Topic 35: Green s Functions II Potential between Two Spherical Shells
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1 Topic 35: ee s Fuctios II otetil etwee Two Spheicl Shells I this topic we develop the tool to e used o idig the potetil o ity chge distiutio etwee two cocetic spheicl suces o dii d ( < ) hvig ity potetils s oudy coditios Fo ou pevious discussio o uiqueess theoes we kow tht the solutio to such pole is uique Howeve it is i geel diicult to oti the solutio uless oe c id the ppopite ee s Fuctio Tht is ou piy gol i this topic Much o the pocedue eployed hee ios tht used i Execise 3- which ivolved the expsio o the ee s Fuctio o ll spce i tes o Legede polyoils The ppoch used i this topic will led to the expsio o the ee s Fuctio i the spce etwee the cocetic sphees i tes o spheicl hoics Fo Topic 33 we kow tht i: (33-6) d o suce S (33-7) the the potetil i the volue V tht is ouded y the suce S is: d V S d (33-8) Thee is o siple ige solutio o this pole so we ust eploy systetic pocedue to id the ee s Fuctio Sice the ee s Fuctio is potetil o speciied chge distiutio withi suce hvig speciied potetil we kow tht oce we hve oud solutio it is the oe d oly solutio y the uiqueess theoe Sice the ee s Fuctio is solutio to Lplce s equtio o we c expess it i tes o spheicl hoics l equtio (3-5) o d : B ( A ( C ust vish o the suces with dii d so we hve tht: ) (35-) ) (35-) Electogetis Topic 35 S Ahle; 9/6/5; : M
2 A ( ) (35-3) d ( ) (35-) Sice is potetil it ust e cotiuous s the sphee o dius : A (35-5) We c stisy this equieet d lso stisy the equieet tht e syetic i d y settig: (35-6) d A (35-7) We c ow coie equtios (35-3) d (35-) ito the ollowig oe: (35-8) whee is the slle o d d is the lge o d To deteie we ipose the coditio tht the electic ield ust e cotiuous s the sphee o dius (except t ): (35-9) d (35-) Electogetis Topic 35 S Ahle; 9/6/5; : M
3 So i (35-) It is esy to idetiy y usig the copleteess eltio o spheicl hoics: K * (35-) whee K is tt Note tht this choice o lso copletes the equieet tht is syetic i d d povides the eeded sigulity t the poit So we ow hve tht: K * We deteie K y usig uss Lw d y equiig the chge etwee sphees with dii slightly lge d slightly slle th e equl to We itegte (35-9) to id the chge eclosed y sphee with dius slightly lge th : d whee we hve used the othogolity o the spheicl hoics Siilly we id the chge withi sphee hvig dius slightly slle th : d The totl chge etwee these two sphees ust e : K So K = d we hve o the ee s Fuctio: Electogetis Topic 35 S Ahle; 9/6/5; : M 3
4 * (35-3) I we let go to zeo d go to iiity we will get the ee s Fuctio o ll spce : * (35-) Copig with equtio (3-39) we oti the dditio theoe o spheicl hoics: whee is the gle etwee d (35-5) * ************************************************************************ Execise 35- A ig o chge hs uio lie chge desity The ig hs dius d lies i the xy ple The totl chge o the ig is The ig is etwee two cocetic gouded etl sphees o dii d ( < ) Fid the potetil etwee the sphees Solutio 35- We use equtios (33-8) d (35-3) The chge desity is give y: The potetil is: * d V Fo the deiitio o the spheicl hoics d ssocited Legede uctios we hve: Electogetis Topic 35 S Ahle; 9/6/5; : M
5 d x x so V d The itegl c e witte: V d d d whee the gete d lesse th sigs ow ee to d : ************************************************************************ Electogetis Topic 35 S Ahle; 9/6/5; : M 5
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