Physics 0, Lctu 5 Today s Topics nnouncmnts: Homwok #3 on Wbssign by tonight Du (with Homwok #) on 9/4, 10 PM Rviw: (Ch. 5Pat I) Elctic Potntial Engy, Elctic Potntial Elctic Potntial (Ch. 5Pat II) Elctic Potntial Fo Vaious Chag Distibutions Point chags, Continuous distibutions (unifom ing, sph, shll) Mo on Conductos in Elctic Filds, Equipotntials Elctic Potntial Engy, Elctic Potntial Wok quid to mov q 0 fom to in E fild: potntial ngy U W " F us id s #" F id s #q 0 " Eid s Point chags: U() k q q 0 U i< j j k q i q j (wok quid to assmbl chag configuation) Elctic potntial diffnc: V U " Eid s q 0 # V " V Exampl: unifom E fild paalll to path ij q 0 (indpndnt of tst chag q 0 ) V "Ed q 1
Exampl: Unifom Elctic Fild In th unifom lctic fild shown: 1. Find potntial at,c,d,g. If a chag q is placd at, what is th potntial ngy U? D C 3. If now a q is at, what is U? V 0 4. If a q is initially at st at G, d will it mov to o? What is its final kintic ngy? G Rcall: Paticls will mov to minimiz thi final potntial ngy Elctic Potntial: Point Chags Chag at oigin: (choos zo of potntial at infinity) V() " Eid s k q " d k q # q Potntial diffnc: V( ) V( ) k q k q Gnal: V( q ) k " Singl chag # V( q ) k i i " i Multipl chags q " P
Elctic Potntial: Continuous Chag Distibutions Finit chag distibutions: usually st V0 at infinity. If chag distibution is known: V k dq H: distanc b/w souc and obs. point Not: scala intgal Mo pcisly (s boad): V( dq ) k # " : fild (obsvation) point : souc point Calculating Elctic Fild, Elctic Potntial Th ways to calculat E fild: Coulomb s Law: (lctu ) Gauss s Law: (lctu 3,4) Potntial: (lctu 4,today) V " Eid s dv Eid s E x "V "x,e "V y "y,e "V z "z E( dq( ) k # ") " 3 " Eid q in # 0 V( dq ) k # " E "V 3
Exampl: Unifomly Chagd Ring Fo a unifomly chagd ing, show that th potntial along th cntal axis is V k Q x a Solution: (also s boad) V kdq k dq x x k a a dq Q Unifomly Chagd Ring: Elctic Fild Find th lctic fild along th cntal axis. ppoach 1: Supposition. (Eg. 3.18) kdq dex de cos# kxq Ex " dex 3 E x 0 du to symmty k xq ( x a ) 3/ ppoach : divativ of potntial k V(x) Q ( x a ) 1/ E x V " x k xq ( x a ) 3/ 4
Exampl: Unifomly Chagd Sphical Shll Fo unifomly chagd sphical shll. gain, us: V "# Eid s V " V Tip: V is th sam insid E0 gion E 0 < R kq E > R Exampl: Unifomly Chagd Sph Show that th potntial of a unifomly chagd sph is: R Hint: mo convnint to us V "# Eids V " V sinc fom Gauss s law: kq E > R kq E 3 R < R 5
Conductos (Lctu 3 Rviw) Fo conductos, lctons insid abl to mov fly Ky points: F lctons insid th conducto mov und th influnc of any applid lctic fild Elcton distibution: additional lctic filds. Evntually (~10 16 s), ach lctostatic quilibium (E0 insid conducto). No pplid Fild E0 In pplid Fild Conductos in Elctostatic Equilibium Rgadlss of shap : Elctic fild insid conducto is zo ll nt chags sid on th sufac Elctic fild on sufac of conducto always nomal to th sufac, magnitud σ/ε 0 Elctic fild also zo insid any mpty cavity within th conducto shap dg lag fild. Sinc E0 insid conducto, Potntial is th sam thoughout th conducto: Equipotntial 6
Chag Distibution On Conducto E high at small adius of cuvatu (mo chag dnsity) E low (lss chag dnsity) Exampl: Chag Distibution On Conductos (I) Th total chag on this conducting sph is 5q. How is th chag distibutd? Evnly distibutd thoughout th body Q sufac 5q, Q body 0 Non of abov Not: Rgadlss of shap, chag sids only on th sufac of a conducto 7
Exampl: Chag Distibution On Conductos (II) Th total chag on this conducting shll is 5q, How is th chag distibutd? (R out R inn ) Q Inn_sufac.5q, Q Out_sufac.5q, Q body 0 Q Inn_sufac q, Q Out_sufac 4q, Q body 0 Q Inn_sufac 0, Q Out_sufac 5q, Q body 0 E0 Not: Rgadlss of shap, chag sids only on out sufac of a conducto if no chag insid cavity (E insid 0). Exampl: Chag Distibution On Conductos (III) Th total chag on this conducting shll (R out R inn ) is 5q. point chag of q is placd at th cnt. How is th chag distibutd? Q Inn_sufac q, Q Out_sufac 6q, Q body 0 Q Inn_sufac q, Q Out_sufac 4q, Q body 0 Q Inn_sufac 0, Q Out_sufac 5q, Q body 0 q 8
Exampl: Chag Distibution On Conductos (IV) Initially, th total chag on this shll is 5q. point chag of q is placd at th cnt, and th shll is thn goundd. How is th chag distibutd? (R out R inn ) Q Inn_sufac q, Q Out_sufac 6q, Q body 0 Q Inn_sufac q, Q Out_sufac 4q, Q body 0 Q Inn_sufac q, Q Out_sufac 0, Q body 0 Q0 vywh. q Chag Distibution on Conductos: Fild lins and Equipotntial Sufacs Not: quipotntials a nomal to fild lins 9
Equipotntials Dfind as: Th locus of points with th sam potntial. Exampl: fo a point chag, th quipotntials a sphs cntd on th chag. Th lctic fild is always ppndicula to an quipotntial sufac V V " Eid s long quipotntial sufac, no chang in V " Eid s #V 0 Thfo, Eid s 0 E d s Elctic fild ppndicula to sufacs of constant V 10