8.022 (E&M) Lecture 4

Transcription

1 Topcs: 8.0 (E&M) Lecture 4 More applcatons of vector calculus to electrostatcs: Laplacan: Posson and Laplace equaton url: concept and applcatons to electrostatcs Introducton to conductors Last tme Electrc potental: Work done to move a unt charge from nfnty to the pont P(x,y,) It s a scalar! r φ ( r ) = E ds w th E = φ Energy assocated wth an electrc feld: Work done to assemble system of charges s stored n E 1 E U = ρφ () r dv = dv 8 π Volume wth charges Gauss s law n dfferental form: Entre space E = 4πρ Easy way to go from E to charge dstrbuton that created t G. Scolla MIT 8.0 Lecture 4 1

2 Laplacan operator What f we combne gradent and dvergence? Let s calculate the dv grad f (Q: dfference wrt grad dv f?) f f f f = ( ˆ x ˆ y ˆ ) ( ˆ x ˆ y ˆ) x y x y f f f = = x y x y f f f f Laplacan Operator G. Scolla MIT 8.0 Lecture 4 3 Interpretaton of Laplacan Gven a d functon φ(x,y)=a(x y )/4 calculate the Laplacan x y a = ( ) = a 4 f = f = s the second dervatve, the Laplacan gves the curvature of the functon G. Scolla MIT 8.0 Lecture 4 4

3 Posson equaton Let s apply the concept of Laplacan to electrostatcs. Rewrte Gauss s law n terms of the potental E = 4πρ E = ( φ) = φ φ = 4πρ Posson Equaton G. Scolla MIT 8.0 Lecture 4 5 Laplace equaton and Earnshaw s Theorem What happens to Posson s equaton n vacuum? φ = 4πρ φ =0 Laplace Equaton What does ths teach us? In a regon where φ satsfes Laplace s equaton, then ts curvature must be 0 everywhere n the regon The potental has no local maxma or mnma n that regon Important consequence for physcs: Earnshaw s Theorem: It s mpossble to hold a charge n stable equlbrum wth electrostatc felds (no mnma) G. Scolla MIT 8.0 Lecture 4 6 3

4 pplcaton of Earnshaw s Theorem 8 charges on a cube and one free n the mddle. Is the equlbrum stable? No! (does the queston sound famlar?) G. Scolla MIT 8.0 Lecture 4 7 The crculaton onsder the lne ntegral of a vector functon over a closed path : F Γ= Fds rculaton 1 Let s now cut nto smaller loops: 1 and Let s wrte the crculaton n terms of the ntegral on 1 and Γ= F ds = = F ds 1 ' F ds ' F ds s F d 1 ' F ds F ds ' = Fds Fds Γ=Γ Γ 1 G. Scolla 1 MIT 8.0 Lecture 4 8 4

5 The curl of F If we repeat the procedure N tmes: = LargeN Γ= Γ = 1 Defne the curl of F as crculaton of F per unt area n the lmt 0 Fds F curl n ˆ lm 0 where s the area nsde The curl s a vector normal to the surface wth drecton gven by rght hand rule the G. Scolla MIT 8.0 Lecture 4 9 F Stokes Theorem = LargeN LargeN LargeN Fds Γ= Γ = Fds = = 1 = 1 = 1 Fds = LargeN In the lmt 0: curl Fn ˆ and d = 1 LargeN LargeN L argen Γ= curl ˆ curl ( ˆ F n = F n ) = curl F curl F d = 1 = 1 = 1 Γ= Fds (defnton of crculaton) ds = curl Fd Stokes Theorem NB: Stokes relates the lne ntegral of a functon F over a closed lne and the surface ntegral of the curl of the functon over the area enclosed by G. Scolla MIT 8.0 Lecture

6 pplcaton of Stoke s Theorem Stoke s theorem: F ds = The Electrostatcs Force s conservatve: The curl of an electrostatc feld s ero. curl Fd Fds = 0 curl Ed = 0 for any surface curl E = 0 G. Scolla MIT 8.0 Lecture 4 11 url n cartesan coordnates (1) onsder nfntesmal rectangle n y plane centered at P=(x,y,) n a vector fled F alculate crculaton of F around the square: b Fy Fd s = F y (, xy, ) ~ y y ( F, xy, ) y a c y y F Fd s = F ( xy,, ) ~ (,, ) F xy y b x d Fy Fd s = F y (, xy, )( y ) ~ F y ( xy,, ) y c a y y F Fd s = F ( xy,, )( ) ~ (,, ) F xy y d d y c P a b y ddng the 4 compone nts: F F y Fds = y y squareyz G. Scolla MIT 8.0 Lecture 4 1 6

7 url n cartesan coordnates () ombnng ths result wth defnton of curl: Fds curl Fn ˆ lm Fds 0 square F F y (curl F ) x = lm = F x 0 F y x y y y 0 Fds = y y square Smlar results orentng the rectangles n // (x) and (xy) planes x ˆ ˆ y ˆ F F y F F x F y F x curl F = x ˆ y ˆ ˆ F y x x y x y F F F x y Ths s the usable expresson for the curl: easy to calculate! G. Scolla MIT 8.0 Lecture 4 13 Summary of vector calculus n electrostatcs (1) Gradent: In E&M: Dvergence: Gauss s theorem: In E&M: Gauss law n dfferental form url: Stoke s theorem: In E&M: φ,, φ x y E = φ F F F x y E d = EdV x y F = curl F = F Fds = E = 0 S V Fd curl G. Scolla MIT 8.0 Lecture 4 E = 4πρ Purcell hapter 14 7

8 Summary of vector calculus n electrostatcs () Laplacan: φ φ In E&M: Posson Equaton: Laplace Equaton: φ = φ = 0 4πρ Earnshaw s theorem: mpossb e to hold a charge n stable equlbrum wth electrostatc felds (no local mnma) l omment: Ths may look lke a lot of math: t s! Tme and exercse wll help you to learn how to use t n E&M G. Scolla MIT 8.0 Lecture 4 Purcell hapter 15 onductors and Insulators onductor: a materal wth free electrons Excellent conductors: metals such as u, g, u, l, OK conductors: onc solutons such as Nal n H O u Free electrons l Na Insulator: a materal wthout free electrons Organc materals: rubber, plastc, Inorganc materals: quart, glass, G. Scolla MIT 8.0 Lecture

9 Electrc Felds n onductors (1) conductor s assumed to have an nfnte supply of electrc charges Pretty good assumpton Insde a conductor, E=0 Why? If E s not 0 charges wll move from where the potental s hgher to where the potental s lower; mgraton wll stop only when E=0. How long does t take? s (typcal resstvty of metals) E E E. G. Scolla MIT 8.0 Lecture 4 17 Electrc Felds n onductors () Electrc potental nsde a conductor s constant Gven ponts 1 and P the φ would be: P φ = nsde the conductor P Eds = 0 snce E=0 nsde the conductor. P 1 Net charge can only resde on the surface If net charge nsde the conductor Electrc Feld.ne.0 (Gauss s law) External feld lnes are perpendcular to surface E// component would cause charge flow on the surface untl φ=0 onductor s surface s an equpotental Because t s perpendcular to feld lnes G. Scolla MIT 8.0 Lecture

10 orollary 1 In a hollow regon nsde conductor, φ=const and E=0 f there aren t any charges n the cavty E=0 Why? Surface of conductor s equpotental If no charge nsde the cavty Laplace holds φ or mnma φ must be constant E=0 onsequence: Sheldng of external electrc felds: Faraday s cage cavty cannot have max G. Scolla MIT 8.0 Lecture 4 19 orollary charge Q n the cavty wl l nduce a charge Q on the outsde of the conductor Q Why? ppy l Gauss slaw to surface nsde the conductor Ed = 0 because E=0 nsde a conductor = 4 π ( ) Ed Q Q nsde Gauss's law Q = Q Q = Q = ( Q conductor s overall neutral ) nsde outsde nsde G. Scolla MIT 8.0 Lecture

11 orollary 3 The nduced charge densty on the surface of a conductor caused by a charge Q nsde t s σ nduced =E surface /4π Why? For surface charge layer, Gauss tells us that E=4πσ Snce E nsde =0 E =4πσ surface Q nduced σ G. Scolla MIT 8.0 Lecture 4 1 Unqueness theorem Gven the charge densty ρ(x,y,) n a regon and the value of the electrostatc potental φ(x,y,) on the boundares, there s only one funct on φ(x,y,) whch descrbes the potental n that regon. Prove: ssume there are solutons: φ 1 and φ ; they w ll satsfy Posson: φ 1 ( r ) = 4 πρ ( r ) φ ( r ) = 4 πρ ( r ) Both φ 1 and φ satsfy boundary condtons: on the boundary, φ 1 = φ =φ Superposton: any combnaton of φ 1 and φ wll be soluton, ncludng φ 3 = φ φ 1 : φ ( r ) = φ ( r ) φ ( r ) = 4 πρ ( ) r 4 πρ( ) r= φ 3 satsfes Laplace: no local maxma or mnma nsde the boundares On the boundares φ 3 =0 φ 3 =0 everywhere nsde regon φ 1 = φ everywhere nsde regon Why do I care? G. Scolla MIT 8.0 Lecture 4 soluton s THE soluton! 11

12 Unqueness theorem: applcaton 1 hollow conductor s charged untl ts external surface reaches a potental (relatve to nfnty) φ=φ 0. What s the potental nsde the cavty? φ=? φ 0 Soluton φ=φ 0 everywhere nsde the conductor s surface, ncludng the cavty. Why? φ=φ 0 satsfes boundary condtons and Laplace equaton The unqueness theorem tells me that s THE soluton. G. Scolla MIT 8.0 Lecture 4 3 Unqueness theorem: applcaton Two concentrc thn conductve sphercal shells or rad R1 and R carry charges Q 1 and Q respectvely. What s the potental of the outer sphere? (φ nfnty =0) What s the potental on the nner sphere? What at r=0? Q R Q 1 Soluton R 1 Outer sphere: φ 1 =(Q 1 Q )/R 1 Inner sphere φ R1 R1 Q Q Q φ1 = Eds = dr= r R R R 1 R Q Q φ = Because of unqueness: φ( r) = φ r < R 1 R R1 G. Scolla MIT 8.0 Lecture 4 4 1

13 Next tme More on onductors n Electrostatcs apactors NB: ll these topcs are ncluded n Qu 1 scheduled for Tue October 5: just weeks from now!!! Remnders: Lab 1 s scheduled for Tomorrow 58 pm Pset s due THIS Fr Sep 4 G. Scolla MIT 8.0 Lecture