24-2: Electric Potential Energy. 24-1: What is physics

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1 D. Iyad SAADEDDIN Chapte 4: Electc Potental Electc potental Enegy and Electc potental Calculatng the E-potental fom E-feld fo dffeent chage dstbutons Calculatng the E-feld fom E-potental Potental of a chaged solated conducto 4-: Electc Potental Enegy When an electostatc foce acts between two o moe chaged patcles wthn a system of patcles, we can assgn an electc potental enegy U to the system. If the system changes ts confguaton fom an ntal state to a dffeent fnal state f does wok W on the patcles But s a consevatve foce wok (W) depends on the end ponts ( and f) and not on the path. As fo othe consevatve foces, the wok W done by electostatc foce esults n change of potental enegy ( U) of the system o we can wte 4-1: What s physcs Consevatve foce lke Gavtatonal foce W c = - U Constuct potental enegy U = mgh 1 U = kx Spng elastc foce Othe example expementally dscoveed by physcsts and engnees s Electcal foce (F e ) We also can constuct a potental enegy wok done by the electc consevatve foce W c = W Fe = - U. Once we have potental enegy we can apply pncple of consevaton of mechancal enegy ( E mech = K + U = 0 ) fo closed system nvolvng the foce and wok enegy theoem (W = K) 4-: Electc Potental Enegy If we assume that the patcle moved fom nfnty (taken as efeence whee U =0) to a fnal poston wthn an E-feld of othe chaged patcle o patcles (whee U f = U) wok done by the electostatc foces F between the patcles dung the move n fom nfnty s W U = U U U o smply W U = f : Nea Eath s suface the E-feld has the magntude E = 150 N/C and s dected downwad. What s the change n the electc potental enegy of a eleased electon when the electostatc foce causes t to move vetcally upwad though a dstance d = 50 m U W F. d qe. d qed cosθ U qed cos180 ( )(150)(50)( 1) dung the 50 m ascent, U of the electon deceases by J 14 J

2 4-3: Electc Potental Wok o U, s dependent on the magntude of the chage, q. We can defne a quantty ndependent of the chage and only has an attbute of the electc feld We can defne the potental enegy pe unt chage at a pont n an electc feld s called the electc potental V (o smply the potental) at that pont Scala quantty 4-3: Electc Potental: wok done by appled foce If a patcle of chage q s moved fom pont to pont f n an electc feld by applyng a foce to t, the appled foce does wok W app on the chage whle the electc feld (electostatc foce) does wok W on t and snce Between any two ponts and f n an E-feld, the potental dffeence V s: and Wok done by appled foce Wok done by E-feld o E-foce Wth U = -W ( V between two ponts n an E-feld) Note that U o W app can be postve, negatve, o zeo dependng on the sgns and magntudes of q and V. 4-3: Electc Potental Settng U = 0 at nfnty V = 0, Hence fnal potental V f = V s: (Potental at some pont n an E-feld) V = J The SI unt fo potental s volt (V) whee Ths unt allow us to expess the E-feld n an othe unt (V/m); Enegy s unt s J = N m N = J / m and Potental unt s V = J / C C = J / V By unt conveson N C J / m Electc Feld unts The Electon-Volt 1eV = e ( 1V ) = ( C)( 1J C) = J C N C = = V C J / V N wok o enegy can be expessed n unts of ev (unt of enegy) m 4-4: Equpotental Sufaces If a patcle of chage q s moved between adjacent ponts ( and f) that have same potental (V = V f ) no wok s done by the E-feld o appled foce to move ths patcle ( V = 0 W= W app = 0 fo any path taken; snce U and hence V depends on end ponts). Such adjacent ponts of same electc potental foms what s called Equpotental Suface. Equpotental suface can be ethe an magnay suface o a eal, physcal suface

3 4-4: Equpotental Sufaces A famly of equpotental sufaces (of potentals V 1, V, V 3, and V 4 ) assocated wth the electc feld due to some dstbuton of chages s show below. The wok done by the E-feld wle movng chaged patcle of chage q s: Fo path I W = 0 ( V = 0) 100 V 80 V 60 V 40 V Fo path II W = 0 ( V = 0) Fo path III W = - q V= -q (V -V 1 ) = -q (80-100) = 0 q Fo path IV W = -q V = - q (V-V1) = -q (80-100) = 0 q Fo the paths III and IV; If q s +ve W s +ve and If q s ve W s -ve the potental between two ponts and f can be calculated f we know along the path between the two ponts. consde the followng abtay path fo test chage q 0 n The wok dw done by electostatc foce dung a dsplacement s ntegatng but thus 4-5: Calculatng the potental fom the feld If we set V = 0 and V f = V Ths s the potental V at any pont f elatve to the zeo potental at nfnty. 4-4: Equpotental Sufaces Equpotental sufaces ae always pependcula to electc feld lnes and thus to pependcula to. We can have a sees of sufaces (as shown) pependcula to and each suface has same potental as shown Fo dffeent E-feld confguatons, we wll have dffeent equpotental suface confguatons as shown 4-5: Calculatng the potental fom the feld: two ponts and f sepaated by a dstance d n a unfom electc Feld. Fnd the potental dffeence V f -V by movng a postve test chage q 0 fom to f a) along the path f (paallel to the feld decton) and b) along the path cf shown. E-feld always dected fom hghe potental to lowe potental V > V f Unfom Feld Pont Chage Electc Dpole

4 4-5: Calculatng the potental fom the feld: a) along the path f (paallel to the feld decton) a) along the path cf a) 4-5: Calculatng the potental fom the feld: Ex: Moton of a Poton n a Unfom Electc Feld. A poton eleased fom est, t undego a dsplacement of 0.5 m n the decton of E-feld as shown. Fnd a) V and U between A and B, b) the speed of the poton at pont B 0 because (o both and c have same potental) s s the length of the lne cf wth sn 45 = cos 45 = d/s As expected, same esult as n (a) because the potental dffeence does not depend on the path 4-5: Calculatng the potental fom the feld: Ex: Two conductng paallel plates sepaated by 0.3 cm, connected to a 1V battey. Calculate the E-feld magntude between the plates 4-5: Calculatng the potental fom the feld: contnued fom pevous Ex: Moton of a Poton n a Unfom Electc Feld. b) the speed of the poton at pont B E between two plates s unfom (constant) V Ed V 1V E = = d E = 4 10 V / m m b)

5 4-6: Potental due to a pont chage To calculate the potental at some pont P that les dstance fom a postve chage q. consde we move a patcle of chage q 0 fom pont P (of potental V) to nfnty (of potental V = 0) along adal decton as shown (smplest path) V V = 0 V E.ds q but E.ds = Eds = Ed = k d d 1 0 V Ed kq kq q V = k f potental V due to pont chage q at adal dstance k Also If q s +ve V s postve and f q s ve V s -ve q Fo two o moe pont chages, the electc potental at pont P can be obtaned by applyng the supeposton pncple Fo example: the electc potental at P fot the chages shown s = = = = d / m V = 4-7: Potental due to a goup of pont chage V = 4 = 1 V = k q1 q q3 q4 V = k( ) q1 + q + q3 + q4 = k = q V Whee s the dstance fom pont P to chage q 1.3 m = = 4-6: Potental due to a pont chage: Fo the gven nsulatng sphee of total chage Q, Fnd the E-potental at ponts B, C, and D Lke fo pont at > R chage V = V D V C at < R kq but E = 3 R kq Ed = R 3 R R ( ) Q V B = k Same followed fo C at dstance R ( = R) fom cente V D = Q V C = k R We can use the E-feld to calculate V D nsde a chaged sold sphee of contnuous chage dstbuton kq V D = ( R ) + V 3 C R kq 3 R R 4-8: Potental Due to the Electc Dpole At P located dstance fom dpole cente, the postve pont chage (at dstance (+)) sets up potental V(+) and the negatve pont chage (at dstance (-)) sets up potental V(-).Then the net potental at P s: wth o

6 4-9: Potental Due to a Contnuous Chage Dstbuton Consde a chage dstbuted contnuously on an extended object Remembe chage denstes The potental at some pont P due to element chage dq s. dq dv = k The total potental due to the chaged object s V = k dq 4-9: Potental Due to a Contnuous Chage Dstbuton: chaged dsc Fo the unfomly chaged dsk of adus R and suface chage densty σ. Fnd the electc potental at a pont P located at a dstance z along the pependcula cental z-axs of the dsk Fo the element chage dq = σda= σ πr dr we have Total potental fo the dsc at P s Fom ntegal tables 4-9: Potental Due to a Contnuous Chage Dstbuton: Lne of chages A od of length L located along the x axs has a total chage q and a unfom lnea chage densty λ. Fnd the electc potental at a pont P located on the y axs a dstance d fom the ogn Fo the element chage dq = λdl= λdx we have Total potental fo the od at P s 4-9: Calculatng the Feld Fom the Potental The potental s defned by E Fo E-feld n x-decton V B A E.ds dv E dx E x dv E.ds x dv dx Fom ntegal tables E Fo E-feld n adal-decton dv E d E dv d

7 4-9: Calculatng the Feld Fom the Potental In geneal, Fo E-feld n thee dmensons (Catesan coodnates) E = E ˆ + E dv ( E dx + E dy + E dz) E x x y x ˆj + E kˆ x z y E y and z y ds = dxˆ + dyj ˆ + dzkˆ E z z 4-9: Calculatng the Feld Fom the Potental: Contnued fom pevous slde c) Calculate V and E x f pont P s located anywhee between the two chages Ex: fnd E x, E y, and E z E x 6xy x E z z E y (3x + y + z) y y Note that at x = 0 V = 0 and 4-9: Calculatng the Feld Fom the Potental: Ex: An electc dpole, along x-axs, centeed at the ogn and sepaated by a dstance a. Calculate a) electc potental at pont P, b) Calculate V and E x at a pont fa (x >> a) fom the dpole, and c) Calculate V and E x f pont P s located anywhee between the two chages a) Assume P s located at dstance x fom the ogn 4-9: Calculatng the feld Fom the Potental: Calculate the E-feld at a pont P located at a dstance z along the pependcula cental z-axs of a dsk of adus R and has chage densty σ. The electc potental at any pont on the cental axs of a unfomly chaged dsk was found befoe The E-potental at z s b) fo x >> a x a and x (Note that V and E ae zeo at x = nfnty)

8 4-10 : Electc Potental Enegy of a System of pont chages The electc potental enegy of a system of fxed pont chages s equal to the wok that must be done by an extenal agent to assemble the system, bngng each chage n fom an nfnte dstance. If we have a chage q potental at P located at 1 s If we bng (fom nfnty by extenal agent) an othe chage q 1 and place t at pont P wok needed s W = q 1 V the potental enegy of the system can be expessed as Ex: fo the chages shown, a) fnd the total potental at P, b) f chage q 3 s bng to pont P (fom nfnty), fnd the change n t s potental enegy, c) the total potental enegy of the system a) 4-10 : Electc Potental Enegy of a System of pont chages: V P = V = k q1 q V P = k( + 1 ) q q q 1 1 = = : Electc Potental Enegy of a System of pont chages fo system of moe than two chaged patcles we obtan the total potental enegy of the system by calculatng U fo evey pa of chages and summng the tems algebacally Ex: fo the chages shown, a) fnd the total potental at P, b) f chage q 3 s bng to pont P (fom nfnty), fnd the change n t s potental enegy, c) the total potental enegy of the system b) 4-10 : Electc Potental Enegy of a System of pont chages: contnued (U = 0 at ) q = = 5 c) q 1 1

9 4-11 : Potental of a Chaged Isolated Conducto Fo conducto n electostatc equlbum, f we consde two ponts A and B on the suface E-feld to the path connectng the two ponts on the suface (snce E s to suface) hence V A = V B The suface of the conducto s an equpotental suface (all the suface has the same E-potental) 4-11 : Potental of a Chaged Isolated Conducto Fo chage conductng sphee of chage Q (the fgue) a) E-potental at the suface s and s equal to potental nsde the sphee b) Outsde the sphee, the E-potental s deceasng wth c) On the othe hand, the E-feld nsde the conducto = 0, But out sde the conducto t s 4-11 : Potental of a Chaged Isolated Conducto If we consde two ponts, A on the suface and B nsde the conducto Snce E nsde = 0 hence V A = V B = V at the suface The E-potental s constant evey whee nsde the condcuto and equals to the potental on the suface 4-11 : Potental of a Chaged Isolated Conducto Popetes Fo conducto n electostatc equlbum E-feld nsde the conducto s zeo. Chages always esde at the oute suface of the conducto. The feld lnes ae always pependcula to suface. Chage densty s hghest at smallest adus of cuvatue. The suface s an equpotental suface (all suface has same E-potental). Insde the conducto, the potental V s constant and equal to the suface value.

10 4-11 : Potental of a Chaged Isolated Conducto Ex: Two sphecal conductos, n equlbum, ae connected by a conductng we as shown n Fgue. Fnd the ato of the magntudes of the electc felds at the sufaces of the sphees. Snce sphees ae connected by a conductng wes they have same E-potental Summay Electc potental s the electc potental enegy pe unt chage. E-Potental s elated to the electc feld. All ponts nsde a conducto ae at the same potental. But 4-11 : Potental of a Chaged Isolated Conducto Cavty wthn a conducto: fo to ponts (A and B) on a conducto wth cavty, f we take and path between the two ponts though the cavty V V B A B A E. ds = 0 snce V=0 fo all paths, E-feld eveywhee n the cavty must be zeo

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