diectin vect f the a (that is d is a unit vect in the diectin f the a. Thus u time t must have dimensins f distance. F example, the line fm A t B can be defined paameticall ( as p( t A B-A t whee 0 < t < B-A and p(t ae all f the pints n the line B-A fm A t B. Hweve, if t is allwed t g bend these bundaies (in a psitive diectin it will define an infinite a in the diectin fm A t B. We can easil detemine the "hit time" t whee u geneal a intesects a paticula line b substituting the paametic a functin int the pint-nmal fm f the line. Since p(t defines a pint n the a at time t and the Pint-Nmal equatin defines all pints p n the line, substituting this value int the Pint-Nmal equatin esults in a functin f t that can be used t detemine the "hit time" t whee the a's pint satisfies the Pint-Nmal equatin f the line: Theefe the hit time is given b ( t np ˆ nˆ s d ns ˆ ndt ˆ (...3 t hit ns ˆ nd ˆ (...4 If nd ˆ 0 the a and the line ( plane d nt intesect because the a and the line ( plane, see belw ae paallel since the nmal n is pependicula t the a diectin d.

10 Vesin. 7/7/008 0 Als, if the hit time t is negative, the a has intesected a line plane in the ppsite diectin, which we fequentl igne. The "hit time" als efes t the distance fm the a's stating pint s t the intesectin pint. T see this, we can evaluate the paametic functin f the a at the hit time t: ns ˆ Pint f intesectin p( thit s dthit s d nd ˆ ns ˆ distance fm s t pint f intesectin p( thit s dthit thit nd ˆ (...4 ( Intesectin with a Plane The pint f intesectin f a a with a plane and the distance t the intesectin fm the stating pint f the a is detemined exactl the same as with a line. The nl diffeence is that nw the Pint-Nmal fm epesents a plane instead f a line. That is, in the equatin np ˆ, n is the nmal t the plane and is the distance fm the igin t the plane in the diectin alng the nmal t the plane..3 Flux Functins.3. Geneal Cnsideatins f Axismmetic Cnfiguatins When studing field lines in the case f an axismmetic cnfiguatin, i.e., when the vects d nt depend n the aimuth angle φ f the clindical cdinate sstem (,φ,, it is useful t cnside thei assciated flux functins. T this end, cnside tw classes f vects: plidal vects, sa V, and tidal vects, sa W. B definitin, a plidal vect V lies in the plane, and thus has tw cmpnents: V and V. In cntast, a tidal vect W has nl ne cmpnent, W φ, pinting alng the aimuth φ, s that its field lines clse n themselves. In the wds, W is divegence fee. If we assume in additin that the plidal vect V is als divegence fee, then it is eas t shw that the cul f a tidal vect geneates a plidal vect (and vice vesa, vi.: whee V V W W ˆ φ W ˆ φ V ˆ V (.3.. Wφ (, φ V [ Wφ (, ] φ ˆ (.3.. We nw define the scala flux functin F(, f V t be flux f V passing thugh a cicle f adius at height cncentic with the -axis, e.g.

11 Vesin. 7/7/008 ( W ˆ da F(, V da V ˆ da (.3..3 suface suface suface Using Stkes Theem, we can tansfm the suface integal t a line integal, giving F(, W dφ π W (.3..4 line φ φ A suface n which F is cnstant is an axiall smmetical shell cntaining the lines f fce f V. Cmpaing equatins (.3.. and (.3..4 shws that V is elated t its flux functin F b V F(, π F(, V (.3..5 π.3. The Time ependence f Field Lines In Magnet-quasi-statics We discuss the cncept f field line mtin in magnet-quasi-statics, and hw t define that mtin in a phsicall meaningful, but nt unique, wa. Cnside the fllwing thught expeiment. We have a slenid caing cuent pvided b the emf f a batte. The axis f the slenid is vetical. We place the entie 9.atus n a cat, and mve the cat hintall at a cnstant velcit V. Ou intuitin is that the magnetic field lines assciated with the cuents in the slenid shuld mve with thei suce, i.e., with the cat. Hw d we make this intuitin quantitative? Fist, we ealie that in the labat fame thee will be a "mtinal" electic field given b V B. We then imagine placing a lw eneg test electic chage in the magnetic field f the slenid, at its cente. The chage will gate abut the field and the cente f gatin will mve in the labat fame because it B difts ( v B / B in the V B electic field. This B dift velcit is just V. That is, the test electic chage "hugs" the "mving" field line, mving at the velcit u intuitin expects. In the me geneal case (e.g., tw suces f field mving at diffeent velcities, the mtin we chse has the same phsical basis. That is, the mtin f a given field line is what we wuld bseve in watching the mtin f lw eneg test electic chages spead alng that magnetic field line. We als use this definitin f the mtin f file lines in situatins that ae nt quasistatic, f example diple adiatin in the inductin and adiatin nes. In this case (but nt in the quasistatic cases the calculated mtin f the field lines is nn-phsical, as thei speed exceeds that f light in sme egins. Hweve, animatins f the field line mtin using the definitin abve ae still useful. F example, the diectin f the diectin f field line mtin s defined indicates the diectin f eneg flw.

12 Vesin. 7/7/008 T calculate field line mtin cnsistent with u definitin abve, we need t insue that u velcit field is flux peseving (Sten, 966; Vasliunas, 97; Rssi and Olbet, 970. F an geneal vect field G(x,,,t, the ate f change f the flux f that field thugh an pen suface S bunded b a cntu C which mves with velcit v(x,,,t is given b d dt G G da da ( G v da ( v G dl t S S S B If we appl this equatin t B(x,,,t and use B 0 and, we have t C (.3.. d dt B da ( v B dl S C (.3.. If we then define the mtin f u cntus s that the magnetic flux thugh the sufaces the bund is cnstant as a functin f time, and cnside cicula cntus and fields with aimuthal smmet, then equatin (.3.. guaantees that thei mtin satisfies v B 0, which is the same as v B / B, assuming that v and B ae pependicula. We can make this assumptin since thee is n meaning t the mtin f a field line paallel t itself. This is just the dift velcit f lw eneg test electic mnples that we efe t abve. This definitin f field line mtin is nt unique (see Vasliunas, 97. Figue.3-: A magnet levitating abve a disk with e esistance.

13 Vesin. 7/7/008 3 Field Lines Oiginating Fm A Singulait: In the situatin that u field lines iginate fm a singulait, cnstucting thei time dependence is staightfwad. Cnside the mtin f the field lines f a magnet levitating abve a disk with e esistance (Figue.3-. The magnet is cnstained t mve nl n the axis f the disk, and the diple mment f the magnet is als cnstained t be paallel t that axis. dd cuents in the disk will epel the magnet, and at sme pint thee will be a balance between the dwnwad fce f gavit and the upwad fce f epulsin. We then cnside small displacements abut this equilibium psitin, which will be peidic. The field lines themselves ae given b avis and Reit (97, and have aimuthal smmet. Hw d we tace the mtin f a field line? We d this b stating u integatin ve clse t the magnet at a cnstant angle fm the vetical axis, fllwing a given field line ut fm that pint. T animate a line, we use the same stating angle at eve pint in the scillatin. The field line taced ut will be diffeent when the magnet is at diffeent distances fm the disk. But cnside the flux inside an pen suface whse bunding cntu is defined b the intesectin f a hintal plane and the field line when tated aimuthall. This pen suface will have cnstant flux inside it, since B 0. Since we alwas stat fm clse t the singulait at the same angle, this cnstant flux will be the same f eve instant f time. Since the left hand side f (.3.. is e b cnstuctin, the ight hand side must als be e, and b smmet the integand must be e as well. Theefe u field line mtin as we have cnstucted it eflects the dift mtin f lw eneg test electic chages spead alng it. Field Lines Nt Oiginating Fm A Singulait: In this case the cnstuctin f the time evlutin f the field lines is me cmplicated, and we use the flux functins defined abve. Ou flux functin f the magnetic field F(,, t B (,, t da (.3..3 suface (cf is nw time-dependent. If we chse successive field lines as time evlves such that the have the same (cnstant value f the flux functin, then (.3.. is again satisfied, and again in smmetic situatins the field line mtin s defined is such that v B 0, equivalentl, v B / B. Anthe wa f stating this is that if we lk at iscntus f the flux functin F(,, t, then these iscntus tace ut the time-dependent mtin f the field lines. Althugh the deivatin that we have sketched abve is elegant, it is pehaps als eassuing t d this in a manne that shws the same thing in me detail, althugh in a much clumsie wa. We cnstuct such a pf in Appendix 8.. As an example f this pcess, cnside the time dependence f a field line when we have tw ings f cuent (Figue.3- sepaated b a vetical distance f 0. The adius f the bttm ing is 0, with a diple mment vect f.0 ẑ, and the adius f the tp ing is 5, with a diple mment that vaies in time. The innemst field line in

14 Vesin. 7/7/008 4 Figue.3- cespnds t a diple mment vect f the tp ing f 0. ẑ, at a flux functin value f 0.5. The middle field line cespnds t the same flux functin value f 0.5, but f a diple mment vect f the tp ing f 0.5 ẑ. The utemst field line is again at the same value f the flux functin, but f a diple mment f the tp ing f.0 ˆ. The sequence f field lines is what we expect f the time evlutin f this field line as the diple mment vect f the tp ing gws in time fm 0. ẑ t.0 ẑ. That is, this wuld be the path taced ut b lw eneg chaged paticles spead alng the field line as the diple mment f the tp ing gws in time. Figue.3-: Iscntu levels f the flux functins f tw ings.4 Rtatinal namics (N. eb Hw d we d the dnamics when an bject can tate? Hee is an veview..4. Chsing a Cdinate Sstem and Specifing the State Pick a cdinate sstem fixed in the bd with igin at the Cente f Mass (COM: x B, B, B. Specif the inetia tens IB in these cdinates. Pick cdinates s that it is eas t calculate I B. If u chse ppel, I B can be diagnal. Cmpute and ste its invese I. B When dealing with a bd that can tate, we specif state f the bject b specifing: x psitin f COM in wld cdinates (vect p linea mmentum (vect Q ientatin in wld cdinates (quatenin L angula mmentum (vect.4. Oientatin f the Bd

15 Vesin. 7/7/008 5 Thee numbes ae needed t specif the ientatin f a bd. Thee ae a vaiet f was t expess these. a These can be thee ule angles, which ae cnvenient f man pupses, but awkwad t descibe t the uninitiated. b Altenativel, we can descibe an ientatin b specifing an axis f tatin (a unit vect û and an angle f tatin A (using the ight-hand ule abut this axis. This desciptin is edundant since it invlves 4 numbes, but nl f the 3 cmpnents f the unit vect ae independent. c Finall, ientatin can be specified b a 3-b-3 tatin matix R defined s that a pint in the bd ( x,, has wld cdinates: B B B x x R B B B x Such a desciptin is edundant since it invlves nine quantities, but R is an thgnal matix s nt all its cmpnents ae independent cm (.4.. Methd (b is usuall implemented b specifing the fu numbes in a unit quatenin Q. F tatin abut a wld axis (diectin specified b unit vect u ˆ b an angle A, Q is defined as: Q [cs( A, sin( A u] ˆ [ Q, v] [,,, ] Q Q Qx Q Q (.4.. Once Q is specified, then R can be cmputed via Q Q QxQ QQ QxQ QQ R QxQ QQ Qx Q QQ QQx (.4..3 QxQ QQ QQ QQx Qx Q As an example, the quatenin [cs45, sin45 k ˆ ].707 [,0,0,] epesents a 90 tatin abut the -axis, sending xb int and B int x. If this Q is used t cmpute R, we get Quatenin Algeba 0 0.

16 Vesin. 7/7/008 6 F a geneal quatenin Q, we define the fllwing: nm f Q Q Q Q Q Q Q 0 v 0 x multiplicatin: Q * Q Q Q - Q Q, Q Q Q Q Q Q v v v v v v The cnvenient featue f quatenins f descibing tatin is that the ientatin f a bd afte tw successive tatins abut tw diffeent axes can be specified b a single quatenin Q Q * Q. net F u unit tatin quatenin Q, define a quatenin angula speed : [ 0, ] W ω (.4.3. Then, it can then be shwn that dq ½ * ½ [- u ˆ sin( A, cs( A WQ ω ω sin( A ω u ˆ ] (.4.3. dt Afte ientatin has been specified we can cmpute the mment f inetia in wld cdinates: R I T I R B I R I T B R ( An angula velcit is defined b ω ω ê, whee e ˆ is a unit vect and ω epesents the ate f tatin abut e ˆ (sign detemined b the ight hand ule. Given an angula velcit, the angula mmentum is defined b L I ω ( quatin f Mtin The state vect x ( t p Y Q L (.4.4. evlves accding t the equatin

17 Vesin. 7/7/008 7 v d ( t F Y dt W*Q τ (.4.4. whee steps: τ is the tque. S, stating with the cuent state Y, we ca ut the fllwing a cmpute v p / m b nmalie Q (s that it cntinues t epesent a tue tatin; then cmpute R. c cmpute I R I R T B d cmpute ω I L, and then W using (.4.3. These quantities, cmbined with the fce F and tque τ, specifies the ate f change f Y. Then we use an O slve t evlve the state vect. lectstatics. Tw imensinal lectstatics (and Magnetstatics.. Geneal Cnsideatins It is well knwn that in tw Catesian dimensins that slving ptential pblems in electstatics due t a discete numbe f line chages has man cespndences with the the f analtic functins f a cmplex vaiable (Mse and Feshbach 953. In paticula, cnside the analtic (except f a discete numbe f singulaities at the lcatin f the line chages functin G(Z f the cmplex vaiable Z x i (Z is nt the spatial cdinate, whee x and ae the Catesian cdinates f the tw-dimensinal pblem. If we can find an G(Z whse eal pat is the electstatic ptential Φ f the pblem, then the electic field lines ae given b the iscntus f the imagina pat f G. F cmpleteness, we sketch wh this is tue. Let G(ZU(x,i V(x,, whee U and V ae eal functins f x and. F G(Z t be analtic at a pint Z, its deivative must exist and be the same whethe we appach the pint Z in the cmplex plane alng the x-axis alng the -axis. That is, we must have

18 Vesin. 7/7/008 8 dg( Z G( x Δx, G( x, G( x, Δ G( x, lim lim (... d( Z Δx 0 Δx iδ 0 iδ Using the definitin f G in tems f U and V, and equating eal and imagina pats in equatin (..., we have U(x, x V(x, and U(x, V(x, x (... fm which it can easil be deduced that bth U and V ae slutins t Laplace's equatin in tw-dimensins, almst evewhee. Nw suppse we find an analtic (except f a discete numbe f singulaities functin G(Z such that the eal pat f G(Z is the slutin t a tw-dimensinal electstatic ptential pblem with discete suces. That is, f this G(Z, u ptential satisfies Re [ F] U φ. The electic field lines due t this ptential ae given b φ. Cnside the iscntus f Im [ F ] V. Let Y(x be an iscntu f V(x,. Then the change in V(x, when we mve alng Y(x must be e. That is which means that dv ( x, Y ( x dx V x V dy dx 0 (...3 dy dx V x V U U φ φ / / / (...4 x x x whee we have used equatin (... abve t eplace the patials f U with patials f V, the fact that U φ b assumptin, and φ. quatin (...4 is exactl what we equie f a cuve defining an electic field line. Thus if the eal pat f G(Z is equal t the electstatic ptential, then the iscntus f the imagina pat f G(Z ae paallel t the electic field lines. Nte that the scala functin V ( x, Im F( Z is in sme cases elated t the flux functin we discussed f example in.3, as ma be seen fm the fact that [ Im F ( Z ˆ ] [ V ( x, ˆ ] (...5 F example, when the sstem is smmetic abut the -axis, the imagina pat f G(Z is ne half f the flux f passing thugh a ectangle f width x in the x-diectin, lcated at height n the -axis, and centeed n that axis, pe unit length in the - diectin. Me imptantl, as can be shwn b explicit cnstuctin, the time evlutin f the electic field lines in elect-quasi-statics ae the same as the time evlutin f the iscntus f the imagina pat f G(Z. We nw give thee useful examples f these functins in tw-dimensinal electstatics, and ne elated example in tw-dimensinal magnetstatics.

19 Vesin. 7/7/ The lectstatic ( Magnetstatic iple and Line f iples Cnside a tw-dimensinal electic diple, that is a diple fmed b taking a line chage λ and a line chage - λ a vect distance d apat, with d pinting fm the negative t the psitive line chage. We let d g t e and λ g t infinit in such a wa that the pduct p λd (... ges t a cnstant. The quantit p is the tw-dimensinal electic diple vect. The electic ptential f a tw dimensinal electic diple, assuming the diple is at the igin, is Φ p X πε ( x πε p x p x ( x (... whee X x i j. We can wite the ptential in equatin (... as ( p ip Z Re πε x x Φ (...3 Theefe u electic field lines f this pblem ae iscntus f V ( x, Im πε ( p ip Z p p x [ p X] x x x πε x πε x (...4 These expessins als hlds f a tw dimensinal magnetic diple. These expessins can be genealied t a line f diples with a given ientatin. Cnside the fllwing cnfiguatin. A line f diples extends alng the x- axis fm d/ t d/. Alng this line thee is a line f tw diples with diple ˆ ˆ mment pe unit length p pxi p j. The electic ptential f this situatin can be deived b integatin f (... t ield ( x d/ d Φ ( x, pxln pactan ( x d/ x d / 4 (...5

20 Vesin. 7/7/008 0 and the electic field cmpnents can be cmputed b taking the negative gadient f this functin. The field lines f this cnfiguatin ae iscntus f the fllwing functin, b integatin f (...4. d ( x d / V( x, pxactan p ln x d /4 ( x d/ ( Cnstant lectic Field As a secnd example, cnside the electic field lines f a cnstant field electstatic ptential in this case is Φ [ ( x x ] Re[ ( x i Z]. The X Re (..3. whee again X x i j, s that the field lines ae given b the iscntus f ( x i Z] x x [ X V ( x, Im[ ] (..3. If we want t find the field lines f a tw-dimensinal diple in a cnstant field, we simpl add the tw functins abve apppiate t the tw ptentials t get the apppiate functin f this case...4 Line f Chage Finall, f u thid example, cnside the electic field lines f a line chage. The electstatic ptential f this case is λ Φ λ ln( x Re ln( Z πε π ε (..4. s that the field lines ae iscntus f λ λ V ( x, Im ln( Z θ πε (..4. πε When thee ae tw me line chages pesent, ne has t be caeful using the functin in equatin (..4. because f the banch cut in θ as θ uns fm 0 t π...5 Line f Cuent

21 Vesin. 7/7/008 Althugh stictl speaking we ae nl cnsideing electstatics in this sectin, we pint ut that we can d much the same thing in magnet-quasi-statics in tw dimensins. If the cuent densit J is e except at discete lcatins (that is, we have a finite numbe f line cuents unning in the -diectin, then B 0 almst evewhee, and we can wite f egins awa fm the discete suces that B Φ B (..5. whee φb is the magnet-quasi-static ptential. Since B 0, we have Φ B 0 (..5. In tw dimensins, finding the magnetic field lines in magnetstatic pblems with a discete numbe f line cuents can be aided b using the the f cmplex vaiables, as abve. In paticula, cnside the analtic (except f a discete numbe f singulaities at the suces functin G(Z f the cmplex vaiable Z x i, whee x and ae the catesian cdinates f the tw-dimensinal pblem. Just as abve, and f the same easns, if we can find an G(Z whse eal pat is the magnetstatic ptential φ B f the pblem, then the magnetic field lines ae given b the iscntus f the imagina pat f G(Z. F example, cnside the magnetic field f a line cuent at the igin. The magnetstatic ptential f this case is μ I μ I Φ B θ Re i ln( Z (..5.3 π π since ˆ μ I μ θ ˆ I B Φ B θ θ π x θ π (..5.4 whee θˆ is the unit vect in the aimuthal diectin, ight-handed abut the axis. Thus the field lines ae given b the iscntus f μ I μ I V ( x, Im i ln( Z ln( π (..5.5 π. Thee imensinal lectstatics.. Pint Chage in a Unifm lectic Field Suppse we have a pint chage at the igin f a spheical pla cdinate sstem. The electic field f this pint chage evewhee except at the lcatin f the chage can be witten as

22 Vesin. 7/7/008 pint ( csθ q (, t x ˆ φ 4πε sinθ (... We want t wite this expessin in a cdinate independent fm. T begin with let us mve the pint chage up the -axis b a distance p. Then ( ˆ p csθp csθ ˆ ˆ x ( p p (... ˆ ˆ ˆ ϕ and since sin Θ the expessin (... becmes q ( csθ (, ˆ pint t x ϕ 4πε sin θ x ( p (...3 The cnstant electic field in the -diectin can be witten as (, t ( sinθ ˆ x φ (...4 The electstatic flux functin (cf equatin (.3..4 can theefe be witten as F ( csθ π ( sinθ q (, θ chage ch ag e chage ε ( Pint Chage Being Chaged B a Line Cuent We cnside the situatin whee we have a pint chage at est at the igin f u cdinate sstem with a chage Q(t which is vaing with time. The inceasing chage is being supplied with cuent b a line cuent which is aanged alng the axis, caing cuent I dq/dt. In the quasi-elect-static appximatin, if ˆ is the spheical pla unit vect, the electic field is given b and the magnetic field is given b Qt ( (, t, ˆ (... 4π ε (

23 Vesin. 7/7/008 3 (,, ˆ I μ B t φ (... 4 π ( / We can deive the expessin f B given abve b cnsideing the line integal f B aund a cicle f adius centeed n the -axis a distance up that axis, and appling Ampee s Law including the displacement cuent tem. The xb magnetic mnple dift velcit in these cssed and B fields is given b (θ is the spheical pla angle B ˆ dq csθ VdB, (, t, c θ (...3 Q dt sinθ The field lines ae adial, and since the velcit f the field lines ae given b the equatin abve, we can wite f the θ dependence f a given field line that dθ dq csθ dt Q dt sinθ (...4 This equatin can be integated t shw that the time dependence f the angle θ denting a paticula adial field line is given b ( θ Q( t cs ( t cnst (...5 This is nt supising given the flux functin in (...3 f the pint chage...3 Flux f a Pint Chage thugh a Lp (fllwing N. eb..3. Flux thugh the Lp A cicula lp f adius a lies in the x- plane with is cente at the igin. A pint chage with chage q is lcated in the x- plane at psitin x ( t x xˆ ˆ d( xˆsinα ˆcs α P P P (..3. with chage q. The field pint is lcated b the vect. Let the angles ( θ, ϕ be the usual spheical pla angles f the field pint, and ( θ p, ϕ p the usual spheical pla angles in a cdinate sstem centeed n the pint chage, The vect ptential f the pint chage is given b (see...

24 Vesin. 7/7/008 4 ( θ p cs (, q ˆ pint t ϕ 4πε x sinθp x x A p (..3. Figue.-: The ientatin f the cicle with espect t the pint chage B Stkes theem, the magnetic flux f the field f the pint chage thugh the lp can be expessed as a line integal whee the integatin is aund the lp and Φ A x d l (..3.3 ( dl a dϕ φˆ a dϕ xˆ φ ˆ [ sin cs ] φ (..3.4 In the plane f the lp n its cicumfeence the field pint can be witten as [ ˆ cs ˆ sin ] a x φ φ (..3.5 and thus x t x a d a d P ( [ ˆ ( cs φ sin α ˆ sin φ ˆ cs ] ( α (..3.6 x t ( a cs φ d P sin α a sin φ d cs α a d ad cs φ sinα (..3.7 and

25 Vesin. 7/7/008 5 csθ p ˆ ( xp ( t d csα (..3.8 x P ( t a d adcsφ sinα Meve, ˆ x t ˆ ˆ xˆ a d ˆ a ˆ d φp ˆ x x ( P ( ( t ( P [ ( csφ sin α sinφ csα] ˆ ( P( t (..3.9 [ ˆ ( acsφ dsin α xˆ asinφ] ˆ φ p (..3.0 a d sin α adsinαcsφ q ( csθ p a dϕ A( x dl ˆ ϕ [ ˆ ˆ p x x] (..3. Φ 4πε sinθp xp q ( csθ p [ ˆ ( acsφdsin α xˆ asinφ] Φ adφ[ xˆ sinφ ˆ csφ] 4πε sin θ p x p a d sin α adsinαcsφ (..3. ( cs p aa [ dcs sin ] π q θ φ α Φ dϕ 4 sin 0 πε θ p x (..3.3 p a d sin α adsinαcsφ ( p ( p π q csθ aa [ dcsφsinα ] Φ dφ 4 0 πε cs θ a d adcsφ sinα a d sin α adsinαcsφ (..3.4 ( csθ p ( p ( a d adcsφ sinα dcsα ( a d ad d csθ csφ sinα csα The expessin in backets in (..3.4 is (..3.5

26 Vesin. 7/7/008 6 ( ( a d adcsφsinα dcsα [ ] a d adcsφsinα dcsα aadcsφsinα a d ad csφ sinα a d sin α ad sinαcsφ (..3.6 T check sme limits befe pceeding, take the case whee α 0, and the pint chage is thus n the axis and a distance d awa fm the igin. In this limit the integal (..3.4 becmes ( a d d ( a d d π q a q a d d Φ dφ ( πε a d ε a In de t put the integal int cel fm (see.., we intduce a diffeent angula vaiable: φ π β. Then, using csφ sin β cs β andsin φ sin β cs β, we wite the pevius expessin as sin Similal, x x β [ μ μ ( a x ] cs β[ μ μ ( a x ] μ sin β cs β x (4...7 x [( x a ] cs β ( x a [ ] x a ax csφ sin β (4...8 Afte defining the quantities R R ± ( x ± a and k c (4...9 R the flux thugh the ing can be witten as Φ π 0 μ μa Aφ adφ 4π R c cs β c sin β c π 3 3 π 3 [ cs β kc sin β ] sin β cs β dβ. (4...0 The last tem vanishes because f anti-smmet and the emaining dimensinless cnstants ae μ x μ ( x a μx μ ( a x c and c (4... μa μa Thus, the flux integal is

27 Vesin. 7/7/008 7 μ μa Φ π R 3 π 0 c cs β c sin [ cs β k sin β ] Cmpaing this integal t the definitin f the cel functin (..., we see that μ Φ π μa c ( k, k, c c cel c c R, 3 β 3 dβ (4... (4...3 The flux thus depends nl n the cmpnents f vects ling in the x- plane as shwn in the figue. All quantities can be intepeted gemeticall (see Figue 4.-. In equatin (4...3 we define the quantities R ± x c ( x ± a xˆ ˆ ± a xˆ ± (4...4 R k c (4...5 R μ R μa ± ˆ (4...6 Figue.- 3 Magnetstatics 3. Magnetic Field Lines f a Pint 3 iple The ptential and field f a single 3 diple at the igin with diple mment m is given b

28 Vesin. 7/7/008 8 μ mn μ m Φ M 4π 4π 3 μ 3nmn ( 3.. The Vect Ptential in Clindical Cdinates B Φ M 3 (3.3. 4π m Figue 3.-: A 3 magnetic diple F a pint 3 diple with diple mment M, the vect ptential A is given b μ X M ˆ μ M sinθ (, ˆ μm A φ φ ( π X 4π 4π ( 3/ 3.. Cmpnents f the Magnetic Field in Clindical Cdinates S [ A(, ˆ φ ] B (, (3... and B (, μ 3 [ A(, ] M 5 / 4π ( (3... μ ( B (, [ A(, ] M 5 / 4π ( ( The Flux Functin in Clindical Cdinates The flux functin f a magnetic diple (cf and 3... is μ π M F(, 4π ( 3 / ( Magnetic Field Lines f A Cicula Lp Of Cuent

29 Vesin. 7/7/ The Vect Ptential in Clindical Cdinates Cnside a lp with adius a caing cuent I, with magnetic diple mment M π a I. The vect magnetic ptential A can be witten as (Jacksn, Pgs μ ˆ φ ˆ 0Ia A(, A(, φ 4π π 0 a csφ dφ a csφ (3... Figue 3.-: Lp f cuent whee φ is the aimuthal angle abut the axis f the lp, and and ae clindical cdinates. If we change vaiables using φ β π, and define the nmalied distances / a and / a, then this equatin becmes Iμ 0 A(, π π / π / cs β dβ cs β (3... which (using cs β cs β sin β can be witten as A(, π whee F futue use, we als define Iμ 0 ( π / 0 (sin β cs cs β ( k c β dβ sin β (3...3 k c ( (3...4

30 Vesin. 7/7/ ( k kc (3...5 With this definitin, and using the definitin f cel given in (..., we have μ M k A(, / 4π π a cel( k,, 0 c, (3...6 T shw the cespndence between this expessin and the expessin abve in (3... f a pint diple, we take the limit that we ae fa fm the ing, which cespnds t the limit that k <<. In this limit, π cel( kc,,, k 6 (3...7 and using this expessin it can be shwn that (3...6 educes t ( Cmpnents f the Magnetic Field and the lectic Field The cmpnents f the magnetic field ae given b The adial cmpnent is thus (using (3... [ A(, ˆ φ ] B (, (3... B μ 0Ia (, [ A(, ] 4π 0 which afte sme manipulatin can be witten as π a csφ dφ a csφ (3... B (, πa Iμ 0 π / (sin β cs [ ( ] 3 / 0 ( cs β sin β k c β dβ 3 / (3...3 Using u definitin f cel in (... and (3...4, and M π a I, we can wite this as 3 μ0 M k B (, cel( k,,, 3 3/ c 4 c k (3...4 π πa Nw, the cmpnent f B is given b

31 Vesin. 7/7/008 3 B μ 0Ia (, [ A(, ] 4π 0 which afte sme manipulatin can be witten as π a csφ dφ a csφ (3...5 [ k ( k ] μ0 M k B (, cel( k,,, (,,, 3 3 / c c c cel kc kc 4π π a (3...6 T shw the cespndence between these expessins and the expessins abve in (3.., we need t take the limit that we ae fa fm the ing, which cespnds t the limit that k <<. In this limit, (3...7 is tue, and we als have in this limit that 3π cel( kc, kc,, k 6 (3...7 Using these limiting expessins it can be shwn that u cmpnents in (II.B..4 and (II.B..6 educe t thse f a 3 pint diple given in (II.A. fa fm the lp. A useful limiting fm f A(, when k is small is μ0 Ia k A(, 4 (3...8 π a a 4 In a situatin whee we have a ing f cuent mving with velcit v and/ with a time changing cuent in the ing, the nn-elativistic expessins f the magnetic field ae just thse given abve, and the electic field is given b the mtinal electic field and the induced electic field, e.g. A μ di a k t 4π dt a a 4 0 v B v B ˆ φ ( The Flux Functin in Clindical Cdinates The flux functin f a cicula lp f wie is just (see.3..4 and μ0 4Mk / F(, cel( k,,, ( / c μ0ki a cel( kc,,, ( π a and if we use (3...5 and (3...7 we see that the abve equatin educes t (3..3. f a pint diple when we ae fa fm the lp. It is instuctive t plt this functin in the plane f the lp, fm the cente f the lp t clse t its adius.

32 Vesin. 7/7/008 3 Lp Flux Functin f < a Flux /a Flux Figue 3.-: Lp flux functin f < a Figue 3.- gives such a plt, f a diple mment vect f unit and a adius f unit. The values f the flux functin scale invesel as lp adius f a fixed value f M. That is, as the adius inceases, the value f the flux functin deceases in pptin, at a given value f /a. We als shw in Figue 3.-3 the dependence f the flux functin n adius in the plane f the lp f > a. Figue 3.-4 shws the field line with a flux functin value f.0, f a lp with unit diple mment and unit adius. This field line csses the plane f the lp at 0.53 and als at 3.6, as we wuld expect given the flux cuves in Figues 3.- and Lp Flux Functin f > a Flux /a Flux Figue 3.-3: Lp flux functin f > a.

33 Vesin. 7/7/ Figue 3.-4: Field line f a lp with a flux functin value f Magnetic Field f a Line f 3 Magnetic iples 3.3. The Ptential f a 3 iple The ptential and field f a single 3 diple at the igin with diple mment m is given b μ mn μ m Φ M 4π 4π The Ptential f a Line f 3 iples μ 3nmn ( B Φ M 3 ( π m We assume the line stats at the igin and lies alng the x-axis, ending at x L. μ We have (dpping the 4 L π L ( x( (( 3/ x x m x x m m Φ M dx (3.3.. L L ( x x dx dx Φ m m m ( ( 3/ 3/ ( x x (( x x (3.3.. M x ( x x dx (( x x (( x x (( x L ( x L 3/ / / / 0 (3.3..3

34 Vesin. 7/7/ L L L L dx dx / ( / 3/ 3/ (( x x ( ( ( x x / dη tan η ( x x / ( cs η dx ( / tan ( x L/ dx ( / ( / 3/ (( x x ( / cs η tan / ( tan η x ( dx ( ( ( x x ( ( dη / tan ( x L/ ( sinη csηdη ( ( ( / tan ( x L/ 3/ / / tan x/ tan x/ ( η sinη / η η ( L dx ( x L x 3/ ( ( x L x (( x x ( S inseting ( and ( int (3.3.. Φ M m x ( x L x ( ( ( ( ( ( / / m m x L x x L x ( T check limits, let L g t e. Then ( m m L x Φ M ml x / x ( x ( x x mlx ( m m L x x 3/ 3/ ( x ( x ( x (3.3.. mlx ( m m L x x x 3/ 3/ ( x ( ( x

35 Vesin. 7/7/ Φ M ( x ( x 3/ m x m m L In the limit that L ges t infinit, we have m ( m m x x Φ M ( x / ( x (3.3.. ( We can genealie this t abita ientatins f the line. If t is the unit vect alng the diectin f the line, and the line begins at the igin, then ( m tt ˆ( ˆ ( tm ˆ t( t μ ˆ ˆ m t t Φ M 4π ( ˆ ˆ ( The Field f a Line f 3 iples alng the x-axis field. We g back t the expessin given in ( t get the cmpnents f the B x m 3 ( B ( m m ( m x x / ( x x ( x ( ( m m x 3/ ( ( x ( m ( m x x m m 3/ x ( ( m m ( m x x x B m 3 ( m 3 ( (

36 Vesin. 7/7/ B ( m m ( m x x / ( x x ( x ( ( m m x 3/ ( ( x ( m ( m x x m m 3/ x ( ( m m ( m x x x B m 3 ( m 3 ( ( O all tgethe B x m 3 ( ( ( m m ( ( m m ( m x x x B m 3 ( m 3 m x x x B m 3 ( m 3 ( T put this in cdinate independent fm, define t ˆ( t ˆ t ˆ ( ˆ t m m t t m ( ( w t ˆ m t t ˆ t m ( m B ˆ x 3 t x w m m t B B ( t m ( tˆ( tˆ tˆ( tˆ ( t m tˆ( tˆ tˆ tˆ ( ( ( ( ( ˆ ( ˆ t t t t t t t ( ˆ ( w m m t ˆ t t t t t t t (

37 B Vesin. 7/7/ ( mt ˆ ( ˆ t tm ˆ m t t t( t m t t B 3 4 t ( ( ˆ t m t 3 t t ( ( ( ˆ ( mt ˆ ˆ tm ˆ ˆ ˆ ˆ ˆ ˆ t t t m m t t t t t B t 3 m 3 t ( ( ˆ ( ˆ ( ˆ ( ( ( ˆ t { } ˆ ( mt ˆ ˆ t t tm t t m ˆ ˆ t t t t t m m B 3 t ( μ whee we have dpped the 4 π fact. 3.4 Tw Magnetic iples Inteacting Tw magnetic diples inteact. The tque n a diple sitting in the magnetic field due t diple is τ m B (3.4. and similal f the tque n due t the field f. The fce n diple sitting in the magnetic field B due t diple is which in cmpnent fm is ( F m B (3.4. j i j F m B i i x, i x,,, j x,, (3.4.3 The field B at a pint X due t a diple m lcated at X is given b μ 3( ˆ nmm nˆ B whee X X 3 n ˆ ( π and the tens gadient f B is given b

38 Vesin. 7/7/ μ 3 B 4 ( 5( 4π m n I mn nm m n nn (3.4.5 F tw inteacting diples, equatin ( then becmes μ 3 4π { } ( ( ( 5 ( ( F m n m 4 m n m m m m n m n n (3.4.6 The fce n diple sitting in the magnetic field BB due t diple is ppsite and equal t the fce given abve, because f the diectin f the nmal n eveses when the index changes.

39 Vesin. 7/7/ The Falling Ring 4.. The quatin f Mtin 4 Faada s Law Figue 4.-: Gemet f the falling ing We have a 3 diple with diple mment μ μẑ. It mves n the axis f a cicula lp f adius a, esistance R, inductance L, with inductive time cnstant L/R. It mves dwnwad unde the influence f gavit. The equatin f mtin is d m dt db mg μ d (4... whee B is the field due the cuent I in the ing (psitive in the diectin shw in Figue 4.-. The expessin f B is B μ Ia 3/ ( a (4... s that equatin (4... is d m dt μμ Ia mg d d 3/ ( a (4...3 This is the apppiate equatin f bth the situatin f the ing at est and the magnet mving, the magnet at est and the ing mving--the mass m switches fm the mass f the magnet t the mass f the ing, depending n the situatin.

40 Vesin. 7/7/ d m dt 3μμIa mg ( a 5/ ( etemining an quatin f I fm Faada's Law Faada's Law is d d di dl diple L dt B da dt B da (4... dt and if J, then dl J dl I dl / A IR, with R dl / A (4... s that di d I R L B diple da (4...3 dt dt We need t detemine the magnetic flux thugh the ing due t the diple field. T d this we calculate the flux thugh a spheical cap f adius a with an pening angle given θ given b sinθ a / a (this is the same as the flux thugh the ing because B 0. Figue 4.-: iple flux thugh a pla cap

41 Vesin. 7/7/008 4 The flux thugh a spheical cap nl invlves the adial cmpnent f the diple field, given b B μ μ csθ 3 (4...4 π Ou expessin f the flux is thus which can be integated t give B diple B diple da μμ μμ da cs d cs θ θ Inseting (4...5 int (4... ields μ μ csθ π 3 π ( cs θ sinθdθ μμ μμ sin θ a ( a 3/ (4...5 IR L di dt d dt μμ a ( a 3/ (4...6 We assume that μ is cnstant, with μ M, s that (4...6 becmes IR L di dt 3μa M d ( a 5/ dt (4...7 quatins (4...4 and (4...7 ae u cupled dina diffeential equatins which detemine the dnamics f the situatin imensinless Fm f the quatins We nw put these equatins int dimensinless fm. We measue all distances in tems f the distance a, and all times in tems f the time a / g. Let t I mga t I, wheei (4..3. a a / g I μ M

42 Vesin. 7/7/008 4 The time a / g is ughl the time it wuld take the magnet t fall unde the influence f gavit thugh a distance a stating fm est. The cuent I is ughl the cuent in the ing that is equied t pduce a fce sufficient t ffset gavit when the magnet is a distance a abve the ing. In tems f these vaiables, u equatins (4...4 and (4...6 ae I L R d dt 3 ( M g di 3 μ 3 a dt mga R I ( 5/ g a d 5/ ( dt (4..3. ( We intduce the fu dimensinless paametes α R L a g μ M β LI a ( μ M Lmga 3 λ L μ a g μa a R λα ( and we define the educed flux ate functin F(' as 3 F( ( 5/ ( Nte that we can wite the efeence cuent I as Ia M mga μ M 4 λβ ( The paametes have the fllwing phsical meanings. The quantit α is the ati f the fee fall time t the inductive time cnstant--if α is ve lage, inductive effects ae negligible. The quantit β is ughl the ati f the cuent due t inductin alne, μ assuming the esistance is e ( Φ diple / L with M Φ diple, t the efeence cuent a I. With these definitins, u equatins becme d F( I dt di d αi β F( dt dt ( (4..3.8

43 Vesin. 7/7/ If we define the speed v d / dt, then we can wite thee cupled fist-de dina diffeential equatins f the tiplet (, I, v, as d / dt v ( di d αi β F( dt dt ( Cnsevatin f neg dv F( I (4..3. dt We assume that μ is cnstant. If we multipl (4...4 b I, afte sme algeba, we find that d v and (4...7 b dt d [ mv mg LI ] I R dt (4..4. which expesses cnsevatin f eneg f the falling magnet plus the magnetic field f the ing. The dimensinless fm f this equatin is d α v I I dt 4 β β (4..4. Suppse the esistance f the ing (the supecnducting case is e (i.e., α 0. In this case, (4..4. becmes, with ne integatin I C β ( 3/ ( If we impse bunda cnditins that I' 0 when t' 0, with '' and v' v' at t' 0, then this is I ( ( β 3/ 3/ ( Using this equatin, (4..4. f the cnsevatin f eneg becmes

44 Vesin. 7/7/ v ( β 4 3 / ( 3 / ( 0 ( Numeical Slutins A magnet falls thugh a cppe ing. At the ing, the speed f the magnet deceases. When the magnet is thugh the ing, the magnet esumes fee fall. We shw a numeical slutin t equatins ( thugh (4..3. abve, apppiate t this case. The initial cnditins (, I, v f the slutin pltted belw ae (,0,0. The values f ( α, β ae (0,00. Belw we plt psitin as a functin f time and cuent as a functin f time (using u dimensinless paametes. '(t' I'(t' Figue 4.-3: Numeical Slutins f the Falling Ring quatins The behavi f these slutins is what we expect. When the magnet eaches a distance f abut a fm the ing, it slws dwn, because f the inceasing cuent in the ing, which epels the magnet. As the magnet passes thugh the ing, the cuent eveses diectin, nw attacting the magnet fm abve, which als slws the magnet. Finall the magnet falls fa enugh that the cuent in the ing ges t e, and the magnet is again in fee fall. These ae appximate slutins nl, using an xcel speadsheet with an ule integatin scheme. In the final animatins, we a futh de Runge-Kutta scheme t integate the equatins with high accuac The Tplg f The Field Hw d we plt the field cnfiguatin given slutins f and I'? The abslute cuent is given b (cf. equatins ( and ( I M I I I λβ a (4..6.

45 Vesin. 7/7/ Hw much feedm d we have in chsing the abslute value f the cuent nce we have slved u dimensinless equatins? And in paticula hw des that feedm affect the tplg f the magnetic field? One measue f the shape f the ttal field is the ati f the field at the cente f the ing due t the ing t the field at the cente f the ing due t the magnet when the magnet is a distance a abve the ing, i.e., B diple cente f ing B ing at cente f ing when dipleat a μi Ia a μ M (4..6. M 3 π a whee we ae dpping numeical facts. Cleal when this ati vaies the veall shape f the ttal field must va. If we use (4..6. in (4..6. we have cente f ing B ing at cente f ing when dipleat a B diple a a M I I I ( M M λβ a λβ The meaning f equatin ( is that the veall shape f the magnetic field tplg is ttall detemined nce we make the ne emaining chice f the dimensinless cnstant λ, defined in equatin ( which up t this pint we have nt chsen (we have nl picked values f α and β t slve u dimensinless equatin. Once that chice is made, we have n additinal feedm t affect the field tplg. 4. Magnetic iple Mving Nea a Cicula Cnducting Lp (N. eb 4.. Flux thugh the Lp A cicula lp f adius a lies in the x- plane with igin cente at the igin. A diple is lcated at psitin x ( ˆ x ˆ t x with magnetic mment μ ( t with abita ientatin. The vect ptential f the diple is given b (see 3... A ( x μ μ ( x x 3 4π x x (4...

46 Vesin. 7/7/ Figue 4.-: The gemet f the ing and diple f the flux calculatin B Stkes theem, the magnetic flux f the diple field thugh the lp can be expessed as a line integal Φ A x d l (4... whee the integatin is aund the lp. F integatin pupses, expess the integand in clindical cdinates. The unit vects ae: ( ˆ csφ xˆ sinφ ˆ φˆ sinφ xˆ csφ ˆ (4...3 xˆ csφ ˆ sinφ ˆ ˆ sinφ ˆ csφ ˆ (4...4 The magnetic mment is μ μ ˆ μ φˆ μ φ φ μ sinφ μ csφ. In the plane f the lp x μ ˆ, whee μ csφ μ sinφ and μ x x aˆ and d l a dφ φˆ (4...5 In the numeat f the flux integand we have μ ( x x φˆ μ ( aˆ x xˆ ˆ φˆ μ [ aˆ x ( csφ ˆ sinφ ˆ ˆ ] φˆ μ ( a x csφ ( μ μ x csφ μ a μ sinφ μ (4...6 x

47 Vesin. 7/7/ In de t put the integal int cel fm (see.., we intduce a diffeent angula vaiable: φ π β. Then, using csφ sin β cs β andsin φ sin β cs β, we wite the pevius expessin as sin Similal, x x β [ μ μ ( a x ] cs β[ μ μ ( a x ] μ sin β cs β x (4...7 x [( x a ] cs β ( x a [ ] x a ax csφ sin β Afte defining the quantities R ± ( x ± a and R R the flux thugh the ing can be witten as Φ π 0 μ μa Aφ adφ 4π R c cs β c sin β c π 3 3 π 3 [ cs β kc sin β ]. (4...8 k c (4...9 sin β cs β dβ (4...0 The last tem vanishes because f anti-smmet and the emaining dimensinless cnstants ae μ x μ ( x a μx μ ( a x c and c (4... μa μa Thus, the flux integal is μ μa Φ π R 3 π 0 c cs β c sin [ cs β k sin β ] Cmpaing this integal t the definitin f the cel functin (..., we see that μ Φ π μa c ( k, k, c c cel c c R, 3 β 3 dβ (4... (4...3 The flux thus depends nl n the cmpnents f vects ling in the x- plane as shwn in the figue. All quantities can be intepeted gemeticall (see Figue 4.-. In equatin (4...3 we define the quantities R ± x c ( x ± a xˆ ˆ ± a xˆ ± (4...4 R k c (4...5 R μ R μa ± ˆ (4...6

48 Vesin. 7/7/ Figue 4.-: Gemetic intepetatin f R and R Limits F a diple n and aligned with the -axis, k c and c ±, s the expessin f the flux educes t μ μa Φ (4... a 3 ( This expessin agees with pevius esults (see quatins f Mtin If the initial ientatin f the diple is such that its velcit vect and magnetic mment lie in the x- plane, then the fce n it will als lie in the x- plane and the tque n it will be in the diectin. Thus, the diple will emain in the x- plane as it tanslates and tates (assuming that this is a stable situatin. In this case the equatins f mtin ae elativel simple. We must specif as paametes f the diple its mass m, its mment f inetia I ; and the magnitude f its magnetic mment μ. F the lp we must als specif a esistance R and self-inductance L. The state f the magnet and cil can be specified b ( x, v, α, ω, IL, whee α is the tatin angle f the diple abut the -axis and ω its angula velcit, and I L.is the cuent in the lp. In tems f α, the magnetic mment is μ sin α xˆ csα ˆ. Then the state s evlutin is detemined b the fllwing equatins:

49 Vesin. 7/7/ dx dα Kinematics: v, ω (4..3. dt dt dv ( μ BL Fce equatin: g (4..3. dt m dω μ B L ˆ μ BLx μ xbl Tque equatin: ( dt I I di L dφ( x R Faada s law: I L ( dt L dt L whee Φ the flux thugh the lp due t the diple, I L the cuent in the lp, BL the magnetic field f the lp. xpessins f L B in tems f I L have alead been btained, s the deivatives indicated abve can be explicitl calculated. In de t descibe me geneal mtin, a me cmplicated state vect is equied. The mment f inetia must be eplaced b an inetia tens I specified in a cdinate sstem attached t the diple; the ientatin f the diple can be specified b a unit quatenin Q; and the angula mmentum L must be teated as a vect. Initial ientatin f the diple is witten as Q [cs( A, sin( A u ˆ] [Q,Q,Q,Q ] x, whee A angle f tatin f the diple bd cdinates abut u ˆ. The state f the magnet and cil is thus ( x, v, L,Q, I L and the evlutin f this state is detemined b the fllwing equatins: dx Kinematics: v ( dt dv ( μ BL Fce equatin: g ( dt m Q Q QxQ QQ QxQ QQ Rtatin matix: R QxQ QQ Qx Q QQ QQx QxQ QQ QQ QQx Qx Q ( T Gemet: I R I R ( dl Tque equatin: τ μ BL ( dt Angula velcit: ω I L, W[0, ω ] ( dq Kinematics: sin( A cs( A, sin( A ½W * Q ½ [- ω uˆ ω ω uˆ ] dt (4..3. di L dφ( x R Faada s law: I L (4..3. dt L dt L

50 Vesin. 7/7/ whee Φ the flux thugh the lp due t the diple, BL the magnetic field f the lp, Q the ientatin f the diple, I the bd-centeed inetia tens f the diple, I the diple mment f inetia in wld cdinates. xpessing the cnstants in clindical cdinates, ( μ μ μ ( a ± μ a ± μ c± ( μa μa Unde cnstuctin. 5 Cicuits 6 The isplacement Cuent Unde cnstuctin. 7 Radiatin 7. lectic iple Radiatin 7.. The lectic and Magnetic Fields The electic field f a time-vaing electic diple p(t is given b [ 3ˆ( n p nˆ p] [ 3ˆ( n p nˆ p ] ( p nˆ nˆ (, t 3 4πε 4πε c 4π ε c quasi - static diple inductin adiatin (7... whee the dt abve a vaiable indicates diffeentiatin with espect t time, and the electic diple mment vect and its time deivative ae evaluated at the etaded time t et t / c. With sme algebaic efft, it can be shwn that this expessin can be witten as

51 Vesin. 7/7/008 5 p p (, t nˆ 4πε c (7... Nw when we ae nt at the igin we ae in vacuum, s that we have (, t B (, t (7...3 c t But if we take the time deivative f (7... and multipl b /c, we easil have (, t p p c t c 4πε c nˆ (7...4 Cmpaing the tw equatins abve, we see that we must have μ p p B (, t 4π c nˆ ( The Flux Functin f a iple Oiented Alng The Z-Axis as If the diple mment p is alwas in the -diectin, we can wite the electic field p ( t / c p( t / c (, t sinθ ˆ φ (7... 4πε c quatin (7... and the develpment in.3 abve impl that the electic field lines f a simple adiating electic diple sstem in this case ae given b the iscntus f p ( t / c p( t / c F(, θ, t π sinθ A(, θ, t π sinθ sinθ (7... 4πε c p ( t / c p( t / c F(, θ, t π sin θ 4πε c (7...3 and this is the flux functin f such a diple, in the sense defined in.3. If we define the dimensinless vaiables we can ewite (7...3 as t t T (7...4 ct

52 Vesin. 7/7/008 5 θ π πε θ sin ( ( 4,, ( t p t p ct t F (7...5 whee nw the dt abve a vaiable indicates diffeentiatin with espect t the dimensinless time vaiable An Oscillating lectic iple Let us cnside a paticula case f the situatin abve. Suppse the diple mment is a cnstant plus a sinusidall vaing functin f time, with peid T. That is, cs( ( T t p p t p π (7..3. in which case (7...3 becmes θ π πε θ sin / ( cs / ( sin 4,, ( π π π T c t p p c T c t p T t F (7..3. in tems f u dimensinless vaiables ( ( ( π π π θ π πε θ sin ( cs ( sin 4,, ( t p p t p ct t F ( Figue 7.- belw shws the field lines f at t 0 as defined b (7..3.3, f tw values f the flux functin, -3 and 3.

53 Vesin. 7/7/ Figue 7.-: Field lines and flux functin f an scillating electic diple A Rtating lectic iple Let is lk at the electic and magnetic fields f a tating electic diple which is iented pependicula t its axis f tatin. We have [ ω ˆ ω ˆ] p( t p cs tx sin t [ xˆ ŷ] p ( t ωp sinωt csωt [ x ŷ] p( t ω p csωt sinωt ˆ (7..4. (7..4. ( We ae ging t lk at the fields nl in the x plane at 0. Thus nˆ csφ xˆ sinφ ˆ ( Using the expessins f p and n abve, we easil have that [ ˆ ˆ] [ ˆ ˆ] pn p csωtx sinωt csφx sinφ p cs( φω t (7..4.5

54 Vesin. 7/7/ [ ˆ ˆ] [ ˆ ˆ] pn ω p sinωtx csωt csφx sinφ ωp sin( φω t ( [ ] [ ] p n ω p sinωtxˆ csωt ˆ csφxˆ sinφ ˆ ωp ˆcs( φω t ( [ ] [ ] p n ω p csωtxˆ sinωt ˆ csφxˆ sinφ ˆ ω p ˆsin( φω t ( [ 3 ˆ( ˆ ] 3p cs( φ ωt ( csφ ˆ sinφ ˆ p ( csωt ˆ sinωt ˆ npn p x x ( [ 3 npn ˆ( ˆ p] p xˆ[ 3cs( φ ωt csφ csωt] p ˆ[ 3cs( φωtsinφsinωt] [ 3 npn ˆ( ˆ p] ωp sin( φ ωt ( csφxˆ sinφ ˆ ωp [ sinωtxˆ csωt ] ( ˆ (7..4. [ 3 npn ˆ( ˆ p ] ω p x[ 3sin( φ ωt csφ sinωt ] ωp ˆ [ 3sin( φωtsinφcsωt] [( p nˆ nˆ] ω p ˆsin( φ ωt ( csφxˆ sinφ ˆ S we have f the electic field (7..4. ( [( p nˆ nˆ] ω p sin( φωt [ sinφxˆ csφ ˆ] ( (, t [ 3cs( φω csφ csω ] ˆ[ 3cs( φω sinφsinω ] p xˆ t t p t t 4πε [ 3sin( cs sin ] ˆ [ 3sin( sin cs ] [ xˆ ˆ] sin( sin cs ω p φ ωt φ φ 4πεc 3 ω px φ ωt φ ωt ωp φ ωt φ ωt 4πεc ( μ ωsin( φωt cs( φωt B (, t ˆω p 4π c ( Linea Antenna (S. Olbet and N. eb

55 Vesin. 7/7/ Ntatins, efinitins, Basics Fllwing cmmn cnventins let, B, A, φ, c and j stand, espectivel, f electic field, magnetic field, vect ptential, scala ptential, chage densit and electic cuent densit. Let us als intduce and H such that Recall that the speed f light c is elated t ε and μ b B ε H (7... c ε μ With these ntatins Maxwell quatins acquie the cmpact fm μ B 0 (7... c B H j t t Let s define vect and scala ptentials A and φ such that (7...3 A φ (7...4 t B A (7...5 With the auxilia cnditin φ A ε μ 0 (7...6 t impsed, ne can then eadil shw that Maxwell s quatins lead t φ c φ εμ t ε (7...7 A t i Ai ε μ μ j i (7...8 whee the subscipt i indicates ne f the Catesian cmpnents f A j. T pve the abve, ne needs the identit ( A ( A A (7...9 B vitue f equatins (7..., (7...3, (7...4 and (7...9, ne has t A t c ( A (7...0

56 Vesin. 7/7/ In de t find the electic field f a given cuent sstem it is mathematicall me expedient t slve fist equatin (7...5 f A and then use equatin (7...0 t wk ut the expessin f. In fact, the geneal slutin f equatin (7...5 is knwn t be ( x, t μ 3 j A d x dt δ ( t tet (7... 4π x x whee is the etaded time. x x t et t (7... c Cente-fed Linea Antenna: Vect Ptential La a linea antenna f length l alng the -axis with its cente at the igin f a Catesian cdinate sstem. Let the chage be fed thugh cente hamnicall s that we can put f the cuent densit in the antenna j j 0 j x I sin kl k δ iω t ( x δ ( e (7...3 whee ω is the cicula fequenc and k is the wave numbe s that ω k (7...4 c We chse the phase f the phsical cuent densit s that it is the imagina pat f ( Inseting equatin (7...3 int equatin (7... ields A x A 0 ( iω t l / μ exp (7...5 et A sin I 4 d kl k π x x l / T simplif ntatin it is cnvenient t eplace time and space vaiables, t and x, b cespnding dimensinless quantities, t wit: a kl (7...6 ω t t ; kx x; kx x. With the dimensinless clindical cdinates and φ such that ( sin ( x cs φ φ

57 Vesin. 7/7/ we then have f u gemetical setup A a exp( itet d sin( a a ( μ I 4π (7...7 Because f the mi smmet abut the (x -plane, we can cnvet the integatin in equatin (7...7 t the psitive half abve this plane, ( 0. With ppe adjustments f signs we thus can eplace equatin (7...7 b A μ Ie 4π a it d sin exp( ir ( a R R exp( ir (7...8 whee we have put f sht ( ± ; x R ± (7...9 Cente-fed Linea Antenna: Clindical Cmpnents f We ae nw ead t wk ut explicit expessins f and. Using equatins (7...8 and (7...0 and taking advantage f axial smmet, ( φ 0, we get i A A (7...0 ω i A (7... ω Cnside sme functin K cntaining in its agument ( ± ; cleal, diffeentiating with espect t can be exchanged with that with espect t, vi., K( ± K( ± ± (7... With this in mind, ne can shw that diffeentiating an integal expessin f the fm F leads t the fllwing esults a ( d sin( a [ K( K( ] (7...3

58 Vesin. 7/7/ F a d cs( a [ K( K( ] (7...4 F F K( a K( a csa K( (7...5 Appling this t equatin (7...0 we find, with sme supise, that the -cmpnent f educes t the elementa fm whee i e iτ exp[ i( ] exp[ i( ] csa (7...6 μ I ω 4π ( ± a ; ; τ t ; ± Unftunatel, the -cmpnent f is nt educible t an elementa expessin. (7...7 Caing ut the diffeentiatin in equatin (7... fist with espect t, then making use f elatin (7...4 and the fact that R ± R± we find the fllwing integal expessin f a it ( ie d cs a [ G( R G( R ] (7...8 whee we have put f sht exp( iξ G ( ξ ( iξ ( ξ In pactical applicatins we need nl the eal ( imagina pats f equatins (7...6 and ( ing the necessa algeba, we aive at desied answe whee ( Q τ Q csτ i ( Q csτ sinτ sin ( Q ( P τ P csτ i ( P csτ sinτ sin P (7...3 a d cs( a [ g( R, g( R, ] P (7...3

59 Vesin. 7/7/ with and with ξ sin( ξ cs( ξ g( ξ, ( ξ a d cs( a [ g( R, g( R, ] P ( ξ cs( ξ sin( ξ g( ξ, ( ξ g Nte in passing that ( ( ξ, g ξ, and g ( ξ g, ( ξ,. Finall cs( cs( cs a Q ( sin( sin( Q ( Cente-fed Linea Antenna: Asmpttic xpessins It is f sme inteest t wk ut asmpttic expessins f equatins ( and ( We have f whee : R ± csθ, ± acsθ ± ± cs θ ; sinθ. ; Retaining nl the leading tem in pwes f (/ we find sinθ a P d cs( a sin( cs θ P 0 0. The integal is elementa; integating b pats ne gets a csθ d cs( a sin( csθ a a 0 cs( csθ cs sin θ [ ] Next, we find that

60 Vesin. 7/7/ Q θ [ cs( acs cs a] ; Q 0 and, theefe, sin( t ctθ ; [ cs( acsθ cs a]. We can cnvet these esults t spheical cdinates and θ. The cmpnents f -field in these cdinates ae elated t thse given in clindical cdinates b s that θ sinθ csθ csθ sinθ cs( a csθ cs a sin( t 0 sin θ θ, which agees with fmulas fund in the liteatue (see, e.g., J.. Jacksn, page Flux Functin f Linea Antenna Recall that ne f the Maxwell equatins in a cuent-fee egin is c B c dtb t Axial smmet f the antenna makes B a tidal vect; hence, accding t equatin(.3..4, the flux f the -field, F, is (7... F(,, t π c dtb (,, t (7... Futheme, ecalling that the B-field is deivable fm the vect ptential A and that f a linea antenna we have B ϕ B A A A φ A B B 0 (7...3 whee, in tems f educed (dimensinless vaiables defined in (7...6, μ A Ie 4π a it d sin Intducing Het s Supeptential Z defined b exp( ir ( a R R exp( ir (7...4

61 Vesin. 7/7/008 6 Z (,, t dt A (,, t i A (,, t (7...5 we thus have f the flux f the electic field f the antenna F c π Z (7...6 and f the cmpnents f the electic field F c (7...7 π F c (7...8 π Nte that F 0 which shws that the lines f the -field ae defined b the equatin F(,,t cnstant Singula Pints f -Field f Linea Antenna Using clindical cdinates and dimensinless vaiables (, with the antenna alng the -axis, we have f the -cmpnent f the electic field whee Q csτ Q sinτ (7..3. τ t ; (7..3. cs( cs( cs a Q ( sin( sin( Q ( and again ( ± ± a ( π antenna a length. In the equatial plane f the antenna, ( 0, λ

62 Vesin. 7/7/008 6 Q Q cs sin ( ( cs a ( whee a. ( Nte that ma be e-witten as ( sinδ csτ csδ sinτ Q ( τ δ Q Q Q sin ( Q whee δ actan. ( Q In the equatial plane, 0, f f which 0 since vanishes evewhee, s singula pints ccu at values 0 thee. The cnditin 0 0 ccus when the phase f is a multiple f π, that is, when τ δ nπ f an intege n. Thus, the lcatins f the f the singula pints satisf the equatin Q t actan nπ Q ( Tpical gaphs f t (f n0 as a functin f f values f a between 0 and π/ ae shwn in Figue 7.-. Nte that in geneal, t has ne maximum and ne minimum. Since t depends nl n the ati f Q t Q, the Q i s can be eplaced b Q UQ, whee U is an nn-e functin f. We chse U s that i i Q Q cs sin ( ( csa (7..3. The esnant antenna (aπ/ deseves special attentin. Since cs a 0, we have Q π t actan nπ ( ( nπ ( nπ Q actan ct (7..3. s that

63 Vesin. 7/7/ π π π t nπ nπ ( inveting this t get as a functin f t, t t π ( ( whee t t nπ. S, at t 0, the singula pints ae at 0, π, etc actan(q /Q.4. a aπ/4 aπ/ Figue 7.- t vs (a.00, π/4, π / Cmputing the phase velcit Cnside sme cnstant value f the phase ( τ δ d d ( τ δ 0. Then, ( ields the expessin f the phase velcit. In detail, we have Q t actan ( Q and its ttal deivative with espect t t: d Q actan ( d Q

64 Vesin. 7/7/ v phase Q Q Q Q ( F the special case f the esnant antenna (aπ/ we can est t the fmula t t π which ields ( Q Q π v phase. ( Nte that in geneal when W ( Q, Q Q Q Q Q Q Q the phase velcit v phase ( ( dt/d 0. This happens at a well-defined place and time t t c. In Figue 7.-, the gaph f t eaches its minimum value t c at c. c The sign f v phase can be used t define tw egins, an ute egin ( > c whee v phase is psitive and an inne egin ( < c whee v phase is negative (s that pints f cnstant phase mve twad the antenna. The tplg f the electic field nea the singula pints in the inne egin ( < c is diffeent fm the utside egin ( > c. B analing the slpe equatin f a singula pint, vi. d Q csτ Q sinτ 0 (7..3. d P csτ P sinτ 0 ne can shw with the help f L Hspital s ule that the singula pints in the inne egin ae X-pints and thse in the ute egin ae O-pints. Geneal iscussin f Singula Pints The field line equatin f a tw-vaiable case is d (, (7..3. d, Assume that thee ae pints S in the (, ( plane whee bth and vanish at the same time making equatin (7..3. an indeteminate fm. T ascetain the shape f the field lines nea these singula pints, expand and in pwe seies in d and d S,. The incements abut a given singula pint ( s s ae taken t be f the fist de. Thus, d d s s (7..3.3

65 Vesin. 7/7/ d d d d d d S S S S S ( s t fist de S S S d d d d d d,,,, ( whee, f bevit, S j i j i x, ( Slving ( f S d d, we get ( 0,,,, ( ±,,,,,,, ( Assuming that all the deivatives f ae well-behaved, then, if the disciminant,,,, Δ ( is psitive, the singula pint S is called an X-pint since the field lines appea t css each the in the limit f small d and d; thewise it is called an O-pint since the lines fm infinitesimal lps ( pssibl spials encicling the pint S. Singula Pints f the Linea Antenna The abve discussin is quite geneal f a tw-vaiable (tw- dimensinal tplg. Tuning t the specific case f a linea antenna, we immediatel ealie that thee ae an infinite seies f singula pints, bth alng the -axis (the pla axis f the antenna and alng the -axis (the equatial plane f the antenna.

66 Vesin. 7/7/ Alng the Pla ( Axis f the Antenna Inspectin f the expessin f shws that is an dd functin in bth and, s that we ma put (, t Ψ, ( whee Ψ is even in bth and and is well-behaved evewhee utside the antenna. Me explicitl, we have (, t Ψ (, csτ Ψ (, sinτ Ψ (7..3.3, Meve, inspectin f shws that it is an even functin f bth and Nw, alng the -axis, f which thee is a singula pint. Q, csτ Q (, sinτ ( ( 0 f 0. Thus, wheeve 0, that is, at all values f tan ( t ( 0, ( 0, Q ( Q xamine ( nea such a singula pint n the -axis. That is, let ε whee ε is assumed t be f secnd de s that ε << d. Since is even in,, ges t e as ges t e. Futheme, ( t, ε Ψ 0, s, ( s that,, in equatin ( is f secnd de cmpaed t,, η ( pvided η is finite and well-behaved as 0. This is s f the linea antenna. Thus, equatin ( educes t η ± η ε Ψ 0, (, t ( which implies that in the limit ε 0, 0. In the wds, the singula pints alng the -axis ae X-pints with tw sepaatix lines cssing at 90, ne line hintal and the the vetical. Alng the quatial Plane ( 0

67 Vesin. 7/7/ F finite values f, as 0, 0 s, wheneve 0, thee is a singula pint. This will happen wheneve tan ( t (,0 (,0 Q ( Q Nw, f 0, 0 and 0 (because is even in. Theefe, equatin,, ( educes t ±,, 0 ( It emains t wk ut in detail the tw deivatives at 0. Using equatin ( we have f cnstant t:, cs Q Q τ Q Q sinτ ( Qi with Qi. 0 Since and, b vitue f equatin ( , 0 csτ Q Q Q sin, Q Q Q Q Q τ, ( Q Q equatin ( becmes Q Q Q Q. ( Numeical calculatins shw that f small, is negative and becmes psitive f > c, whee c is detemined fm the cnditin 0. Meve, ne finds that is negative f all < c and bend. Thus, equatin ( implies that f all < c, the singula pints in the equatial plane f the antenna ae f the X-tpe and becme O-pints f > c. Figue 7.- shws these tw classes f singula pints f the esnant antenna case.,,,

68 Vesin. 7/7/ Figue 7.-: Singula pints f the esnant antenna.