LAPLACE S EQUATION IN SPHERICAL COORDINATES. With Applications to Electrodynamics
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1 LALACE S EQUATION IN SHERICAL COORDINATES With Appitions to Eetodynmis We hve seen tht Lpe s eqution is one of the most signifint equtions in physis. It is the soution to pobems in wide viety of fieds inuding themodynmis nd eetodynmis. In you ees s physis students nd sientists, you wi enounte this eqution in viety of ontets. It is impotnt to know how to sove Lpe s eqution in vious oodinte systems. The oodinte systems you wi enounte most fequenty e Ctesin, yindi nd sphei po. We investigted Lpe s eqution in Ctesin oodintes in ss nd just begn investigting its soution in sphei oodintes. Let s epnd tht disussion hee. We begin with Lpe s eqution: ( We n wite the Lpin in sphei oodintes s: ( + (sinθ + ( sinθ sin θ φ ( whee θ is the po nge mesued down fom the noth poe, nd φ is the zimuth nge, nogous to ongitude in eth mesuing oodintes. (In tems of eth mesuing oodintes, the po nge is 9 minus the titude, often temed the otitude. To mke ou initi utions itte simpe, et s ssume zimuth symmety; tht mens tht ou pmete does not vy in the φ dietion. In othe wods, / φ, so we n wite the Lpin in ( bit moe simpy. Assuming zimuth symmety, eq. ( beomes: ( + (sinθ sinθ ( This is the fom of Lpe s eqution we hve to sove if we wnt to find the eeti potenti in sphei oodintes. Fist, et s ppy the method of sepbe vibes to this eqution to obtin gene soution of Lpe s eqution, nd then we wi use ou gene soution to sove few diffeent pobems. To sove Lpe s eqution in sphei oodintes, we wite:
2 Fist Step: The Ti Soution ( + (sinθ sinθ (4 The fist step in soving pti diffeenti equtions using sepbe vibes is to ssume soution of the fom: R( Θ( θ ( whee R( is funtion ony of, nd Θ(θ is funtion ony of θ. This mens tht we n set: ( ( θ R Θ ; R( ( θ Θ (6 Substituting the etionships in (6 into (4 podues: Θ( ( θ R ( R ( + (sinθ Θ ( θ sinθ (7 If we mutipy eh tem in (7 by nd then divide eh tem by R( Θ(θ, we obtin: d d ( R ( + (sinθ Θ ( θ R( d Θ( θ sinθ dθ (8 Notie tht the deivtes in (8 e no onge pti deivtives. This is beuse the method of sepbe vibes hs podued two tems; one is soey funtion of nd the othe is soey funtion of θ. Seond Step: Septing ibes Eqution (8 ows us to septe Lpe s eqution into two septe odiny diffeenti equtions; one being funtion of nd the othe funtion of θ. As we hve disussed in ss, we eize tht eh tem on the ight hnd side of (8 is equ to onstnt. This mens we n septe (8 into: d ( R( d d R ( ( + nd (sinθ Θ ( θ ( + Θ( θ sinθ dθ (9 We now hve two diffeent odiny diffeenti equtions whih we wi sove. We eize tht the podut of soutions wi ow us to use eq. ( (ong with ppopite boundy onditions to detemine the soution to Lpe s eqution. You my wonde we we hoose to wite the seption onstnt s something s non-obvious s (+.
3 The eson is tht we hve ed hed in the sipt, nd know tht by witing the seption onstnt in this wy we wi podue we known diffeenti eqution whose soution we edy know. Notie tht seption onstnt is positive in one eqution (the di pt nd negtive in the othe (the ngu pt; this is neessy so tht the sum of equtions is zeo s equied by Lpe s eqution. The di eqution Let s stt by soving the di eqution of eq. (9. We mutipy though by R( nd epnd the deivte to find: d R dr + ( + R ( d d This is fiy simpe empe of Fobenius (see diffeenti eqution. This is so n empe of n Eue (o Cuhy diffeenti eqution. See fo moe detis bout soving Eue s eqution. Using eithe the method of Fobenius o methods of Eue s equtions, we n find the soution to eqution (: ( + R ( A + B ( whee A nd B e onstnts whih wi be detemined one we ppy speifi boundy equtions. The ngu eqution We sove the ngu potion of eqution (9 by mutipying though by Θ(θ nd epnding the deivtive to obtin: d Θ osθ dθ + + ( + Θ dθ sinθ dθ ( This is tuy diffeenti eqution you e vey fmii with, though pehps not in this et fom. Refe bk to the soutions fo the vey fist homewok set of the semeste, nd eview gin the soutions to pobems 4 nd. You wi see tht the eqution you hve deived in ( is just the we known Legende eqution. We know tht the soutions to the Legende eqution e the Legende poynomis, (os θ. Thid Step: Constuting the ompete soution
4 Hving septed Lpe s eqution into two odiny diffeenti equtions, we n use the esuts bove to substitute into eq. ( to eize tht the gene soution to Lpe s eqution in sphei oodintes wi be onstuted of sum of soutions of the fom: ( + (, θ ( A + B ( Fom ou epeiene with Lpe s eqution in Ctesin oodintes, we know tht the fu soution wi be onstuted by tking sum of soutions of the fom of (; in othe wods, ou gene soution to Lpe s eqution in sphei oodintes is: ( + (, θ ( A + B (4 Now, we need e boundy onditions to detemine the vues of the oeffiients A nd BB. Appying Boundy Conditions Fist Empe (Bos pp Let s see how we n use (4 s the stting point to detemine soution to Lpe s eqution with speifi boundy onditions. Fo this pupose, et s use the empe in Bos pp Without ny oss of mening, we n use tk bout finding the potenti inside sphee the thn the tempetue inside sphee. So, et s ssume thee is sphee of dius, nd the potenti of the uppe hf of the sphee is kept t onstnt +, nd the potenti of the owe hf of the sphee is hed t. How n we detemine the potenti t ny point inside the sphee? Fist, et s wite the boundy onditions s:, < θ < π / o < osθ <, π / < θ < π o < osθ < Remembe tht the noth poe of the sphee oesponds to θ, nd θ π/ in the equtoi pne. Now, et s ook bit moe osey t (4. We e sked to find the potenti t ny point inside the sphee. This egime inudes, of ouse, the point, nd we n ook t (4 nd eize tht the soution diveges t uness BB. Appying the neessity fo meningfu physi soution to this pobem ows us to set oeffiients B B to zeo, so tht (4 simpifies to: (, θ A (
5 Now, we use the boundy ondition fo the sufe of the sphee. When, we know tht in the uppe hf sphee nd in the owe hf sphee. This mens we n wite ( s: (, θ A fo < osθ < (6 The epession in (6 shoud ook fmii to us: we e seeking to wite funtion (in this se the funtion equs the onstnt in tems of n infinite seies. We hve seen how to do this using both Fouie seies nd Legende oynomis. We know tht ou funtion n be epnded in seies if nd ony if we n epnd tht funtion in tems of ompete set of othogon funtions. Fouie seies e possibe beuse sin nd os epesent ompete set of othogon funtions on (-π, π; epnsion in tems of Legende poynomis is possibe sine we hve ened tht Legende poynomis e ompete set of othogon funtions on (-,. Thus, we n epnd ny funtion f( on (-, s: whee the oeffiients, e detemined by: f ( ( (7 + f ( ( d (8 We n see tht eqution (7 ppies to eq. (6 with f(, nd A, o A (9 A we hve to do now is detemine the vues of the oeffiients fom (8, substitute these vues into (9 nd then use those vues of A in ( to detemine the ompete soution to the potenti inside the sphee. We n detemine seve of the oeffiients esiy by diet integtion; in ft this is done in Bos on p. 8. Using these Legende oeffiients with f( nd substituting into (6 we obtin n epiit epnsion of ou soution fo (, θ: 7 (, θ [ ( ] ( 4 6 nd you n epnd the vious Legende poynomis epiity in tems of osθ if you wish, but thee is ey no need to go beyond the epession s it is witten in (.
6 Seond Empe Conside sphee of dius tht hs potenti on its sufe given by: (, θ os θ ( nd we e sked to find the potenti t points eteio to the sphee. We go bk to eq. (4 nd begin to ppy boundy onditions. Fist, we eize tht A must go to zeo sine n get vey ge, owing us to simpify (4 s: ( + (, θ B ( Now, we ppy the boundy ondition ( nd obtin: (, ( + θ B os θ ( This is just nothe fom of eq. (7. Hee, the funtion f( is os θ, nd the -(+ oeffiient BB stnds in the pe of. So, ou tsk now is vey fmii: ompute the oeffiients using (8, use these to detemine the vues of B B, nd substitute these vues of BB into ( to find ou ompete soution. Let s begin by finding the oeffiients. We n set os θ; sine θ vies fom to π then vies fom - to, whih is vey onvenient in uting Legende oeffiients sine the Legende poynomis e ompete, othogon set on (-,. With this substitution, we wi ute ou oeffiients fom: + + f ( ( d ( d ( + d (4 The fin integ on the ight is petty esy to do; Legende poynomis e, we, poynomis, nd mutipying them by just podues nothe poynomi whih is esy to integte between these imits. But et s think bit moe nd mke ou ives even esie. We e tht Legende poynomis e even funtions fo even vues of, nd e odd funtions fo odd vues of. This mens tht fo odd vnish sine the integnd in (4 beomes the podut of n even funtion ( nd n odd funtion ( ( fo n odd. This mens the integnd in (4 is odd wheneve is odd, nd the integ of n odd funtion between imits symmeti with espet to the oigin vnishes. Let s ompute oeffiients:
7 ( d d ( ( d ( d You wi find tht highe inde oeffiients vnish; does it mke sense tht this funtion is epessibe in tems of ony ( nd (? Thee e ony two tems whih wi ontibute to the seies epnsion of, nmey the nd tems. We emembe fom befoe tht we use ou vues of to find the vues of BB tht substitute bk into (; eq. ( tes us tht: B ; B, + so B (6 Using these vues of BB in ou gene soution ( gives us the ompete nswe to this pobem: (, θ B + B [( + ( ] (7 Thid Empe Let s sy now tht we wnt to find the potenti outside sphee of dius whose sufe is hed t potenti given by os(θ. We know tht sine we e deing with eteio points ou soution wi be of the fom of eq. (, nd tht we wi hve to find the oeffiients BB. The poess we foow is identi to the empe immeditey bove, eept now f( os(θ the thn os θ. We sw in the empe bove how we oud simpify ou utions by eizing we oud set os θ; we woud ike to epess ou uent f( in tems of os θ, but we wi hve to do itte tig nd geb mnipution to ompish this. Let s stt by witing os(θ os(θ+θ. We now epnd this s: os(θ + θ os θ osθ sin θ sinθ (os θ sin θ osθ sin osθ[ {os θ ( os θ } ( os θ ] 4os θ osθ whee os θ. θ osθ 4 (8 Now it is fiy stightfowd tsk to find the neessy oeffiients to sove ou pobem. We foow the empe of eq. (4, now with f( (4, nd sove fo :
8 ( (4 ( (4 d d 8 ( [ (4 7 ( (4 7 d d These esuts te us tht: 8 ; 4 4 B B nd we use the fom of (4 to wite the fin nswe s: ] (os 8( (os ( [ (os (os, ( 4 4 θ θ θ θ θ B B + +
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