Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When the lock flls veticlly down, the gvittionl foce does positive wok, nd the potentil enegy (U) deceses. h W U In this cse, W U U U U mgh When is positive (lock flling), U U U>, nd is negtive. The potentil enegy deceses. When the lock is lifted up, the gvittionl wok is negtive, nd is positive. Gvittionl foce is consevtive foce. U PHYS 53 8W
The wok done y consevtive foce hs 4 popeties.. It is evesile. 2. It is independent of the pth etween the stt nd end points. 3. When the stt nd end points e the sme, totl wok is zeo. 4. The wok cn e expessed s diffeence in potentil enegy. The electic field foce is consevtive, nd the ides ove pply to the wok done in moving chge in n electic field. Fo positive chge moving in the diection of E, U deceses, nd moving opposite to E, U inceses. The opposite is tue fo negtive chge. Fo consevtive foce, the chnge in K.E. is minus the chnge in P.E. K K ( U U ) o K + U K + U Fig. 23.3 PHYS 53 8W 2
Cn esily show tht ecuse the electosttic foce is consevtive, the wok done y this foce is independent of pth. F When foce cts on pticle tht moves fom point to point, the wok done is given y line integl, W l Fd. l Fcosφdl d is the incementl displcement long the pticle s pth, nd φ is the ngle etween F nd l d t ny point. Fig. 23.6 In the figue, nd do not lie on the sme dil line. W Fcosφdl qed since cosφdl d So, the wok done duing smll PHYS 53 8W 3
displcement independent of the pth. dl depends only on the chnge d in the dil distnce, () Electic potentil enegy in unifom field. + + + + + q F d - - - - - + - x Wok done ove some distnce d is W Fd qed Wok done y the field is positive, so the potentil enegy of the pticle deceses. In going fom to, the decese in P.E. is q Ex () Electicl P.E. of two point chges. Conside positive test chge q distnce fom positive chge q. PHYS 53 8W 4
Hee the foce is not constnt with distnce s in the unifom field. If the foce moves the pticle fom to long dil line W qq Fd. 2 4πε W U qq πε ( 4 ) d The wok, nd hence chnge in P.E., depends only on the end points. Hee the wok is positive nd the P.E. chnge is negtive. Recll, the wok done does not depend on the pth etween the end points. Fig. 23.5 Suppose we hve infinity s ou efeence point, whee U nd ing q fom infinity to point. PHYS 53 8W 5
Then qq ( ) U 4πε U Thus, the potentil enegy U of test chge q t ny distnce fom chge q is 4πε qq (electic potentil enegy of two point chges q nd q.) U (c) Electic potentil enegy of sevel point chges. Recll the esultnt electic field is the supeposition of ll individul fields. The wok done to move q to point is the sum of the wok done ginst ech field. So the P.E. of chge q t point is the lgeic sum of the potentils of ech pi of chges involving q. Fig. 23.8 PHYS 53 8W 6
U q q q2 q3 q ( + + +...) 4πε 4πε 2 3 q i i i If q is moved to nothe point, then U is given y the sme expession with the distnces i mesued to point. The wok done in moving q fom to is U U The wok done in inging togethe ll the chges in Fig. 23.8 (not just q ) is given y U 4πε i< j Electic Potentil qq i ij j (i<j so tht we include ech pi only once) Potentil is the potentil enegy pe unit chge. The symol is V, nd the unit is the volt. V U q V, o J/C. PHYS 53 8W 7
Refe ck to Fig. 23.3 on slide 2, nd use the equtions on slide. The wok done y the field in moving the positive chge fom to, W U U The electic field does positive wok, nd the potentil enegy deceses. W U U The potentil is given y V V V q q q The wok done y the field in moving the chge fom to is positive, And so V > V. () Potentil of single point chge. Recll the expession fo the electic potentil enegy of two point chges, q nd q. Hence V U q 4 πε q Wht is the efeence point hee? PHYS 53 8W 8
() Potentil due to collection of point chges. Recll the expession fo the electic potentil enegy fo point chge q nd collection of chges q i. Hence V U q 4 πε q i i i Potentil t point distnce i fom q i. Potentil hee is clculted with espect to the sme point in (). This point is infinity, whee V. (c) Potentil due to continuous distiution of chge. V 4πε dq Recll, potentil is the wok pe unit chge. Suppose positive chge q is moved fom to. W qed. l is the distnce fom dq to the point whee V is eing evluted, gin with espect to inifinity. PHYS 53 8W 9 q
V V Ed. l E cosφdl In this cse W is negtive, nd hence V > V. We see fom this tht the units of E cn e V/m s well s N/C. PHYS 53 8W
Polem 22.48 A solid conducting sphee with duis R cies positive totl chge Q The sphee is suounded y n insulting shell with inne dius R nd oute dius 2R. The insulting shell hs unifom chge density ρ. () Find the vlue of ρ so tht the net chge of the entie system is zeo. () If ρ hs the vlue found in pt (), find the mgnitude nd diection of electic field in ech of the egions < < R, R < < 2R, > 2R. Plot gph of the dil component of E s function of. (c) Do the esults gee with the genel ule tht the electic field is discontinuous only t loctions whee thee is thin sheet of chge? PHYS 53 8W
Exmples of clculting Electic Potentil Polem 23.32 A totl electic chge of 3.5 nc is distiuted unifomly ove the sufce of metl sphee with dius of 24. cm. If the potentil is zeo t point t infinity, find the vlue of the potentil t the following distnces fom the cente of the sphee: () 48. cm, () 24. cm, (c) 2. cm. Recll tht the electic field due to unifomly chged sphee is given y E 4πε q 2 (sme s fo point chge t the cente). The potentil due to point chge is V 4πε q Fig. 23.7 PHYS 53 8W 2
Polem 23.33 A unifomly chge thin ing hs dius 5. cm nd totl chge of 24. nc. An electon is plced on the ing s xis distnce of 3. cm fom the cente of the ing nd is constined to sty on the xis of the ing. The electon is then elesed fom est. () Descie the susequent motion of the electon. () Find the speed of the electon when it eches the cente of the ing. PHYS 53 8W 3
Polem 23.34 An infinitely long line of chge hs line chge density 5.x -2 C/m. A poton (mss.67x -27 kg. nd chge.6x -9 C) is 8. cm fom the line nd moving diectly towd the line t.5x 3 m/s. () Clculte the poton s initil K.E. () How close does the poton get to the line of chge? PHYS 53 8W 4