Laplace Potential Distribution and Earnshaw s Theorem
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1 Laplae Potential Distibution and Eanshaw s Theoem Fits F.M. de Mul Laplae and Eanshaw
2 Pesentations: Eletomagnetism: Histoy Eletomagnetism: Elet. topis Eletomagnetism: Magn. topis Eletomagnetism: Waves topis Capaito filling (omplete) Capaito filling (patial) Divegene Theoem E-field of a thin long haged wie E-field of a haged disk E-field of a dipole E-field of a line of dipoles E-field of a haged sphee E-field of a polaized objet E-field: field enegy Eletomagnetism: integations Eletomagnetism: integation elements Gauss Law fo a ylindial hage Gauss Law fo a haged plane Laplae s and Poisson s Law B-field of a thin long wie aying a uent B-field of a onduting haged sphee B-field of a homogeneously haged sphee Laplae and Eanshaw
3 Laplae and Eanshaw. Eleti Field equations: Gauss Law and Potential Gadient Law. Laplae and Poisson: deivation 3. Laplae and Poisson in dimension 4. Chage-fee spae: Eanshaw s Theoem Finite-Elements method fo Potential Distibution 5. Laplae and Poisson in and 3 dimensions Laplae and Eanshaw 3
4 Eleti Field Equations Gauss: integal fomulation: Id. diffeential fomulation: E(, y, z) (, y, z) S E ds e... Ee... d E... Potential: integal fomulation: Id. diffeential fomulation: E e B A E dl (, y, z) e... ee Laplae and Eanshaw 4 B A......
5 Eleti Field Lines and Equipotential Sufaes E(, y, z) (, y, z) E E = onst. Laplae and Eanshaw 5
6 Laplae and Poisson: deivation Gauss: E(, y, z) (, y, z) Potential: E Laplae / Poisson : e e y E... z Laplae and Eanshaw 6 = : fee spae (Laplae) : mateials (Poisson)
7 Laplae and Poisson in dimension = : fee spae (Laplae) : mateials (Poisson) Calulate () fo = by integation of Laplae equation E d d d d E ( ) ' - Bounday onditions: at and at : ( ) ( ) Laplae and Eanshaw 7
8 Laplae and Poisson in dimension E - = : fee spae (Laplae) : mateials (Poisson) Calulate () by integation of Poisson s equation. d d ( ) E Laplae and Eanshaw 8 Assume =onst.: Bounday onditions at and Paaboli behaviou '
9 Laplae and Poisson in dimension a E - a = : fee spae (Laplae) : mateials (Poisson) ( ) Assume =onstant: Bounday onditions at and Speial ase: = ; = and = a ; = Calulate () and E() a ( ) Laplae and EEanshaw ( ) ( a) 9 a ' a
10 Laplae and Poisson in dimension a ( ) a Assume =onst.: Bounday onditions: at and Speial ase: = ; = and = a ; = /a / Laplae and Eanshaw
11 Laplae in dimension = : fee spae (Laplae) : mateials (Poisson) E d d d d ( ) ' E Bounday onditions at and : - ( ) ( ) Laplae and Eanshaw
12 Laplae in dimension: Eanshaw d d d d E ( ) ' Eanshaw: E If no fee hage pesent, then: Potential has no loal maima o minima. - Consequenes:. is linea funtion of position. at eah point is always in between neighbous Laplae and Eanshaw
13 Laplae in dimension: Eanshaw Eanshaw: If no fee hage pesent, then: Potential has no loal maima o minima. Consequenes:. is linea funtion of position. at eah point is always in between neighbous Numeial method fo alulating potentials between boundaies:. Stat with zeo potential between boundaies. Take aveages between neighbous 3. Repeat and epeat and... Laplae and Eanshaw 3
14 Laplae in dimensions: Eanshaw Potential =f (,y) on S? S? y Eanshaw: If no fee hage pesent, then: Potential has no loal maima o minima. Solution of Laplae y (, y) will depend on boundaies. Patial diffeential equation Laplae and Eanshaw 4
15 Laplae / Poisson in 3 dimensions Spatial hage density: =f (,y,z) Potential = f (,y,z)? Bounday onditions:, and 3 = f (,y,z) z 3 y Solution of Laplae/Poisson: y z will depend on boundaies. 4-D plot needed!? Laplae and Eanshaw 5 (, y) Speial ases: ylindial geomety spheial geomety
16 Laplae / Poisson in 3 dimensions z Speial ase : Cylindial geomety y With z Laplae and Eanshaw ln 6 If dependene only: and boundaies will be f (). Thus: will be f () only Eample: = at and at, and = Calulate = ( ) d d d d ( ).ln ' boundaies : ( ) d d ( and ':onst.) ( ) ln d d ln ln
17 Laplae and Eanshaw 7 Laplae / Poisson in 3 dimensions Speial ase : Spheial geomety sin sin sin If dependene only: and boundaies will be f (). Thus: will be f () only Eample: = at and at, and = Calulate = ( ) ( and ':onst.) '. ) ( d d d d d d d d / / / / ) ( ) ( : boundaies With z y the end
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