Classical Electrodynamics

Transcription

1 A Fst Look at Quantum Physcs Cassca Eectodynamcs Chapte 4 Mutpoes, Eectostatcs of Macoscopc Meda, Deectcs Cassca Eectodynamcs Pof. Y. F. Chen

2 Contents A Fst Look at Quantum Physcs 4. Mutpoe Expanson 4. Mutpoe Expanson of the Enegy of a Chage Dstbuton n an Extena Fed 4. Eementay Teatment of Eectostatcs wth Pondeabe Meda 4.4 Bounday-Vaue Pobems wth Deectcs 4.5 Moecua Poazabty and Eectc Susceptbty 4.6 Modes fo Eectc Poazabty 4.7 Eectostatc Enegy n Deectc Meda Cassca Eectodynamcs Pof. Y. F. Chen

3 A Fst Look at Quantum Physcs 4. Mutpoe Expanson Locazed chage dstbuton Let x, ' x' x ' ( x ') ( x x ') x It s ust the expanson esut fo / x x ' unde spheca coodnates. The potenta gven by a ocazed chage dstbuton ( x ') s ( x ') dx' ( x) 4 x x ' O Consde >, we know that ' ' ' 'cos ' ' x x / [ ( ) ( )cos ] * Accodng to addton theoy fo spheca hamoncs, 4 P(cos ) Ym( ', ') Ym(, ) m 4 ' * Thus, Y ( ', ') (, ) m Ym x x' m Smay, we can use Tayo expanson to expand unde Catesan coodnates. Tayo expanson fo snge vaabe functon: / x x ' df d f ( ) ( ) xa( ) ( )... xa f x f a dx x a! dx x a (expand aound x=a) P (cos ) Cassca Eectodynamcs Pof. Y. F. Chen

4 f f Consde vecto fed : f( xa) f( a) x xx... x x x!, xx If we expand / xx' at x' x ' ' ' x x x x!, ' ' x x x ( ) ' ( ) ' '... xx' x' x x xx' xx' ( ) ( ) x ' ' ' x x x ( xx') ( y y') ( zz') ( ') ( ') ( ') xx aˆ y y aˆ zz [( xx') ( y y') ( zz') ] [( xx') ( y y') ( zz') ] [( xx') ( y y') ( zz') ] / / / ( xx') [( xx') ( y y') ( zz') ] ( xx')( y y') ( xx')( zz') ( ) 5 5 5, x' x ' xx' xx' xx' xx' ( y y')( xx') ( y y') xx' ( y y')( zz') xx' xx' xx' ( zz')( xx') ( zz')( y y') ( zz') xx' xx' xx' xx', x' x' 5 x' xx', x' x ' xx', Fom above dscusson, we know that ( ) ; ( ) Theefoe, x ' ' '... (The esut unde Catesan coodnates) x x', 5 x x!, Cassca Eectodynamcs Pof. Y. F. Chen

5 Fo a chage dstbuton ( x '), the scaa potenta obseved at x can be egaded as the contbutons fom dffeent pats of mutpoe expanson. ( x') d x', ( x) ( x'){ x ' ' '...} ' x 5 x d x 4 x x' 4 q, { ˆ ( ') ' ' ( ') ' ' '...} x x d x x x 5 x d x 4, Defne: q ( x') d x' (monopoe) ; p ( x') x ' d x' (dpoe) ; Q x ' x ' ( x') d x' (quadupoe), q Theefoe, ( x) { p Q,.....} 4 6 (, ) ˆ 5, Futhemoe, fom the symmety of Q, we know that Q Q Q Q Q Q Q Q Q Ony 6 ndependent components Q Q Q It s a educbe tenso unde Catesan bass epesentaton. We can use pope tansfomaton and constant to eque Q and defne Q Q an educby taceess quadupoe tenso : Q ( x ' x ' x', ) ( x') d x',whch satsfy Q Q Q [( x ' x ' x ' ) x' ] ( x') d x' If we expand the scaa potenta unde spheca coodnates: ( x') d x' 4 ' * ( x) ( x')( Y ( ', ') (, )) ' m Ym d x 4 x x' 4 m Cassca Eectodynamcs Pof. Y. F. Chen

6 4 4 Y (, ) ( x) ( ) ( ' ( x') Y ( ', ') d x') Y (, ) ( ) q 4 4 * Defne: qm ' ( x') Ym( ', ') d x' * m m m m m m d d ( m)! P x x P x x x m m dx! dx We know that Ym(, ) P (cos ) e,whee 4 ( m)! m m ( m)! m P ( x) ( ) P ( x) ( m)! P ( x) Y(cos ) / 4 P ( x) x Y(cos ) / 4 cos / P ( x) ( x ) Y(cos ) / 8 sn (cos sn ) P ( x) x Y(cos ) 5 / 4 ( cos ) P ( x) x( x ) Y (cos ) 5 / 4 cos sn (cos sn ) / P ( x) ( x ) Y(cos ) 5 / 96 sn (cos sn ) m m m m/ m/ ( ) ( ) ( ) ( ) ( ) m m q ( x') /4 d x' q /4 q ( x') ' /4 cos ' d x' /4 ( x') z' d x' /4 pz q ( x') ' /8 sn '(cos ' sn ') d x' /8 ( x')( x' y')' d x' /8 ( px py ) q ( x') ' 5/4 ( cos ' ) d x' 5/6 ( x')( z' ' ) d x' 5/6Q q 5 / 4 ( x') ' cos 'sn '(cos ' sn ') d x' 5/ 4 ( x') z'( x' y ') dx' 5/4 ( Q Q) q 5/ 96 ( x') ' sn '(cos ' sn ' sn 'cos ') d x' 5/ 96 ( Q Q Q ) Cassca Eectodynamcs Pof. Y. F. Chen

7 The dscusson above hep us to fnd the eaton between mutpoe moment unde spheca coodnates q m and mutpoe moment unde Catesan coodnates. Consde eectc feds expessed by mutpoe expanson wth gven,m : Ym(, ) E qm () Unde spheca coodnates : E qm Y (, ) m () m E qm Y sn () sn If thee s a dpoe p at x, then the eectc fed Ex ( ) obseved at x ( nˆ pnˆ) p Ex ( ), whee nˆ s the unt vecto fo ( xx ). 4 x x (, ) The esut can aso be deved fom the scaa potenta contbuted by the dpoe moment. ˆ dpoe( x) p ; p ' ( ') ' x x d x 4 If we shft the coodnates and et ( x x) x x dpoe( x) p 4 x x p ( x x )( x x ) p Edpoe( x) dpoe( x) dpoe( x) { } x 4 xx xx, 5, p xx p xx x x p xx x x p x x xx xx xx ( ) ( )( ) ( )( ) { 4 p( x x) p( x x) p( x x)( x x) p x x xx xx xx p( x x)( x x) p( x x)( x x) p( x x) p } xx xx xx xx Cassca Eectodynamcs Pof. Y. F. Chen m

8 [ p( xx ˆ ˆ )]( xx) p n( pn) p Edpoe( x) dpoe( x) { } 5 4 x x xx 4 xx ( p ) p If x, Edpoe( x) { } 5 4 p cos p p cos aˆ p And f p pzˆ, x Edpoe( x) { } When the obsevng pont s vey cose to the eectc dpoe, then the esut must be modfed. Assume an effectve chage dstbuton to mode the eectc dpoe s n the foowng fom: a Cassca Eectodynamcs Pof. Y. F. Chen ( ) cos Cacuate the dpoe moment contbute by ths chage dstbuton : dp zdqzˆ & zacos ; dqa sn cosd / / 4 p acos ˆ ˆ ˆ cosa sn d z 4a cos snd z a z a / cossnd d a cossnd And de cos ˆ ˆ cos ˆ ˆ z z E z z 4 a 4 4 a Thus, the eectc fed contbute fom the dpoe moment at the ogn (cente of dpoe) s p 4 E ( )/( a )

9 Theefoe, the tota contbuton fom the eectc dpoe moment s nˆ( pnˆ) p 4 E dpoe( x) [ p ( x x )] 4 x x 4. Mutpoe Expanson of the Enegy of a Chage Dstbuton n an Extena Fed The nteacton enegy between a ocazed chage dstbuton and an extena scaa ( potenta can be expessed as x) ( x ') W ( x) ( x) d x ( ) If the scaa potenta s sowy vayng nea the chage dstbuton ( x), then the tota nteacton enegy can be expessed by mutpoe expanson: Expand the extena scaa potenta aound a pope ogn ( x) () x xx... x x x, xx E Use E E ( x) () x E() ()... x, x E E ( x) () xe() ( xx, ) ()... Fo extena potenta (fed) 6, x x xx Substtute back nto ( ) E W ( x) () dx ( xxdxe ) () ( x)( xx ) dx ()... x, 6, dpoe wth fed E q () p E() Q, ()... 6, x monopoe wth potenta quadupoe wth the fed gadent Cassca Eectodynamcs Pof. Y. F. Chen

10 4. Eementay Teatment of Eectostatcs wth Pondeabe Meda E ext E E E tot ( n matte ) ext nd End If the chage dstbuton n the matte won t nduce hghe mutpoe moment densty othe than dpoe moment unde the extena fed, then the potenta n a sma egon v Matte n the matte woud be: ( x) P( x') ( xx') P ( xx, ') [ v' v'], P Np (dpoe moment pe unt voume) 4 x x' x x' v ( x) P( x') ( xx') ( xx, ') [ dx' dx' ] 4 xx' xx' contbuton fom fee chage contbuton fom the bound chage n matte ( xx') ( x) x dx Px dx We know that ( ) '( ), then ( ) [ ' ( ') '( ) '] xx' x x' xx' 4 xx' xx' Use ( f A) ( f) A f( A) ( f) A ( f A) f( A) Px ( ') '( Px ( ') ) dx' ( ' Px ( ')) dx' nda ˆ xx' xx' ( ' Px ( ')) dx' xx' x x' ( x) ' P( x') ( x) ' P( x') Theefoe, ( x) [ dx' dx'] [ dx' 4 xx' xx' 4 xx' We can egad the tota chage n the matte at ths tme to be tot f b, whee b P Cassca Eectodynamcs Pof. Y. F. Chen

11 Accodng to Gauss's aw: E [ P] tot f Then, ( EP) f D f, whee DEP s the eectc dspacement. Fo sotopc and nea deectc, P e E Thus, D EP ( ) E E E, whee ( ). e e The bounday condton fo deectc: D f D Accodng to Maxwe's equaton: A B d E t m D nds ( D D) na Qf ( D D) n f d d E m ( E ) nds m E d m B nds d d C d t E ˆ t E t ( E E) n d d ( DD) n f xs xs, f f n n ( E ˆ E) n xs xs <Exampe > (,, d) q ' mage chage z q d q souce chage (,, d) (Azmutha symmety) Q. Fo a pont chage q n deectc, evauate the scaa potenta unde ths case. We can use mage chage method to sove ths pobem. Assume the nduced chage q'& q at the mage poston and enfoce the bounday condton to get the vaues. () Regon : z, the effectve chage seen n ths egon ae q, q' q q' (,, z) [ ] 4 ( zd) ( zd) Cassca Eectodynamcs Pof. Y. F. Chen

12 () Regon : z, the effectve chage seen n ths egon ae q, q q" Let q'' qq (,, z) 4 ( zd) Consde the bounday condton : () xs xs( z ) ( z ) ( q q') q" ( qq') q" (a) 4 d 4 d () xs xs z z n n z z ( zd) q ( zd) q' q"( zd) [ ] / / z / z( qq') q" (b) 4 [( zd) ] [( zd) ] 4 [( zd) ] q' ( ) q" ( q q') q" Sove (a) & (b) q" ( ) q, q' ( ) q" ( ) q ( q- q') q" q ( ) q" q ( )/( ) [ ], z 4 ( zd) ( zd) Theefoe, (,,z) q [ ( )], z< 4 ( z d).. Cassca Eectodynamcs Pof. Y. F. Chen

13 We can use P b to cacuate the nduced suface chage ( PP ˆ ˆ ) naqb b ( P P) n A Fom D E EP P ( ) E ; P ( ) E d And on the suface ˆ E E b ( E- E) n q ( )/( ) E (,, z) [ ]ˆ a / / 4 [( zd) ] [( zd) ] (Fo z ) q z d ( z d)( )/( ) Ez (,, z ) [ / / ] a ˆ z z 4 [( zd) ] [( zd) ] q E (,, z) [ ( )]ˆ a / 4 [( zd) ] (Fo z ) q z d Ez (,, z) [ ( )]ˆ a / z z 4 [( zd) ] qd qd qd Ez() Ez() ( ) ( ) ( ) / / / 4 ( d ) 4 ( d ) ( d ) q d b ( ) ( d ) /.. Cassca Eectodynamcs Pof. Y. F. Chen

14 4.4 Bounday-Vaue Pobems wth Deectcs Consde a sphee wth deectc constant ε ocates n an unfom deectc matte ε and be affected by an extena eectc fed E E a E Ea ˆz a ˆz P z Unde ths condton, the scaa potenta at ( ) E cos (bound chage poazaton due to sphee) Snce the case satsfes azmutha symmety, ts genea souton n spheca coodnates s (, ) [ A B ] P(cos ) n (Fo a), (, ) A P(cos ) (eque souton convege) ( ) (Fo a), out (, ) [ B C ] P (cos ) When, the contbuton fom sphee can be negected: ( ) E cos ( ) out ( ) m [ B C ] P (cos ) Ecos Accodng to the othogonaty of Legende poynoma, as ony contbutes. ( ) out (, ) B C P (cos ) & B E n(, ) A P (cos ) Cassca Eectodynamcs Pof. Y. F. Chen

15 Consde the bounday condton: ˆ s s ( D D) n, f = ( E ˆ E) n s n n ( ) n a out a s, A cos a[ EC ]cos a A E Ca () ( ) (), A P(cos ) a C P(cos ) a A Ca () n out ( a a), Acos [ E ( ) C cos a] A [ E Ca ] () ( ) (), A P(cos ) a ( ) C P(cos ) a A ( ) Ca (v) Fom () A Ca A Ca &(v) A ( ) Ca Ca [ ( ) ] ( ) ( ) ( ) ( ) It must be vad fo any C, A (fo ) A E Ca Fom (),() A E Ca A [ E Ca ] C Ea ( ) E( ) Ca ( ) A E E( ) E (, ) E cos ( ) E cos n(, ) ( )cos E Theefoe, we have out (, ) E cos ( ) E cos n a a out (, ) Ecos ( ) E cos due to the spped fed due to the bound chag e poazaton Cassca Eectodynamcs Pof. Y. F. Chen

16 q p( xx') xx Compae wth mutpoe expanson, ( x) [ Q...] 5 4 xx' xx', xx' a We can see ( ) E cos as the contbuton fom eectc dpoe moment p cos a Thus, ( ) E ˆ cos p4 a ( ) Eaz 4 P ( ) E And, ( ˆ po P P) n, & E E on the suface P ( ) E ˆ ˆ po ( E E EE) n ( EE) n We can aso cacuate the eectc fed: n n E ˆ [ ˆ a Ecos Ecos ( )] ( ) Ecosa E aˆ [ E sn E ( )sn ] ( ) E snaˆ n n n out Substtute E and E nto, a a po po [( ) E cos ( ) E cos ] ( ) E cos cos a out out a E a E E out out a E a E E ˆ [ cos ( ) cos ] ˆ [ sn ( ) sn ] Reca the ntena fed fo a sphee wth suface chage dstbuton cos E aˆ z p Then n ths case ˆ ˆ ( ) E and P ( ) Eaz az (4/) a Cassca Eectodynamcs Pof. Y. F. Chen

17 . Cassca Eectodynamcs Pof. Y. F. Chen

18 4.5 Moecua Poazabty and Eectc Susceptbty We know that n nea deectc, E Ea ˆz dense meda a P P E n aout a (a) Accodng to bounday condton : D D D P e n out n(, ) A cos Fom the exampe n secton 4.4 : out (, ) Ecos C cos P P a Theefoe, (, ) ( E ) cos & (, ) E cos ( ) cos n z (b) out If the medum s composed by hgh densty moecue, then the tota contbuton to the eectc fed n ths egon s E. nea E p E (ntena fed) We can cacuate the ntena eectc fed of the spheca egon n an apped fed E Ea and ˆz suounded by poazaton P // E : (B.C. (a)) A E Ca A E Ca (B.C. (b)) A E Ca PA E P P P A ( E ) & C ( AE) a a n P P P.S. In amophous & smpe cubc matte, Enea E ( ˆ ˆ n E )cos a ( E ) az E nea P e Ep Substtute En nto P ee P e( E ) P( ) ee P Consde the tota poazaton, P N pmo And pmo mo ( EE ) mo ( E ) (aveaged poazaton pe moecue), whee s mcoscopc moecue poazabty mo Cassca Eectodynamcs Pof. Y. F. Chen

19 P P N mo Thus, P Nmo ( E ) e( E ) P( Nmo ) NmoE P ( ) E N N ( / ) mo e We have, ( ), ( ) [ ] (we have used ) e N mo N ( / ) e e N mo mo 4.6 Mode fo the Moecua Poazabty Effectve spng mode: ee ee The equaton of moton: F kxm xee x P mo ex m k e m e And Pmo mo E mo m m If the moecue s composed by on wth dffeent mass and vaence, then thee ae assume atom fx dffeent spng constant ω. Cacuate the thema aveage fo P mo (The weghtng can be decded fom the Botzmann facto) The nteacton enegy between eectc dpoe and the extena eectc fed s W pe pecos pe cos ( pcos )exp( ) d kt B -snd dy Pmo Let pe cos exp( ) d cos y kt B pey pey pe pe pe pe pey kt B kbt kt B kbt kt B kbt kbt kt B kbt kbt py exp( ) dy p[( ) e y ( ) e ] p[( )( e e ) ( ) ( e e )] kbt pe pe pe pe pe pe pey kt B kbt kbt exp( ) dy ( )( e e ) kt pe B Cassca Eectodynamcs Pof. Y. F. Chen

20 pe pe pe pe kt B kbt kbt kt B kbt kbt [( )( ) ( ) ( )] p e e e e pe pe kt B kt B Theefoe, Pmo P{coth( ) } P cos pe pe kt pe pe B kbt kbt ( )( e e ) pe It s caed Langevn-Debye fomua. x x e e ( xx /) ( xx /) x x x x If x s vey sma ( - ) ( )( x ) x x e e x ( xx / x /!) ( xx / x /!) x x 6 x 6 pe Thus, when s vey sma, kt B P mo P E kt B Usuay, the poazaton n moecue may be ognated fom () pemanent poazaton () nduced poazaton negect x mo pemanent P kt B nduced Cassca Eectodynamcs Pof. Y. F. Chen