Lecture 3. Electrostatics

Transcription

1 Lecue lecsics In his lecue yu will len: Thee wys slve pblems in elecsics: ) Applicin f he Supepsiin Pinciple (SP) b) Applicin f Guss Lw in Inegl Fm (GLIF) c) Applicin f Guss Lw in Diffeenil Fm (GLDF) C Fll 7 Fhn Rn Cnell Univesiy Field f Pin Chge (GLIF) Cnside pin chge f Culmbs siing By symmey, elecic field cn nly pin in he dil diecin d dv Suund he chge by Gussin sufce in he fm f spheicl shell f dius By symmey, he -field mgniude n he sufce mus be unifm nd pining in he dil diecin Using Guss Lw: ( π ) π π ˆ C Fll 7 Fhn Rn Cnell Univesiy

2 C Fll 7 Fhn Rn Cnell Univesiy The Pinciple f Supepsiin in lecmgneics Mwell s euins e LINAR nd llw f he supepsiin pinciple hld Suppse f sme chge nd cuen densiies,, ne hs fund he nd fields, nd Suppse f sme he chge nd cuen densiies,, ne hs ls fund he nd fields, nd The supepsiin pinciple sys h he sums,, e he sluin f he chge nd cuen densiies, ( ) ( ) nd s ( ) ( ) nd nd A Simple Pf ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) nd C Fll 7 Fhn Rn Cnell Univesiy Tw Pin Chges (SP) We knw h single chge pduces dil field given by: ˆ π Cnside Tw Chges ˆ ˆ π π Tl -field ny pin is he vec sum f he cnibuins fm ech chge TOTAL TOTAL

3 λ Culmbs/m y ds A Chge Ring (SP) θ Quesin: Wh is he l elecic field pin P disnce fm he ing cene n he -is? Symmey Agumen: The l cnibuin he -cmpnen nd ls he y- cmpnen f he field pin P e bh e (why?) The -cmpnen f he cnibuins fm ech elemen ds cn be dded lgebiclly ge he finl nswe: ( y, ) λ π π ( ) cs ( θ ) P λ π π ( ) F disnces >>, he field behves like h f pin chge wih l chge λ π λ π ( y, ) >> π C Fll 7 Fhn Rn Cnell Univesiy λ Culmbs/m A Line Chge (SP) d' L L ' y Quesin: Wh is he l elecic field pin P disnce fm he line cene n he -is? Symmey Agumen: The l cnibuin he -cmpnen nd ls he y- cmpnen f he field pin P e bh e (why?) The -cmpnen f he cnibuins fm ech elemen d cn be dded lgebiclly ge he finl nswe in n inegl fm: ( y, ) L L λ d' λ L π ( ' ) π ( L ) P ( ) F disnces >> L, he field esembles h f pin chge wih l chge λl λ L ( y, ) >> L π C Fll 7 Fhn Rn Cnell Univesiy

4 Field f n Infinie Chge Plne (GLIF) sufce chge densiy (unis: Culmbs/m ) y Symmey Agumen: The chge disibuin is symmeic w nd diecins Theefe, if ny pin hee is n -field cmpnen in he diecin, hee mus ls be -field cmpnen in he diecin Since he field cnn hve -cmpnen pining in bh nd diecins he sme ime, hee cnn be -cmpnen f he field Similly, hee cnn be y-cmpnen f he -field S he field cn nly hve n -cmpnen A Dw Gussin sufce in he fm f cylinde f e A piecing he chge plne Tl elecic flu cming u f he sufce A Tl chge enclsed by he sufce A By Guss Lw: C Fll 7 Fhn Rn Cnell Univesiy A A Sufce Chge Densiy Bundy Cndiin (GLIF) Suppse we knw he sufce nml elecic field n jus ne side f chge plne wih sufce chge densiy Quesin: Wh is he sufce nml field n he he side f he chge plne??? Sluin: Dw Gussin sufce in he fm f cylinde f e A piecing he chge plne Tl flu cming u f he sufce ( ) A Tl chge enclsed by he sufce A?? By Guss Lw: ( ) A A A ( ) ( ) This n eemely impn esul h eles sufce nml elecic fields n he w sides f chge plne wih sufce chge densiy C Fll 7 Fhn Rn Cnell Univesiy

5 Tw Oppsiely Chged Plnes (SP) When he chge plnes e bugh gehe he field cnibuins fm ech plne uside he plnes cncel ech he nd he fields inside he plnes einfce ech he C Fll 7 Fhn Rn Cnell Univesiy Field f Unifmly Chged Sphee I (GLIF) vlume chge densiy (unis: Culmbs/m ) By symmey, he -field mus nly hve cmpnen in he dil diecin (ie in he ˆ diecin) F : Dw Gussin sufce in he fm f spheicl shell f dius Tl flu cming u f he sufce ( π ) Wk in spheicl Tl chge enclsed by he sufce π c-dines By Guss Lw: Check he Answe (des he sluin sisfy Guss Lw?): Les clcule he divegence f he -field in he nswe bined ( ) Using he fmul f divegence in spheicl cdines C Fll 7 Fhn Rn Cnell Univesiy 5

6 Field f Unifmly Chged Sphee II (GLIF) F : Dw Gussin sufce in he fm f spheicl shell f dius Tl flu cming u f he sufce ( π ) Tl chge enclsed by he sufce π π By Guss Lw: π Check he Answe (des he sluin sisfy Guss Lw?): Les clcule he divegence f he -field in he nswe bined ( ) Using he fmul f divegence in spheicl cdines C Fll 7 Fhn Rn Cnell Univesiy Field f Unifmly Chged Sphee III Skech f he -field: Ne h he nn-e field divegence is included in he skech C Fll 7 Fhn Rn Cnell Univesiy 6

7 Field f Unifmly Chged Sphee IV (GLDF) Using Guss Lw in Diffeenil Fm: F : F : ( ) ( ) Muliply bh sides by nd Inege fm (whee ) ( ) ( ) d ( ) sme s befe! d Muliply bh sides by nd Inege fm (whee ) ( ) d ( ) ( ) π ( ) ( ) π sme s befe! C Fll 7 Fhn Rn Cnell Univesiy Field f Chge Diple (SP) Cnside Tw ul nd Oppsie Chges We e ineesed in he field in he plne f he w chges disnce fm he cene f he pi, whee >> d Wk in spheicl c-dines θ C Fll 7 Fhn Rn Cnell Univesiy P T T d d d d cs( θ ) T ˆ d cs( θ ) d d π π cs( θ ) π cs( θ ) ˆ d d T θ sin( θ ) sin( θ ) d sin( θ ) π π π T d π ( cs( θ ) ˆ sin( θ ) ˆ θ ) cs( θ ) cs( θ ) 7

8 Field f Chge Diple Skech f he -field: C Fll 7 Fhn Rn Cnell Univesiy C Fll 7 Fhn Rn Cnell Univesiy 8