Lecture 6. Capacitance

Transcription

1 Lectue 6 Capacitance In this lectue u will lean: Capacitance Capacitance f Sme Simple Stuctues Sme Simple Slutins f Laplace s Equatin Capacitance Cnsie tw metallic bjects cnnecte b a vltage suce ttal chage Q ttal chage Q The ttal chage n bth metal bjects have the same magnitue but ppsite sign The chage is pptinal t the applie vltage: Q The cnstant f pptinalit is calle the capacitance C : Q C Capacitance has units Culmbs/lts Faas Q Me geneall, capacitance is als efine as: C 1

2 Capacitance f Paallel Plates Fm pevius lectues: aea A σ L ε ε A Q E φ φ 0 σ R ε C Q Q ε A Fm pevius lectues: Capacitance f Cncentic Clines E ( ) b ln a b a σ ut ε bln σ in ε aln ( b a) ( b a) Q chage pe Since the stuctue is infinite in the ziectin (i.e. in the iectin cming ut f the slie) ne can nl talk abut capacitance pe unit length (units: Faas/m) C capacitance pe unit length ( π b) ε unit length Q b ln ( b a) Q ln π ε ( b a)

3 Capacitance f Paallel Metal Wies I Cnsie tw infinitel lng metal wies cnnecte b a vltage suce (the wies ae infinitel lng in the ziectin) metal wies f aius a >> a Nee t fin the capacitance pe unit length between them une the assumptin that >> a Line Chage Revisite Stat fm the line chage. λ φ( ) ln π ε λ Culmbs/m 3

4 Then cnsie the line iple Line Chage Diple Revisite λ Culmbs/m λ Culmbs/m λ φ( ) ln π ε Capacitance f Paallel Metal Wies II Lk at the Efiels f the line iple: λ Culmbs/m λ Culmbs/m 4

5 Capacitance f Paallel Metal Wies III Daw imagina clines f aius a aun each wie: Nte that the Efiel lines ae appimatel (but nt eactl) nmal t the suface f the clines (as lng as >> a) λ Culmbs/m λ Culmbs/m The ttal electic flu pe unit length cming ut f the left cline is λ The ttal electic flu pe unit length cming ut f the ight cline is λ Capacitance f Paallel Metal Wies I Nw if ne eplaces the clines an the line chages with metal wies f aius a that ca λ an λ culmbs f chage pe unit length then the Efiel utsie emains appimatel unchange >> a λ Culmbs/m λ Culmbs/m The ttal chage caie b the left wie is λ The ttal chage caie b the ight wie is λ λ φ( ) ln π ε 5

6 Capacitance f Paallel Metal Wies Nw cme back t the iginal pblem with a vltage suce: metal wies f aius a φ( ) λ ln π ε Ptential at Ptential at the the λ left cline ln π ε a λ a ight cline ln π ε >> a Capacitance f Paallel Metal Wies I Finall calculate the capacitance pe unit length: metal wies f aius a >> a Ptential iffeence Capacitance pe λ π ε ln a Q unit length C Q λ π ε λ π ε ( ) ln a ln a 6

7 Slutins f Laplace s Equatin in Spheical Cinates I Fllwing ae the slutins f φ( ) 0 Spheicall Smmetic Pint Chage Slutin: φ( ) A q z A Cnstant Unifm Electic Fiel Slutin φ( ) E cs( θ ) E z θ E ( ) φ( ) E z ˆ Slutins f Laplace s Equatin in Spheical Cinates II A Pint Chage Diple Oiente Alng zais Slutin: φ ( ) cs A ( θ ) q z θ q A Cnstant Ptential Slutin (Tivial Slutin): φ( ) A 7

8 Slutins f Laplace s Equatin in Clinical Cinates I Fllwing ae the slutins f φ( ) 0 Clinicall Smmetic Line Chage Slutin: φ( ) A ln( ) λ A Cnstant Unifm Electic Fiel Slutin φ( ) E cs( φ ) E E ( ) φ( ) E ˆ φ Slutins f Laplace s Equatin in Clinical Cinates II A Line Diple Oiente Alng Ais Slutin: cs A φ( ) ( φ ) λ λ φ λ λ cs ln ln π ε π ε cs φ( ) ( φ ) ( φ ) λ cs π ε ( φ ) A Cnstant Ptential Slutin (Tivial Slutin): φ( ) A 8