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1 . Two concentic sphees hve dii, b (b nd ech is divided into two heisphees by the se hoizont pne. The uppe heisphee of the inne sphee nd the owe heisphee of the oute sphee e intined t potenti V. The othe heisphees e t zeo potenti. Deteine the potenti in the egion b s seies in egende poynois. Incude tes t est up to. Check you soution ginst known esuts in the iiting cses b, nd. The pobe is syetic Φ, θ A B cosθ ( ( ( ( θ utipy both sides by cos then integte (, θ ( cosθ d( cosθ Φ A B - V ( x dx A B b ( ( ( Ab Bb d d ( ( ( V x dx V x dx ( x x ( x x ( x! dx! dx (!! ( x dx k! ( ( b ( A Vk b & B Vk b b b (!! Φ (, θ V cos θ! b b b b V ( b b( b 7V ( b b ( b b ( cosθ ( cos ( b 6 b θ ( b If b V 7V Φ (, θ ( cosθ ( cosθ 6 If V 7V Φ (, θ ( cosθ ( cosθ b 6 b. ( ( ( (

2 A spheic sufce of dius hs chge unifoy distibuted ove its sufce with density Q/, except fo spheic cp t the noth poe, defined by the cone θα. ( Show tht the potenti inside the spheic sufce cn be expessed s Φ Q 8ε [ ( cosα ( cosα ] ( cosθ whee, fo - (cosα -. Wht is the potenti outside? (b Find the gnitude nd the diection of the eectic fied t the oigin. (c Discuss the iiting fos of the potenti (pt nd eectic fied (pt b s the spheic cp becoes ( vey s, nd ( so ge tht the e with chge on it becoes vey s cp t the south poe. ( (b Φ (, ε (, θ σ x x d Q * (, (, ε α Y θ ϕ Y θ ϕ θ dθ dϕ Q Q cos ( α (cos θ (cos θ θ dθ (cos θ ( x dx 8ε 8 α ε ( x Q (cos θ ( x ( x 8ε ( Q 8ε ( ( x ( x x cos ( α ( cosα ( cosα ( cosθ E Φ ˆ θˆ θ Φ Q ( cosα ( cosα ( cos θ 8ε ( Q Q ( cosα ( cosα ( cosθ cos α cosθ ε ε Q αcosθ 6ε

3 Φ Q ( cosθ cosθ( cosθ ( cos ( cos ( α α θ θ 8ε ( cos θ cos θ ( cosα ( cosα ( cosθ cosθ ( cosθ Q θ 8ε cos θ cos θ Q Q cos ( α cos θ 6ε θ α θ 6ε Q E α ( cosθˆ θθˆ 6ε (c cp vey s ( i Cp vey s Q Φ 8ε α ( cosα ( cosα ( cosθ Q 8ε ( ( ( cosθ Q Q ( ( ( cosθ 8ε ε Q E α ( cosθˆ θθˆ 6ε α ( ii Cp vey ge Q Φ cosα cosα cos θ 8ε ( ( ( Q Q ( ( ( cos θ α 8ε ε Q E α ( cosθˆ θθˆ 6ε α. A thin, ft, conducting, cicu disc of dius is octed in the x-y pne with its cente t the oigin, nd is intined t fixed potenti V. With the infotion tht the chge density on disc t fixed potenti is popotion to /, whee is the distnce out fo the cente of the disc. ( Show tht fo the potenti is Φ (, θ, φ V ( ( cosθ

4 (b Fond the potenti fo. (c Wht is the cpcitnce of the disc? ( δ( cos θ Θ( θ (, θ k δ( z Θ( k δ(cos θ Θ( θ k ( θ ( θ Y ( θϕ, (cos θ, ziuthy syetic, * Y (, (, θ ϕ Y θ ϕ x x (cos (cos θ θ Φ (, θ ε (, θ x x d x δ(cos θ ( θ (cos (cos ( θ ε k Θ θ θ d θ dθ dϕ ( θ k ( Γ k ( (cos θ ε d ( (cos θ ε Γ( n Γ( k ( ( ( ( n n ε Γ( ( n!! n!! n( n(cos θ, n, n n n n k n ( n(cos θ ε n n k tn ε k k εv Φ (, θ V tn k ε ε n n V n Φ (, θ ( n(cos θ n

5 (b fo (, θ Φ (, θ d x ε x x kδ(cos θ Θ( θ (cos (cos ( θ ε θ θ d θ dθ dϕ ( θ k k ( (cos ( (cos θ d ε θ ( ε d ( k ( (cos θ d d! ε ( ( ( tn ( ( ( V Φ (, θ V c osθ ( (c Q C d V Q C V.6 Two point chges q nd q e octed on the z xis t z nd z-, espectivey. ( Find the eectosttic potenti s n expnsion in spheic honics nd powes of fo both nd. (b Keeping the poduct q p/ (c constnt, tke the iit of nd find the potenti fo. This is by definition dipoe ong the z xis nd its potenti. (d Suppose now tht the dipoe of pt b is suounded by gounded spheic she of dius b concentic with the oigin. By ine supeposition find the potenti eveywhee inside the she. ( Φ ( x ε q x x q ε q x x, * * ( Y (, φ Y ( θ, φ Y (, φ Y ( θ, φ the pobe is ziuthy syetic, ony tes suvive.

6 * * Y (, ϕ ( cos & Y (, ϕ ( cos ( q q Φ ( x Y ( θϕ, ( ( cosθ ( ε ε q ε Φ ( x q cos, ( θ ( c ( θ ( ε os, (b (c & q q Φ ε (, θ ( cosθ ( ( cosθ q pcosθ ( cosθ ε ε odd ε s d A ( cosθ ε ΦΦ Φ ( b Φ p Φ (, θ cosθ ε b b p cosθ.7 Thee point chges (q, -q, q e octed in stight ine with seption nd with the idde chge (-q t the oigin of gounded conducting spheic she of dius b, s indicted in sketch (Jckson p. 7. ( Wite down the potenti of the thee chges in the bsence of the gounded sphee. Find the iiting fo of the potenti s, but the poduct q Q eins finite. Wite this tte nswe in spheic coodintes. (b The pesence of the gounded sphee of dius b tes the potenti fo b. The dded potenti cn be viewed s cused by the sufce-chge density induced on the inne sufce t b o by ige chges octed t b. Use ine supeposition to stisfy the boundy conditions nd find the potenti eveywhee inside the sphee fo nd. Show tht in the iit, Φ ( Q ε ( θ, φ ( cosθ, 5 b 5

7 this pobe is sii to (.6, we use sighty diffeent ethod hee: on z-xis the potenti is q q q q Φ ( x ε x x x x x x ε xx xx q, ϕ θϕ, ϕ, θϕ, ε, ziuthy syetic, ony exist q Φ ( x ε q ( ( ( cosθ ε When * * ( Y ( Y ( Y ( Y ( ( ( ( ( cosθ q Φ ( x ( ( ( cosθ ε n q q n n ε even nε n q cosθ Q cosθ Q cos θ ε ε ε ( cosθ ( cosθ ( ( ( (b the sufce chge on the sphee poduce n ext contibutionφ to the potenti within the sphee. s q Φ ( x ( ( ( cosθ A ( cosθ ε q ( cos ( cos, θ A θ ε even Φ (, θ q ( cos ( cos, θ A θ ε even q Φ ( b, θ ( cosθ Ab ( cos θ, b ε b even b, A, A odd q n q ( cosθ A b ( cos θ A ε n n n n n n n n b n ε b

8 n q n n ( cos θ, n n ε n b Φ (, θ n n q n n ( cos θ, n n ε n b n n n Q n n n cosθ 5 n b ε b q, Φ (, θ ε 5 Q ( cosθ ε b ( ( cosθ.9 A hoow ight cicu cyinde of dius b hs its xis coincident with the z xis nd its ends t nd. The potenti on the end fces is zeo, whie the potenti on the cyindic sufce is given s V(φ,z. Ug the ppopite seption of vibes in cyindic coodintes, find seies soution fo the potenti nywhee inside the cyinde. φ with b.c. Φ Φ Φ (, φ, (, φ, ( b, φ, z V ( φ, z b.c. Q(φ Q(φ Φ (, φ, z I z { A ( φ B cos( φ } n n n A n, B n to be deteined fo b.c. t b n n V z I b z An Bn ( ϕ, { ( ϕ cos( ϕ } n n n nb n b dz dϕv ( ϕ, z ( ϕ z I A δ I A n n n n n n An dz dϕv ϕ nb I (, z ( ϕ z n Bn dz dϕv( ϕ, z cos( ϕ z nb I n

9 .7 The Diichet Geen function fo the unbounded spce between the pnes t z nd z ows discussion of point chge o distibution of chge between pe conducting pnes hed t zeo potenti. ( Ug cyindic coodintes show tht one fo of the Geen function is i( φ φ nz nz n G( x, x e I K n (b Show tht n tentive fo of the Geen function is G i( φ φ ( x, x dke J ( ( k J k ( The diffeenti eqution is G δ ( δ ( φ φ δ ( z z ( kz h[ k( z ] h( k eign function in z diection nz i, inφ diection e φ nz G A ( n,, z, φ e n pug into diffeenti eqution iφ n nz iϕ A n e n δ ( δϕ ( ϕ δ ( zz n z i ϕ n dz e dϕ A n n z i ϕ n z δ ( e An gn (, e n g n (, δ( i ϕ n AI ( k, n gn(, When AI( k BK( k, k BK ( k, k g (, ( n δ k g n (, d δ ( d gn(, kbk ( k ka I ( k K( k I( k A & B k I ( k K ( k I ( k K ( k k I ( k K ( k I ( k K ( k ( (

10 Use the etion : I( k K ( k I ( k K( k k I( k K( k, K( k I( k A & B gn(, gn(, I( k K( k I( k K( k, n n G I K nz nz i( ϕϕ e n (b do expnsion in & φdiections G G dk δ dk A k iφ ( z, z,, φ e J ( k k Ak z ( δ ( φ φ δ ( z z iφ ( z, z,, φ e J ( k iφ δ k z ( dk k A e J ( k δ( φφ δ( z z ( dk k δ e e dφ A J ( k J ( k d J ( k d e δ( φφ dφδ( z z i φ iφ i φ k z iφ δ( k k i dk k A φ ( ( ( ( ( ( k J k J k d J k e δ z z dk k A k J k e δ z z z k z iφ k iϕ A ( ( k kj k e δ zz A z k gk( z e J( k k g ( ( k z kδ zz z t (Se s ( ψ (, ψ ( ( kz ψ ( k( z ψ h, h CW h ( kz,h( k( z k k h( k C C h( k h( kz h( k( z i( ϕϕ G dk e J( k J( k h( k. ( Fo the esuts of obe.7 o fo fist pincipes show tht the potenti t point chge q between two infinite pe conducting pnes ged t zeo potenti cn be witten s q ( nz nz n Φ z, K ε n whee the pnes e t z nd z nd the chge z on the z xis t the point. (b Ccute the induced sufce-chge densities σ ( nd σ ( on the owe nd uppe ptes. The

11 esut fo σ ( is q ( n nz n σ ( n K n Discuss the connection of this expession with tht of obe.9b nd.9c. (c Fo the nswe in pt (b, ccute the tot chge Q on the pte t z. By suing the Fouie seies o by othe ens of copison, check you nswe ginst the known expession of obe.. ( ( x q δ( δ( ϕ ϕ δ( z z i( ϕϕ nz nz n n G( x, x e I K n ( x G( x, x Φ d x ( φ is syety, ony te ε q nz nz n n I K δ( d ε n q nz nz n n I K δ( d ε n q nz nz ε n n n n n I K δ( d I K δ( d q nz nz n n I( K I ε n K ( q nz nz n K ε n (b σ nˆ ε Φ ( zˆ Φ z z q n nz n q nz n σ n cos( n K ( n K n n nˆ ε Φ zˆ Φ z z q n nz n q K n n nz n n K

12 (c nz nz Q n K d d n K d q n q n n ϕ ( ϕ n n nz i z q n nz q n e q i ( I ( In e n n n n z z z q n i i i qz I e e e. The geoety of two-diension potenti pobe is defined in po coodintes by the sufces φ, φ nd, s indicted in the sketch. Ug seption of vibes in po coodintes, show tht the Geen function cn be witten s G (, φ,, φ obe.5 y be of use. φ φ φ Fo., fo, the ngu soution is Q ( φ δ ( φ φ G φ φ 8 (, φ,, φ δ ( Expnd G φ φ δ G ( φ φ φ φ (, φ,, φ g (, g Fo,, g, (, g (, φ φ 8 δ A ( B B, g, g (, A ( (, B ( ( is invint unde, exchnge

13 [ ] C g integte coss the jup ( ε ε ε ε δ d d g d 8 s 8 ' ' g g [ ] (,,, 8 G g C C C φ φ φ φ