Remember: When an object falls due to gravity its potential energy decreases.

Transcription

1 Chapte 5: lectc Potental As mentoned seveal tmes dung the uate Newton s law o gavty and Coulomb s law ae dentcal n the mathematcal om. So, most thngs that ae tue o gavty ae also tue o electostatcs! Hee we want to study the concepts o wok and potental as they apply to the electc eld. In the study o mechancs we talk about wok done by (o on the gavtatonal eld. ample: The wok (W done by gavty when a 1 kg mass (m alls a dstance (d o 1 mete s: Wmgd(1kg(9.8m/s (1m9.8 Joules We also talk about the potental enegy o an object. In geneal : Wok F d Fd cosθ ample: A 1kg mass sttng 1m above the eaths suace has a potental enegy (U o: Umgd(1kg(9.8m/s (1m9.8 Joules The elatonshp between wok done by gavty on an object and ts change n potental enegy s: W done by gavty - U -(U nal -U ntal When ou 1 kg object alls 1m ts potental enegy deceases: U U nal -U ntal 9.8J -9.8J The wok done by the eath s gavtatonal eld s: W done by gavty - U -(-9.8J 9.8J Constant oce : Wok F d Fd cosθ emembe: When an object alls due to gavty ts potental enegy deceases. Imagne nstead o gavty the oce on the object was electostatc. ample: A postve chage ( o 1 C s n an electc eld ( o 9.8N/C that ponts down. How much wok s done by the -eld the chage moves 1 m n the decton o? W -eld d(1c(9.8n/c(1m9.8joules How much does the chage s potental enegy change when t moves 1m n the decton o? U -W done by -eld -(9.8J -9.8J. Kass P13 Sp4 1

2 The lectc Potental Dened It tuns out to be vey useul to dene a uantty called the electc potental (V. The electc eld can be calculated om the electc potental and vsa vesa. The electc potental s just the electc potental enegy pe unt chage: U V electc potental Actually, t s the electc potental deence we want snce n analogy wth potental enegy t s only the change n potental enegy that counts: eample: In ou pevous eample we sad that a 1 kg mass held 1 m above the eath s suace had a potental enegy o 9.8 J. In dong ths poblem we (mplctly assumed that the potental enegy at the eaths suace was J. Moe coectly, we should say the potental enegy deence o a 1 kg mass 1m above the eath s suace s 9.8J. U U U V V V electc potental deence The electc potental deence s a scala uantty. Ths s one o ts vtues. It allows us to calculate the electc eld, a vecto, om a scala! Fnally, we can elate the electc potental deence to the wok done by an electostatc oce: W V V V note the mnus sgn! The unt o potental deence s the volt.. Kass P13 Sp4 1 volt1 joule pe coulomb

3 The lectc Potental Dened contnued In electostatcs we usually set the eeence at nnty (: U. Wth ths eeence V too. We can dene the electc potental V at any pont n an electc eld n tems o the wok (W t takes to move chage om nnty to a pont :. Kass P13 Sp4 V V V W The electc potental s a popety o the electc eld. It s dened ndependent o any chages placed n the electc eld. ample: A postve chage 1C moves 1m n an electc eld o 9.8N/C as n the gue. a The wok done by the -eld s: W -eld dcosθdcos(18-(1c(9.8n/c(1m-9.8j b The chage s potental enegy deence s: U -W done by -eld -(-9.8J9.8J c The electc potental deence s: V U/9.8 volts An mpotant popety o the potental deence s that ts value s ndependent o the path taken to get om pont to pont. a y d Let s calculate the potental deence n gong om to by two deent outes. b the dect oute to : V U/(-W -eld /-dcosθ-dcos(-d the ndect oute to : along path to a: V as no wok s done snce: W -eld ycosθycos(9 along path a to b: V U/(W -eld /-dcosθ-dcos(-d along path b to : V as no wok s done snce: W -eld ycosθycos(7 Thus V s ndependent o the path taken! The electostatc oce s a consevatve oce and theeoe the potental deence s ndependent o path. 3

4 upotental Suaces It s also useul to speak o eupotental suaces o lnes. These ae ponts n space at the same potental. Snce along an eupotental suace we have V -V (duh! no wok s done movng along an eupotental path. upotental lnes and a pont chage. The gue on the ght shows the electc eld lnes (adally outwad and the eupotental lnes (concentc ccles o a postve pont chage. Caeul nspecton o the geomety shows that the lnes o the electc eld and the eupotental ae pependcula to each othe. Ths s tue n all ccumstances. I t wee not tue then thee would be component o the electc eld along an eupotental and theeoe wok would be done movng along an eupotental. But ths would volate the denton o an eupotental! HW Fg. 5-3 The gue on the ght shows a constant electc eld and ts lnes o eupotental. As epected, the electc eld lnes and the eupotentals ae pependcula to each othe.. Kass P13 Sp4 4

5 Calculatng the lectc Potental om the Feld We can get an epesson o the potental deence n tems o the -eld usng wok (W. Hee we move a postve test chage a dstance s n an electc eld. dw F ds ds The total wok done movng the chage a dstance s (om pont to n an electc eld s: W F ds ds Usng the denton o potental deence we nd: V V cosθds θ s the angle between and ds. ds W ds I we choose V we get: V Let s calculate the potental om a pont chage ( electc eld. Ou path takes us om to, denng V : d V V V ds cosθd d V We pcked a path whee and d wee paallel (θ. ds. Kass P13 Sp4 5

6 The lectc Potental (V o a Goup o Chages We can calculate the V due to a goup o pont chages by etendng ou esult o a pont chage: V V potental at a dstance om a sngle pont chage n 1 potental at pont (,y,z om n pont chages It s mpotant to emembe the ollowng about V: We ae assumng that V at nnty. Ths s the potental at some pont n space (, y, z: VV(, y, z s dstance that chage ( s om the pont (,y,z. s always postve. v The chage can be postve o negatve. The sgn o V depends on the sgns o the s v The potental s a scala uantty and we ae usng supeposton to calculate ts value hee. ample: Fou chages ae at the cente o a suae wth sde L as shown n the gue. 1 What s the potental at the cente o the suae? The dstance o each chage to the cente o the suae s L/ 4 C C C C Vcente L / L / L / L / C C -C -C 1. Kass P13 Sp4 What s the potental at, a pont mdway between the two postve chages? The negatve chages ae (5/4 1/ L om. C C C C 4C V [ 4 / 5] L / L / L 5 / 4 L 5 / 4 L 6

7 The Potental Due to a Contnuous Chage Dstbuton We can deve the epesson o the potental due to a contnuous chage dstbuton n a ashon smla to the one used o the electc eld om a chage dstbuton. d V dv πε 4 Some thngs to note about dv: s the dstance to d. It s always postve. d can be postve o negatve dependng on what type o chages we have. V s a scala!! No dot poduct n the ntegal.. Kass P13 Sp4 wanng! V s the potental not Volume So, to nd the potental (assumng t s zeo at nnty we must do the ollowng ntegal: 1 d V ample: Suppose a chage s unomly dstbuted n a ccle o adus as shown n n the gue. What s the potental at the cente o the ccle assumng V? Hee we have a lnea chage densty λ. Note: o ths poblem s constant (adus o ccle. Pevously we ound that dλds, and usng sac length (sθ we get dλdθ. d ds d V 1 1 λ 1 λ θ λ λ(π dθ π 4 π λ ε πε The last step used λ/(π 7

8 The Potental Due to a Contnuous Chage Dstbuton contnued Let s do a moe challengng poblem, one whee the dstance to the chage s not constant. d ample: A thn unomly chaged od o length L wth lnea chage densty λ. The dstance om d to P s: h P Fo a lnea chage densty along a lne we have: dλd h L/ V 1 d 4 1 πε L / λd L / h L/ Fo ou poblem we get:. Kass P13 Sp4 Ths ntegal s gven n App., as #17: h d ln( L / λ d λ λ L / h ( L / V 4 [ln( L / h ( L / ln( L / h ( L / ] ln πε 4 / h πε L L / h ( L / We can smply the above wth a bt o algeba to get: V λ πε L / ln h h ( L / h OK, so ths wasn t so easy. At least we ddn t have to use vectos.. 8

9 Calculatng the lectc Feld om the Potental One o the geat thngs about the potental s that t s elated to the electc eld by a devatve! Pevously, we dened the potental to be: V V V W ds Fo an nntesmal change n the potental: V dv and we can wte: dv ds Let s consde the case whee the electc eld s n the decton ( and ds s also n the decton (dsd. dv We can eaange ths to ead: ds dv d The component o the electc eld s just the negatve o the potental wth espect to! Ths esult can be genealzed o cases whee the electc eld has components along, y, o z: y y d z z. Kass P13 Sp4 9

10 Calculatng the lectc Feld om the Potental contnued Let s ty a ew eamples to see how ths woks. ample: Let s calculate the electc eld o a pont chage om the potental o a pont chage. y-as We want to calculate at (,y a chage s at (,. The potental at (,y due to a pont chage at (, s: (, y V (, y θ -as y Accodng to ou pescpton we can calculate and y usng devatves: y (, y (, y y ( y y ( y 1/ 1/ ( ( y y y 3 / 3 / 1 1 y To check we got the coect answe let s wok the poblem by calculatng the electc eld. We know that the magntude o the electc eld at a dstance away om a pont chage s: The components o the electc eld and y ae: Same as what we got cosθ snθ om takng devatve cosθ y snθ o potental! πε 4 cosθ snθ. Kass P13 Sp4 1

11 Calculatng the lectc Feld om the Potental contnued We can use the elatonshp between the potental and electc eld to show that the potental n a conducto must be zeo. A potental: V(,yc, wth c a constant has electc eld: (, y c y (, y y c y Thus a conducto s an eupotental nsde and on ts suace! Sometmes we don t have an euaton that descbes the potental but nstead we have a bunch o measuements o the potental at deent ponts n space. We can estmate the electc eld stength usng - V/. ample: Suppose we have the ollowng measuements: (m V(volts The electc eld at ~m s: -(-1/( V/m The electc eld at ~1m s: -(- (-5/( V/m s pontng n the decton. Kass P13 Sp4 11