(b) (c) (d) When, where

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1 . Ug Dc delt fucto the ppopte coodte, expe the followg chge dtbuto thee-dmeol chge dete ρ(x). () I phecl coodte, chge ufomly dtbuted ove phecl hell of du R. (b) I cyldcl coodte, chgeλpe ut legth ufomly dtbuted ove cyldcl ufce of du b. (c) I cyldcl coodte, chge ped ufomly ove flt ccul dc of eglgble thcke d du R. (d) The me pt (c), but ug phecl coodte. Soluto : () (b) (c) ρ() d = ρ = δ( R) = θddθdϕ ρ() d = dρ() = R = ρ = δ( R) R ρ( ) = δ( b) λ λ= ρ() ddθ= πb ρ= δ( b) πb ρ θ ( ) = δ ( z) θ ( R ) ( x) whee, f x < =, f x > ρ πr () = δ ( z) θ ( R ) (d) Whe, c ρ( ) = cδ( z) θ( R ) = cδ( coθ) θ( R ) = δ( coθ) θ( R ) = ρ() θddθdϕ= πc θ( R ) d ( co ) d( co ) cr δ θ θ = π π c= ρ() = δ( coθ) θ( R ) πr πr.4 Ech of thee chged phee of du, oe coductg, oe hvg ufom chge dety wth t volume, d oe hvg pheclly ymmetc chge dety tht ve dlly (>-), h totl chge. Ue Gu theoem to obt the electc feld both de d outde ech phee. Sketch the behvo of the feld fucto of du fo the ft two phee, d fo the thd wth = -,+.

2 Soluto :.., f < E d= E = ε, f > ε ρ() =, < 4 π ρ () E d = θddθdϕ 4 π E() =, f < ε 4 ε, f < ε E() =,f > ε π. ρ () = C = C d= C C = ( + ) C C whee <, E() = C ( ) d = = ε + ε ε + ε, whe = ε E () =, whe = 5 ε +.5 The tme-veged potetl of eutl hydoge tom gve by q e α α Φ= + ε Whee q the mgtude of the electoc chge, d, beg the Boh du. Fd the dtbuto of chge (both cotuou d dcete) tht wll gve th potetl d tepet you eult phyclly. Soluto :

3 q α α q α α ρ= ε Φ= e + = e + α α α α e αe e αe α α α α e + = + e αe = α α α α α α α e e ( e ) ( α) e α e α e = + α α α α α α α αe α αe = e αe αe α e α e e = + α α α α α e α αe αe = e + = e δ( ) + = δ( ) + α ( e δ = δ δ =, whe ) q αe qα q e 8π α ρ ε δ = Φ= + = δ.6 A mple cpcto devce fomed by two ulted coducto djcet to ech othe. If equl d oppote chge e plced o the coducto, thee wll be cet dffeece of potetl betwee them. The to of the mgtude of the chge o oe coducto to the mgtude of the potetl dffeece clled the cpctce ( SI ut t meued fd). Ug Gu lw, clculte the cpctce of () Two lge, flt, coductg heet of e A, epted by mll dtce d; (b) Two cocetc coductg phee wth d, b (b>); (c) Two cocetc coductg cylde of legth L, lge comped to the d, b(b>). (d) Wht the e dmete of the oute coducto -flled coxl cble whoe cete coducto cyldcl we of dmete mm d whoe cpctce - F/m? - F/m? Soluto : σ σ Aε AE = E = = E = E = = E d C = = ε Aε ε ε d α tot. tot. () b b εb E = E= = d C ε ε = = ε ε b b () c E πl= E= ε πε L b b L = d= C= = πε πε L πε b L

4 .8 () Fo the thee cpcto geomete Poblem.6 clculte the totl electottc eegy d expe t ltetvely tem of the equl d oppote chge d plced o the coducto d the potetl dffeece betwee them. (b) Sketch the eegy dety of the electottc fled ech ce fucto of the ppopte le coodte. Soluto : () Pe plte: Ad d ε A ε εa ε ε σ W = E = Ad = = = = C Sphecl: ε b bd b W = 4 d C π ε = = = = 8πε 8πε b C Cyldcl: ε b b W = πld = = = C πl εl C (b) Pe plte: ε W () = whe d, d W ( ), othewe < < = Aε Cocetc phee: ε W () = whe b, d W ( ), othewe < < = ε Cocetc cylde: ε W () = whe b, d W ( ), othewe < < = πε L.9 Clculte the ttctve foce betwee coducto the pllel plte cpcto (Poblem.6) d the pllel cylde cpcto (Poblem.7) fo () Fxed chge o ech coducto; (b) Fxed potetl dffeece betwee coducto. Soluto : () Pe plte cpcto: ε ε σ Ad d z W = E = Ad ε = = = A ε εa εa z We chooe the z decto F = W = = eˆ ε A ε A z

5 (b) Pllel cylde cpcto C C πε d =, the cpctce pe ut legth πε L d λ d d Lπε πε = = = = λ λ d λ z W = = = We chooe the z decto C πε πε λ z λ λ πε = W = = eˆ ˆ ˆ z = ez = e z πε πεz πε z d d = d F the foce pe ut legth (c) Pe plte cpcto: Aε Aε Aε C= = W = C = = d d z We chooe the z decto Aε Aε Aε F = W = = eˆ = eˆ z z d Aε ˆz ε = eˆz = e Aε d A Pllel cylde cpcto C πε d = z z z= d, the cpctce pe ut legth πε πε d z d W = C = = We chooe the z decto πε πε πε = = = ˆ ˆ z = e z F W e z z z z = d d d the foce pe ut legth. Pove Gee ecpocto theoem: If Φ the potetl due to volume-chge dety ρ wth volume d ufce-chge dety σ o the coductg ufce S boudg the volume, whle Φ' the potetl due to othe chge dtbuto ρ' d σ', the Φ ρ + σφ d = ρ Φd x + σ Φd Soluto :

6 Ψ ϕ Ψ Ψ dx= Ψ d ϕ ϕ ϕ S ϕ =Φ Let Ψ=Φ ρ ρ Φ σ Φ σ,,, ε ε ε ε Φ= Φ = = = ρ ρ σ σ d Φ Φ dx= Φ Φ ε ε S ε ε Φ + Φ = Φ + Φ ρ σ d ρ σ d S S.4 Code the electottc Gee fucto of Secto. fo Dchlet d eum boudy codto o the ufce S boudg the volume. Apply Gee theoem (.5) wth tegto vble y d Φ=G(x,y), φ= G(x',y), wth yg(z,y)=-δ(y-z) Fd expeo fo the dffeece [ G(x,x')- G(x',x)] tem of tegl ove the boudy ufce S. () Fo Dchlet boudy codto oe the potetl d the octed boudy codto o the Gee fucto, how tht G D (x,x') mut be ymmetc x '. (b) Fo eum boudy codto, ue the boudy codto (.45) fo G (x,x') to how tht G (x',x) ot ymmetc geel, but tht G (x,x')-f(x) ymmetc x ', whee F ( x) = G ( x, y) d y S (c) Show tht the ddto of F(x) to the Gee fucto doe ot ffect the potetl Φ(x). See poblem.6 fo exmple of the eum Gee fucto. Soluto : ψ φ v ( φ ψ ψ φ) dx= φ ψ d, letφ= Gxy (, ) & ψ= Gx (, y) Gx (, y) Gxy (, ) v( Gxy (, ) ygx (, y) Gx (, y) ygxy (, )) dy Gxy (, ) Gx (, y) = dy Gx (, y) Gxy (, ) v ( Gxy (, )( 4 πδ( x y) ) Gx (, y) ( 4 πδ( x y) )) dy= Gxy (, ) Gx (, y) d Gx (, y) Gxy (, ) Gxx (, ) Gx (, x) = Gxy (, ) Gx (, y) dy () Fo Dchlet boudy codto we domd (.4): GD( x, y) = fo y o S GD( x, y) GD( x, y) GD( xx, ) GD( x, x) = GD( xy, ) GD( x, y) dy GD( x, y) GD( x, y) = d y = G ( x, x ) clely ymmetc x D y

7 (b) G ( x, x ) Eq.(.45), = fo x o S S G( x, y) G( x, y) G ( x, x ) G ( x, x) = G ( x, y) G ( x, y) d y = G( xyd, ) y G( x, yd ) y= Fx Fx S S G ( x, x ) ot ymmetc x ButG ( xx, ) G ( x, x) = Fx Fx ( ) G ( xx, ) Fx = G ( x, x) Fx ( ) let G ( x, x ) = G ( x, x ) F( x) & G ( x, x) = G ( x, x) F( x ) G ( x, x ) = G ( x, x) G ( x, x ) = G ( x, x ) F( x) clely ymmetc x () c Φ( x) Eq Φ x = Φ + ρ x G x x + G x x d.(.46), (, ) (, ) ε v G( x, x ) G( x, x ) F( x), F( x) = G( x, y) dy S Φ( x ) Φ x = Φ + x G x x F x + G x x F x d ρ (, ) (, ) ε v Φ( x) Φ( x ) = Φ + ρ( x ) G (, ) (, ) xx dx G xx d ρ x F xdx Fxd ε + v ε v Fx Fx Φ( x ) Fx Fx Φ( x ) = Φ = Φ + Φ ( ) x ρ xdx x dx d ε v v Fx Φ( x ) F( x) Φ( x ) =Φ ( x) + d d =Φ x the ddto of F( x ) to the Gee fucto doeot ffect the potetl Φ( x).6 Pove the followg theoem: If umbe of ufce e fxed poto wth gve totl chge o ech, the toducto of uchged, ulted coducto to the ego bouded by the ufce lowe the electottc eegy. Soluto : q q q q q q E, E, + q + = the toducto of uchged, ulted coducto to the ego

8 Itl electottc eegy : W = ε E E : Fl electottc eegy: W ε E E : : 新加入導體的體積不包含導體的體積 = 不包含導體的體積 P ove : W W = + W W= ε E Edx ε EEdx = ε E Ed x ε EEd x + = ε ( E E E Edx ) EEdx = ε( ( E E E Edx ) ) ε E dx ( E E) = E E + E E E E = E E + E E E E+ E E = ( E E E E) + E ( E E) ( E E E E) = ( E E) + E ( E E) = ε ( E E E E) E E = ε ( E E) + E ( E E) ε E = ε( E ( E E) ) ε E + E E 其中 = φ ( E E) ˆ d + φ ( E E) = A+ B E E E= φ E E= φ E E + φ E E S A= E E ˆ d = E E d ˆ = E E d ˆ + + φ φ φ S S S = = equl potetl o coducto ufce S = ˆ = = = φ ( E E) d φ ( σ σ) d φ ( q q) S S = = = q = q, equl chge o ech coducto (o chge de ) B= φ E E = φ ρ ρ = ( E E) ˆ d φ ( E E) ε( ) ε φ + S dx= A+ B= W W = E E Edx Edx + E Edx = ε E + E E The ulted coducto to the ego lowe the electottc eegy

9 .7 A volume vcuum bouded by ufce S cotg of evel epte coductg ufce S. Oe coducto held t ut potetl d ll the othe coducto t zeo potetl. () Show tht the cpctce of the oe coducto gve by C = ε Φ d x whee Φ(x) the oluto fo the potetl. (b) Show tht the tue cpctce C lwy le th o equl to the qutty C [ Ψ] = ε Ψ whee Ψ y tl fucto tfyg the boudy codto o the coducto. Th vto pcple fo the cpctce tht yeld uppe boud. () W = = j= ε ε = d = d = C = C = C φ j j φ = j= C j j C= ε φ d Becue thee oly oe coducto, o =, (b) Let Ψ ( x, λ) = φ( x) + λf ( x), f ( x) bty fucto Ψ( x, λ) & φ( x) hve me B.C. Ψ ( x, λ) = φ( x) = φ( x) + λf( x) λf( x) = S S S S S [ ] C Ψ= ε φ( x) + λf( x) = C + ελ φ( x) f( x) + ε λ f( x) = ε φ x + ε λ φ x f x + ελ f x φ( x) f( x) = ( f( x) φ( x)) ( f( x) φ( x)) = ˆ ( f( x) φ( x)) d ( f( x) φ( x)) = ˆ ( f( x) φ( x)) d ( f( x) φ( x)) S S = σ S f( x) d ( f( x) φ( x)) = ( f( x) φ( x)) BC.. f( x) =, o coducto ufce S C[ Ψ= ] C ( f( x) φ( x)) + ε λ f( x) = C+ ελ f( x) φ( x) ( o chge o ) = C[ Ψ ] = ελ f( x) ε = λ =, C[ Ψ ] = C mmu λ C[ Ψ ] = C+ ελ f( x) C

10 .9 Fo the cyldcl cpcto of Poblem.6c, evlute the vto uppe boud of Poblem.7b wth the ïve tl fucto,ψ(ρ)=(b-ρ)/(b-). Compe the vto eult wth the exct eult fo b/=.5,,. Expl the ted of you eult tem of the fuctol fom of Ψ. A mpoved tl fucto teted by Coll(pp ). εb C=, whee b> b b ρ Ψ Ψ ( ρ) = Ψ ( ρ) = ρˆ = ρˆ, By.7(b) b ρ b b ε b ε b C[ Ψ= ] ε Ψ ( ρ) d= d = ( b ) ( b ) b C[ Ψ] = C b b b/ =.5 C[ Ψ] (.5) = =.5556 C.5(.5 ) b/ = C[ Ψ] = =.6667 C ( ) b/ = C[ Ψ] = = C ( )

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