//8 H 7 Ectodynics 9-9: AM MW Oin n fo Lctu 8: Stt ding Chpt Mutipo ont pnsion of ctosttic potnti A. Sphic coodints B. Ctsin coodints //8 H 7 Sping 8 -- Lctu 8 //8 H 7 Sping 8 -- Lctu 8 oisson nd Lpc ution in sphic po coodints sin cos y sin sin z cos http://www.uic.du/csss/cs/cs/ttbook/nod.ht //8 H 7 Sping 8 -- Lctu 8
//8 oisson nd Lpc ution in sphic po coodints -- continud Lpc ution fo ctosttic potnti : sin R ( ) Sphic honic functions : sin sin sin sin sin //8 H 7 Sping 8 -- Lctu 8 optis of sphic honic functions (stndd Condon-Shoty convntion) sin dω θφ θφ θ dθ dφ θφ θφ δ δ Coptnss: θφ θ φ ˆ ˆ cos cos Rtionship to Lgnd poynois: θφ cos Rtionship to Associtd Lgnd poynois:! θφ cos! i //8 H 7 Sping 8 -- Lctu 8 Lgnd nd Associtd Lgnd functions Lgnd diffnti ution : d d o d d ( ) d d Associtd Lgnd diffnti ution : ( ) d ( ) d!! / d ( ) d ( ) //8 H 7 Sping 8 -- Lctu 8 ( )
//8 Usfu idntity: ctosttic potnti vnishing fo d d Ep fo isotd chg dnsity θφ θ φ : with θφ θ φ //8 H 7 Sping 8 -- Lctu 8 7 So sphic honic functions: ˆ ˆ ˆ ˆ ˆ ˆ sin 8 cos i sin i sin cos 8 cos i //8 H 7 Sping 8 -- Lctu 8 8 Ep: Gnfo of ctosttic potnti with boundy vu fo isotd chg dnsity Suppos tht d d θφ θ φ θφ θφ d d : //8 H 7 Sping 8 -- Lctu 8 9
//8 Ep: sin cos 8 θφ θφ Suppos V V Φ θφ d d o 8 Φ θφ θφ d d V o 8 V Φ θφ θφ d //8 H 7 Sping 8 -- Lctu 8 Ep -- continud: 8 θφ θφ Suppos V V o 8 Φ θφ θφ d d V sin cos V o Φ 8 V θφ θφ d sin cos V //8 H 7 Sping 8 -- Lctu 8 Ep -- continud: o sin cos Φ V o Φ V V sin cos //8 H 7 Sping 8 -- Lctu 8
//8 Notion of utipo ont: In th sphic honic psnttion - - dfin th ont d dfin th dfin th Q d d ρ( ) d ρ( ) ρ( ) In th Ctsin psnttion - - dfin th onopo ont p i : ρ( ) i dipo ont udupo ont coponnts of th (confind) chg distibuti on p : i Q (i yz) : i ρ( ) : //8 H 7 Sping 8 -- Lctu 8 Significnc of utipo onts Rc gn fo of ctosttic potnti with boundy vu fo isotd chg dnsity d d o outsid th tnt of θφ θ φ : θφ θφ //8 H 7 Sping 8 -- Lctu 8 d : θ φ Mutipo onts continud: d θ φ o outsid th tnt of : θφ = Rtionship btwn sphic honic nd Ctsin fos of utipo onts: p z p ip 8 y Q //8 H 7 Sping 8 -- Lctu 8 zz Q iqy Qyy 88 Qz iq 7 yz
//8 Consid pvious p: 8 θφ θφ V V W pviousy showd tht fo 8 V Φ θφ θφ d 8 θφ θφ sin cos V V Not tht: 8 V 8 p V Lctu 8 //8 H 7 Sping 8 -- Gn fo of ctosttic potnti in ts of utipo onts: o outsid th tnt of p : θφ θφ In ts of Ctsin pnsion : i d i Qi θ φ //8 H 7 Sping 8 -- Lctu 8 7 Ep of utipo pnsion in vuting ngy of vy ocizd chg dnsity () in ctosttic fid () (such s n nucus in th fid du to th ctons in n to). W d d pe i i Qi //8 H 7 Sping 8 -- Lctu 8 8
//8 7 //8 H 7 Sping 8 -- Lctu 8 9 Sip ps of utipo distibutions y z y z ˆ ˆ z y d d p p d p z z ˆ ˆ zz yy d d Q d Q Q z z //8 H 7 Sping 8 -- Lctu 8 Anoth p of utipo distibution sin sin cos : Not tht sin / d d / / //8 H 7 Sping 8 -- Lctu 8 Anoth p of utipo distibution -- continud / / cos fo Lgnd poynois : ; in ts o cos fo Lgnd poynois : in ts ; o c os
//8 Anoth p of utipo distibution -- continud o ; in ts fo Lgnd poynois : cos W i i Qi Ipictions fo ctic udupointction : o ; in ts of Ctsin coodints y z y z cos cos z z y //8 H 7 Sping 8 -- Lctu 8 Anoth p of utipo distibution -- continud Ectic udupointction : W Qi Q Q yy Q zz i i y z o sytic nuci Q Q Q Q Q W zz yy //8 H 7 Sping 8 -- Lctu 8 8