CHAPTER : Boudr-Vlue Problems i Electrosttics: I Applictios of Gree s theorem
.6 Gree Fuctio for the Sphere; Geerl Solutio for the Potetil The geerl electrosttic problem (upper figure): ( ) ( ) with b.c. s s hs the forml solutio: (see Sec..) 3 ( ) ( ) G (, ) d v 4 D S ( ) ( 4 s G D, ) d, (.44) where the Gree fuctio G (, ) is the solutio of (lower figure) D s ( ) ( ) G (, ) 4 ( ) with b.c. G (, ) for o S D G D (, ) c be regrded s the potetil due to uit poit source ( q 4, p. 64) t rbitrr positio iside the sme surfce S, but with the homogeeous b.c. GD (, ) = for o S. D uit poit source GD (, ) GD (, ) S (sme S s i upper figure)
.6 Gree Fuctio for the Sphere (cotiued) Empl e : Use (.44) to fid due to poit chrge q t b i ifiite spce. q ( ) ( b ) with b.c. ( ) I order to use (.44), we first obti the Gree fuctio from GD(, ) 4 ( ) with GD(, ) for o S () The solutio of () is GD (, ) Sub. q ( b ) for ( ) d / for G D (, ) i to (.44) q ( b) 3 G (, ) ( ) ( ) (, ) ( ) D GD d d 4 v 4 s q 4 b 3
.6 Gree Fuctio for the Sphere (cotiued) Emple : ( ) with b.c. ( r ) (,, ) Fid ( ) i the regio r (see left figure). First, fid G (, ) from D the equtio (see right figure) G (, ) 4 ( ) D ) with G (, ) o S. G D D S S (,, ) Note : poits outwrd from the volume of iterest. GD (, ) This equtio hs the solutio (see Sec.. ): (, ) G D (, ) 4 ( ) i regio of iterest ( r ) cos cos Note: G (, ) G (, ) gle betwee d D D (3) 4
.6 Gree Fuctio for the Sphere (cotiued) GD (, ) GD(, ) ( ) ' ( cos ) 3 Sub. (3) ito (.44) 3 G (, ) D ( ) ( ) GD (, ) d ( ) d 4 v 4 s ( ) (,, ) (.9) 4 s d 3 ( cos ) Questios:. I (3), we hve G D (, ) s solutio of / G D GD (, ) 4 ( ). But (, ) ppretl lso stisfies the sme equtio. Does this violte the uiqueess thm.?. C the solu tio of GD (, ) 4 ( ) be writte i the form G (, ) G ( )? wh? 5 D D
. Method of Imges The method of imges is ot geerl method. It works for some problems with simple geometr. Cosider poit chrge q locted i frot of ifiite d grouded ple coductor (see figure). The regio of iterest is d is govered b the Poisso equtio: imge q s s ( ) chrge ( s s ) q q q subject to the boudr coditio regio of regio of ( ). iterest iterest I order to miti zero potetil o the coductor, surfce chrge will illbe iduced d (b q) o the coductor. We m simulte the effects of the surfce chrge with hpotheticl "imge chrge", q, locted smmetricll behid the coductor. T he, 6
. Method of Imges (cotiued) q ( ) 4 imge s s s s chrge d, b smmetr, ( ) stisfies q q q the boudr coditio ( ). regio of regio of Operte ( ) with iterest iterest q ( ) [ ( ) ( )] () I the regio of iterest ( ), we hve ( ). Thus, ( ) obes the origil Poisso equtio q ( ) ( ) This shows tht we must put the imge chrge outside the regio of iterest Sice ( ) stisfies both the Poisso equtio d the boudr coditio i the regio of iterest, it is solutio. B the uiqueess theorem, it is the ol solutio. Note tht the Poisso equtio () d the solutio ( ) re irrelevt outside the regio of iterest. 7
. Poit Chrge i the Presece of Grouded Coductig Sphere Refer to the coductig sphere of rdius show i the figure. Assume poit chrge q B is t r ( ). To fid for r, we put q q imge chrge q t r ( ). The, q 4 q 4 ( ) First, set, or '=, q 4 q 4 ' so tht = Boud coditio requires Note: ' ; hece, q' lies outside q 4 q 4 ( ) the regio of iterest. q q' Net, set so tht RHS. q 4 q 4 ' ( ) ' This gives ' q q q. 8
. Poit Chrge i the Presece of Grouded Coductig Sphere (cotiued) q 4 q 4 This is equivlet to (.) Rewrite ( ) d (.4) of Jckso. q I the regio of iterest ( r ), we hve ( ) ( ). Thus, s i the cse of the ple coductor, stisfies the Poisso equtio d the b.c. It is hece the ol solutio. The E-field lies re show i the figure below. q B q 9
. Poit Chrge i the Presece of Grouded Coductig Sphere (cotiued) Surfce chrge desit o the sphere : The solutio for ( ) c be epressed i terms of sclrs s q ( ) 4 / 4 ( cos ) / ( cos ) where is the gle betwee d. B Guss's lw, the surfce chrge B q q desit t poit B is Er ( ) ( q cos cos ) [ 8 3/ 4 3 ] ( cos ) 3/ ( cos ) q ( ) (.5) 4 3/ ( cos ) ( ) [ ]
. Poit Chrge i the Presece of Grouded Coductig Sphere (cotiued) Totl chrge o the sphere: : The totl surfce chrge c be obtied b itegrtig over the sphericl surfce. B However, it c be deduced from simple q q rgumet: I the regio r, the electric field due to the surfce chrge is ectl the sme s tht due to the imge chrge q. Hece, b Guss's lw, the totl surfce chrge must be q ( q). Force o q: Sice, t the positio of chrge q, the field produced b the imge chrge q is the sme s tht produced b the surfce chrge, the force o q is the Coulomb force betwe q d q. q( q) 3 qq q F = 4 ( ) 4 ( (.6) 4 ) ( )
.3 Poit Chrge i the Presece of Chrged, Isulted, Coductig Sphere (with Totl Chrge Q) If the sphere is isulted with totl chrge Q o its surfce, we m obti i two steps. Step : Groud the sphere sme problem s i Sec.. q 4 q ( ) 4 / with totl surfce chrge q q. Q Step : Discoect the groud wire. Add Q q/ to the sphere so tht the totl chrge o the sphere is Q. The, Q q will be distributed uiforml o the surfce becuse the chrges were lred i sttic equilibrium. Q q due to Qq is ( ) 4 q s
.3 Poit Chrge i the Presece of Chrged, Isulted, Coductig Sphere (cotiued) Hece, the totl is q q Qq ( ) [ ] (.8) 4 / qq ( q/ ) force due to The force o q is the force i (.6) plus 4 3 dded chrge q 3 q ( ) Q F Q q (.9) 4 4 ( ) qq As, F (Coulomb force betwee poit chrges) 4 As, F is lws ttrctive eve if q d Q hve the sme sig. Questio: If there is ecess of electros o the surfce, wh do't the leve the surfce due to mutul repulsio? (See p. 6 for discussio o the work fuctio of metl.) 3
.7 Coductig Spheres with Hemisphere (to be discussed i Sec. 3.3).8 Orthogol Fuctios d Epsios Defiitio of Orthogol Fuctios : Cosider set of rel or comple fuctios U ( ) (,, ) which re squre itegrble o the itervl b. ier product b orthogol, if U, m ( ) U m( ) d U ( )'s re, m b, m orthoorml, if U ( ) U m( ) d m, m Geometricl logue: e, e,d e z re orthoorml set of uit vectors, i.e. e m e = m. B compriso, the dot product e m e is similr to the ier product. But the lgebric set U () c be ifiite i umber. 4
.8 Orthogol Fuctios d Epsios (cotiued) Lierl Idepedet Fuctios : The set of ( )'s re sid to be lierl idepedet if the U ol solutio of U ( ) (for ever i the rge of b) is for. If set of ffuctios re orthogol, the re lso lierll idepedet. Proof : U ( ), uless m b b U ( ) U ( ) d U ( ) U ( ) d m m b ( ) U d for 5
.8 Orthogol Fuctios d Epsios (cotiued) Grm-Schmidt Orthogoliztio Procedure: Orthogolit is sufficiet, but ot ecessr, coditio for lier idepedece, i.e. lierl idepedet fuctios do ot hve to be orthogol. However, the c be recostructed ito orthogol set b the Grm-Schmidt orthogoliztio procedure. Cosider two vectors, e d ( e e), s simple emple. These two vectors re ot orthogol, becuse e( e e), but re lierl idepedet becuse e b( e e ) b. We m form two ew vectors s lier combitios of the old vectors, e e d e e + e + e, d demd e e. e e + e e The ew set, e( e) d e( e), re thus orthogol (s well s lierl idepedet). The sme procedure c be pplied to lgebric fuctios. 6
.8 Orthogol Fuctios d Epsios (cotiued) Completeess of Set of Fuctios : Epd rbitrr, squre itegrble fuctio f ( ) i terms of fiite umber ( N) of fuctios i the orthoorml set U ( ), N f( ) U ( ) (.3) d ddefie the me squre error ( M ) s b N M N f( ) U ( ) d. If there eists fiite umber N such tht for N > N the me squre error M N c be mde smller th rbitrril il smll positive qutit b proper choice of s, the the set U ( )is sid to be complete d the series represettio N U ( ) f ( ) (.33) is sid to coverge i the me to f ( ). 7
.8 Orthogol Fuctios d Epsios (cotiued) Rewrite (.33): f ( ) U ( ) (.33) Usig the orthoorml propert of U ( )'s, we get b U * ( ) f ( ) d (.3) Chge i (.3) to d substitute t (.3) ito (.33) b * f ( ) U U d ( ) ( ) ( ) (.34) f * f ( ) is rbitrr U ( ) U ( ) ( ) (.35) (completeess or closure reltio) Jckso, p. 68: "All orthoorml sets of fuctios ormll occurrig i mthemticl phsics hve bee proved to be complete." "(This sttemet will be illustrted di Sec..9.) 9) 8
.8 Orthogol Fuctios d Epsios (cotiued) Fourier Series : emple of complete set of orthogol fuctios Epoetil represettio of f ( ) o the itervl : ik f( ) f( ) e (4) ik k ; ( ) f e d I (4), f ( ) is i geerl comple fuctio d, eve whe f ( ) is rel, is i geerl comple costt. I the cse f( ) is rel, we hve the relt coditio: * * Proof : f ( ) rel f ( ) f ( ) ik * ik * ik e e e ik e * (sice is lierl idepedet) Questios:. Wh " to " isted of " to "?. Wh k isted of k? 9
.8 Orthogol Fuctios d Epsios (cotiued) Siusoidl represettio of f ( ) o the itervl : ik ik ik f( ) e e e cos cos si si k ki k k cos si k i k A f( ) A cos k B si k, k (5) where Sme s (.36) d (.37) ik ( ) ik A f e e d f( )coskd ( ) cosk i ik ik B i e e d f ( ) ( )si k d ( ) f si i k
.8 Orthogol Fuctios d Epsios (cotiued) Discussio: i : It is ofte more coveiet to epress epress phsicl qutit ( rel umber) i the epoetil represettio th i the siusoidl id represettio, tti becuse the comple coefficiet i ( ) of epoetil term crries twice the iformtio of the rel coefficiet i (A or B ) of siusoidl id lterm. For emple, if (t)=re[e iωt ] is the displcemet of simple hrmoic oscilltor, the comple cotis both the mgitude d phse of the displcemet. I the siusoidl represettio, the sme qutit will be writte (t)=acos(ωt)+bsi(ωt). Epoetil terms re lso esier to mipulte (such s multiplictio d differetitio). This poit will be further discussed i Ch. 7.
.8 Orthogol Fuctios d Epsios (cotiued) Fourier Trsform : If the itervl becomes ifiite ( ), we obti the Fourier trsform (see Jckso p.68). ik f( ) ( ) (.44 A k e dk ) ik Ak ( ) f( e ) d (.45) Chge to i (.45) d sub. (.45) ito (.44) ( ) ik f( ) df( ) e dk ( ) ik( ) e dk ( ) [completeess reltio] (.47) Questio : Does Ak ( ) coti more or less iformtio th f ( )?
.8 Orthogol Fuctios d Epsios (cotiued) ik Questio : Does i f ( ) e hve the sme dimesio s Ak ( ) i f( ) Ake ( ) dk? [ssumig is dimesiol qutit.] ik ( ) Rewri te (.47): ( ) e dk Iterchge d k ik ( k) e d ( kk), [orthogolit coditio] (.46) Let k k d sub. it ito (.46) i ( ) e d Sice ( ) ( ), we m write more geerll, i e d ( ) (6) ik 3
.8 Orthogol Fuctios d Epsios (cotiued) There re two useful theorems cocerig the Fourier itegrl: () Prsevl's theorem : The Prsevl's Prsevls theorem sttes f ( ) d A ( k ) dk (7) Proof : ik f ( ) A ( k ) e dk (.44) (44) Rewrite the Fourier trsform: ik Ak ( ) ( ) (.45) f e d * f ( ) d f ( ) f ( d ) ik * ( ) ( ik d ) A k e dk A k e dk * ( ) dk ( ) ( ik k A k dka k ) de ( ) A k dk ( ) kk 4
.8 Orthogol Fuctios d Epsios (cotiued) () Covolutio theorem : The covolutio theorem sttes ik [ ( ) ( ) ] ( ) ( ) (8) f f d e d A k A k This is clled the covolutio of f ( ) d f ( ) where the fctor follows the covetio i (.44) d (.45). ik Proof : LHS of (8) f( ) d ( ) f e d Let ( dd ) ( ) f d f ik( ) ( ) e d ( ) ( ) ik ik f d f e d A( k) A( k) 5
.9 Seprtio of Vribles, Lplce Equtio i Rectgulr Coordites Lplce equtio i (.48) z Crtesi coordites Let ( z,, )= XYZz ( ) ( ) ( ) (.49) d X d Y d Z X Y d d Z dz (.5) Sice this equtio holds for rbitrr vlues of,, d z, ech of the three terms must be seprtel costt. d X d Y d Z X ; Y; Z subject to d d dz ei ei e z X( ) ; Y( ) ; Z( z) with i z e e i e 6
.9 Seprtio of Vribles, Lplce Equtio i Rectgulr Coordites (cotiued) Emple : Fid iside chrge-free rectgulr bo (see figure) with the b.c. (,, z c) = V(, ) d o other sides. i i X ( ) Ae Be i i X() BA X A( e e ) A'si X ( ),,, z X( ) A si Similrl, Y ( ) Ae Be. Y () d Y( b) give Y( ) A si, m m m i m i m b z z Solutio for Z: Z( z) Ae Be z z Z() BAZ( z) A( e e ) A''sih z m, A si si sih z, (.56) m m m m m 7
.9 Seprtio of Vribles, Lplce Equtio i Rectgulr Coordites (cotiued) To fid A, we ppl the b.c. o the z c ple: m ( z,, c) V (, ) m, V(, ) A si si sih c (.57) 4 bsih Am c m m m m b d dv (, )si si m (.58) Questios:. The method of imges is ot geerl method, but the method of epsio iorthogol fuctios is. Wh?. I electrosttics, ol chrges c produce Φ. I this problem, =, how c there be Φ? 3. C we fid the surfce chrge distributio ( ) o the wlls from the kowledge of Φ isidetheid bo??if so, uder wht coditio? i 8
.9 Seprtio of Vribles, Lplce Equtio i Rectgulr Coordites (cotiued) Discussio: : m, m b Rewrite (.57): V(, ) A si si sih c, m m m z where d m. This is good emple to substtite the followig sttemet o phsics groud: "All orthoorml sets of fuctios ormll occurrig i mthemticl phsics hve bee proved to be complete." (p. 68) I (.57), si( ) d si ( m) re orthogol fuctios geerted i phsics problem. Phsicll, we epect the problem to hve solutio for boudr coditioothesurfcez the z c, i.e. for fuctio V(, ) specified i (.57). Thus, si( ) d d si ( m ) must ech form complete set i order to represet rbitrr V(, ). 9
Homework of Chp. Problems:,, 3, 4, 5, 9, 3, 6 3