Lectue 4 Electic Ptentil In this lectue yu will len: Electic Scl Ptentil Lplce s n Pissn s Eutin Ptentil f Sme Simple Chge Distibutins ECE 0 Fll 006 Fhn Rn Cnell Univesity Cnsevtive Ittinl Fiels Ittinl Cnsevtive Fiels: Vect fiels F f which F 0 e clle ittinl cnsevtive fiels F This implies tht the line integl f un ny clse lp is ze F. s 0 Eutins f Electsttics: Recll the eutins f electsttics fm pevius lectue:. ε E E 0 In electsttics electusisttics, the E-fiel is cnsevtive ittinl (But this is nt tue in electynmics) ECE 0 Fll 006 Fhn Rn Cnell Univesity
Cnsevtive Ittinl Fiels Me n Ittinl Cnsevtive Fiels: If the line integl f F un ny clse lp is ze.. F. s 0. then the line integl f F between ny tw pints is inepenent f ny specific Pth (i.e. the line integl is the sme f ll pssible pths between the tw pints) F. s 0 F. s + F. s 0 pth pth pth B F. s F. s 0 pth B pth pth B F. s F. s pth pth B ECE 0 Fll 006 Fhn Rn Cnell Univesity The scl ptentil: The Electic Scl Ptentil - I ny cnsevtive fiel cn lwys be witten (up t cnstnt) s the gient f sme scl untity. This hls becuse the cul f gient is lwys ze. If F ϕ Then F ϕ ( ) ( ) 0 F the cnsevtive E-fiel ne wites: (The ve sign is just cnventin) E φ Whee φ is the scl electic ptentil The scl ptentil is efine nly up t cnstnt If the scl ptentil φ( ) gives cetin electic fiel then the scl ptentil φ( ) + c will ls give the sme electic fiel (whee c is cnstnt) The bslute vlue f ptentil in pblem is genelly fixe by sme physicl esning tht essentilly fixes the vlue f the cnstnt c ECE 0 Fll 006 Fhn Rn Cnell Univesity
The Electic Scl Ptentil - II We knw tht: E φ This immeitely suggests tht: The line integl f E-fiel between ny tw pints is the iffeence f the ptentils t thse pints E. s ( φ ). s φ( ) φ( ) The line integl f E-fiel un clse lp is ze E. s ( φ ). s 0 ECE 0 Fll 006 Fhn Rn Cnell Univesity The Electic Scl Ptentil f Pint Chge ssumptin: The scl ptentil is ssume t hve vlue eul t ze t infinity f wy fm ny chges Pint Chge Ptentil s E ˆ 4πε D line integl fm infinity t the pint whee the ptentil nees t be etemine 0 E. s ( φ ). s φ( ) φ( ) φ( ) φ( ) E. s E s 4π ε φ( ) 4π ε. 4π ε φ( ) ECE 0 Fll 006 Fhn Rn Cnell Univesity
Electic Scl Ptentil n Electic Ptentil Enegy The electic scl ptentil is the ptentil enegy f unit psitive chge in n electic fiel Electic fce n chge f Culmbs E (Lentz Lw) Ptentil enegy f chge t ny pint in n electic fiel Wk ne by the fiel in mving the chge fm tht pint t infinity Wk ne F. s E. s φ Wk ne n unit chge P.E. f unit chge φ ( ) ( ) [ φ ( ) φ( ) ] φ( ) φ ( ) s Ptentil enegy f chge f Culmbs in electic fiel φ( ) ECE 0 Fll 006 Fhn Rn Cnell Univesity Stt fm:. ε E Use: E φ T get:. ε ( φ ) φ Pissn s n Lplce s Eutin It is nt lwys esy t iectly use Guss Lw n slve f the electic fiels Nee n eutin f the electic ptentil Pissn s Eutin ε If the vlume chge ensity is ze then Pissn s eutin becmes: φ 0 Lplce s Eutin Pissn s Lplce s eutin cn be slve t give the electic scl ptentil f chge istibutins ECE 0 Fll 006 Fhn Rn Cnell Univesity 4
Ptentil f Unifmly Chge Spheicl Shell - I Use the spheicl cinte system F : φ 0 0 φ( ) + F σ ssume slutin: Culmbs/m F must be 0 s tht the ptentil is 0 t F 0 : φ 0 0 ssume slutin: B φ( ) + D Ptentil must nt becme infinite t 0 s B must be 0 ECE 0 Fll 006 Fhn Rn Cnell Univesity Ptentil f Unifmly Chge Spheicl Shell - II Buny cnitins F 0 φ E ( ) D ( ) 0 F φ( ) E ( ) We nee tw itinl buny cnitins t etemine the tw unknwn cefficients n D () t the ptentil is cntinuus (i.e. it is the sme just insie n just utsie the chge sphee) D () t the electic fiel is NOT cntinuus. The jump in the cmpnent f the fiel nml t the shell (i.e. the il cmpnent) is elte t the sufce chge ensity ε ( E E ) ut in σ ε 0 σ ECE 0 Fll 006 Fhn Rn Cnell Univesity σ 5
Sufce Chge Density Buny Cnitin Suppse we knw the sufce nml electic fiel n just ne sie f chge plne with sufce chge ensity σ Questin: Wht is the sufce nml fiel n the the sie f the chge plne? E E?? σ Slutin: Dw Gussin sufce in the fm f cyline f e σ piecing the chge plne Ttl flux cming ut f the sufce ε ( E E) Ttl chge enclse by the sufce σ E E?? By Guss Lw: ε ( E E) σ ε ( E E) σ ( E E ) σ ε This n extemely imptnt esult tht eltes sufce nml electic fiels n the tw sies f chge plne with sufce chge ensity σ ECE 0 Fll 006 Fhn Rn Cnell Univesity Ptentil f Unifmly Chge Spheicl Shell - III F 0 F φ ( 4πσ ) ( ) 4π ε φ ( 4πσ ) ( ) 4π ε σ Sketch f the Ptentil: φ( ) ECE 0 Fll 006 Fhn Rn Cnell Univesity 6
Ptentil f Unifmly Chge Sphee l Pissn n Lplce In spheicl c-intes ptentil cn nly be functin f (nt f θ φ ) F : φ 0 F 0 : φ ε 0 ε ssume slutin: φ( ) + F F must be 0 s tht the ptentil is 0 t ssume slutin: B φ( ) + D + C hmgenus pts By substituting the slutin in the Pissn eutin fin C Culmbs/m pticul slutin Ptentil must nt becme infinite t 0 s B must be 0 Wk in spheicl c-intes C 6ε ECE 0 Fll 006 Fhn Rn Cnell Univesity Ptentil f Unifmly Chge Sphee l Pissn n Lplce F 0 φ( ) D 6 ε F φ( ) Buny cnitins We nee tw itinl buny cnitins t etemine the tw unknwn cefficients n D () t the ptentil is cntinuus (i.e. it is the sme just insie n just utsie the chge sphee) () t the il electic fiel is cntinuus (i.e. it is the sme just insie n just utsie the chge sphee) E () gives: () gives: D 6ε ε ε D ε ECE 0 Fll 006 Fhn Rn Cnell Univesity 7
Ptentil f Unifmly Chge Sphee l Pissn n Lplce F 0 F φ( ) ε 4 π 4π ε φ( ) Sketch f the Ptentil: φ( ) ECE 0 Fll 006 Fhn Rn Cnell Univesity The Pinciple f Supepsitin f the Electic Ptentil Pissn eutin is LINER n llws f the supepsitin pinciple t hl Suppse f sme chge ensity ne hs fun the ptentil φ Suppse f sme the chge ensity ne hs fun the ptentil φ The supepsitin pinciple sys tht the sum chge ensity + ( ) Simple Pf ( φ + φ ) is the slutin f the φ + ε φ ε ( + φ ) ( ) φ + ε ECE 0 Fll 006 Fhn Rn Cnell Univesity 8
Wk in spheicl c-intes Ptentil f Chge Diple Cnsie Tw Eul n Oppsite Chges z We e inteeste in the ptentil t istnce fm the cente f the pi in the plne f the chges, whee >> θ + + P + + cs( θ ) cs( θ ) Ptentil cntibutins fm the tw chges cn be e lgebiclly φ( ) 4π ε + 4π ε 4π ε cs θ cs( θ ) 4π ε ( ) 4π ε + cs( θ ) ECE 0 Fll 006 Fhn Rn Cnell Univesity cs( θ ) φ( ) cs( θ ) 4π ε Fiel f Chge Diple E φ( ) 4πε ( cs( θ ) ˆ + sin( θ ) ˆ θ ) + Sme esult f the E-fiel ws btine in the pevius lectue by supepsing the iniviul E-fiels (the thn the ptentils) f the tw chges ECE 0 Fll 006 Fhn Rn Cnell Univesity 9
Cnsie n infinite line chge cming ut f the plne f slie Ptentil f Line Chge The electic fiel, by symmety, hs nly il cmpnent λ Culmbs/m Dw Gussin sufce in the fm f cyline f ius n Length L pepenicul t the slie Using Guss Lw: ε E ( π L) λ L λ E π ε φ( ) But φ E λ π ε λ φ φ ln π ε Upn integting fm t we get: ( ) ( ) The pblem is tht this slutin becmes infinite t y x Wk in cylinicl c-intes Whee is cnstnt f integtin n is sme pint whee the ptentil is knwn ECE 0 Fll 006 Fhn Rn Cnell Univesity Ptentil f Line Diple Cnsie tw infinite eul n ppsite line chges cming ut f the plne f slie + λ Culmbs/m + y x λ Culmbs/m Using supepsitin, the ptentil cn be witten s: φ( ) λ ln π ε λ ln π ε + + λ ln π ε Questin: whee is the ze f ptentil? The finl nswe es nt epen n the pmete Pints f which + euls - hve ze ptentil. These pints cnstitute the entie y-z plne ECE 0 Fll 006 Fhn Rn Cnell Univesity 0
The D Supepsitin Integl f the Ptentil In the mst genel sceni, ne hs t slve the Pissn eutin: ( ) φ( ) ' ε We knw tht the slutin f pint chge ' sitting t the igin: φ( ) 4π ε T fin the ptentil t ny pint ne cn sum up the cntibutins fm iffeent ptins f chge istibutin teting ech s pint chge ( ') φ ( ) V ' V ' x' y' z' 4π ε ' Check: F pint chge t the igin ( ') δ ( ') δ ( x' ) δ ( y' ) δ ( z' ) ( ') δ ( ') V ' V ' 4π ε ' 4π ε ' 4π ε 4π ε φ( ) ECE 0 Fll 006 Fhn Rn Cnell Univesity ECE 0 Fll 006 Fhn Rn Cnell Univesity