2 dependence in the electrostatic force means that it is also

Transcription

1 lectc Potental negy an lectc Potental A scala el, nvolvng magntues only, s oten ease to wo wth when compae to a vecto el. Fo electc els not havng to begn wth vecto ssues woul be nce. To aange ths a scala el s ene om whch the electc el may be eve. ecall om gavtaton that the gavtatonal oce s a consevatve oce, whch mples the ollowng attbutes o the gavtatonal oce: I. A potental enegy tem may be euce om whch enegy assocate wth the chages spatal conguaton gavtatonal chage = mass can be evaluate. II. The wo one by the el as a chage s move n a close loop wthn that el s eo, an the wo one on the chage n movng along an abtay path epens only on the begnnng an enng ponts. F l III. Path nepenence s A lne ntegal gvng the wo one by the el ung 'chage' moton n a close loop. A smla epenence n the electostatc oce means that t s also consevatve an the above esults hol o the electostatc oce. Gven the oce, a potental enegy tem eves om F U When F s aally epenent only, then U ˆ lectc el s theeoe avalable va the scala electc potental enegy U.

2 Integatng U, an vng by the test chage enes the electc potental eence The electc potental eence s measue n volts, A. olta What s electc potental enegy? Potental enegy s the enegy assocate wth a patcula physcal conguaton. Wth the gavtatonal oce, mass s elevate to a heght above goun level then ths s a conguaton wth moe potental enegy. The analogous electostatc case must somehow nvolve electc chages beng 'elevate'. Conse two electc pole conguatons n an extenal electc el: Havng agee that gven the was establshe by a lne o + chages, then the st conguaton + 'closest' to + chages o has moe 'potental enegy' as ollows:

3 When a mass s elease om some heght, a net oce om gavty on the object acceleates the mass tanseng ts potental enegy nto netc enegy. Fxng the cente o the pole as an axs o otaton, any small petubaton away om hoontal on the top pole leas to a tanston nto the moe stable n conguaton. The n conguaton coespons to the mnmum electc potental enegy mass on the goun an the st conguaton s at hghe potental enegy. lectc potental enegy s enegy o a chage object n an extenal electc el. I we 'elease' the chage, then wo one by the electc el on the chage: W Done_ by_ the_ el U Fo the pole, as the st conguaton moves to 'goun' U P an the el oes postve wo as the pole 'alls' nto the n conguaton. lectc Potental Fom U s the chage on the test object s emove om the poblem when electc potental s ene. That s, epesents the potental enegy pe unt chage ue the electc el:

4 The electc potental s a popety o an exstng electc el, egaless o whethe o not a chage object s place n the el. Dmensonally, has SI unts J C O N C m olt = Joule/Coulomb an electc els have unts o olts/mete. The connecton to a battey o electc outlets as souces o electcty voltage s that the voltage supple s the esult o a chage sepaaton that can establsh an electc el whch may be use to pouce 'cuents',.e., plug-t-n. Once we have the electc potental n a system contanng electc chages scete o contnuous, we can use s to obtan the electc el. The el-mappng laboatoy s an example whee the el was euce usng a set o eupotental suaces. lectc el ecton s pepencula to eupotentals. lectc Potental o a Pont Chage As n the case o gavty we tae electc potental enegy to be eo at = : Then the electc potental at a locaton om the pont chage may be oun: s

5 Q Q Q 4 Fo a +Q establshng the electc el, s a postve scala. Fo a -Q establshng the electc el, s a negatve scala. Fo a system o scete chages: n n Once a metho o calculatng electc el om electc potental s nown, calculatng the electc potental an evng an electc el s much ease then ong the vecto algeba poblem assocate wth a supeposton o electc els om each chage. lectc Potental om a Dpole

6 Cos Cos Fo >> the bnomal expanson o these two tems gves: +x n ~ +nx ] [ Cos Cos Cos An the pouct -+ s just The esultng potental om the pole s 4 PCos Cos Potental o Contnuous Chage Dstbutons Fo elements o eental chage wthn the stbuton, ntegate to obtan the electc potental as ollows:

7 lectc potental pouce by a nte length o chage wth a chage pe unt length / P X Q x x x x x Fom the gue, The acal has x = *Tan x = *Sec x + = *+Tan = *Sec

8 Sec Tan Tan Tan Sec ln ln Tan Sec ln

9 On-Axs lectc Potental om a Chage Ds A ' ' ' ' ' et u ' u ' '

10 u u Note the asymptotc lmts o the vaable Z. Calculatng lectc Fels om lectc Potentals Statng om the enton o electc potental we can eentate on both ses: s ecall the oce s eucble om the potental enegy as U F The gaent o the electc potental gves the electc el: j y x ˆ ˆ ˆ Chage Ds Potental Is

11 ˆ ˆ lectc Potental negy o a System o Pont Chages Gven one chage pouces an electc potental Then the wo one by an extenal agent n bngng anothe chage nto ths egon om nnty s mnus the wo one by the exstng electc el o U U U U At ths pont the new electc potental s Now the wo one by an extenal agent n bngng anothe chage 3 nto ths egon om nnty s mnus the wo one by the new electc el o U U U U 3

12 The new electc potental s then Subseuent teatons lea to the concluson that the assembly o a goup o n pont electc chages n a nte egon o space eues an amount o wo one U U n n j j j j The electc potental enegy o a system o pont chages s eual to the wo neee to assemble the conguaton, all chages ntally an nnte stance apat. lectc Potental o a Chage Isolate Conucto Fom the enton o electc potental an the conton o electostatcs, geneal popetes o a chage solate conuctos may be euce: s

13 Zeo cuent eues nse the conucto snce ee electons othewse low. Gauss' aw then mples that all excess chage on the conucto eses on ts suace. the suace o a conucto s an eupotental o Imples s the conucto suace. electc el s pepencula at Non-unom conuctos wth shap eges have hghe n these aeas. In a statc extenal el, chages on a conucto estbute to mantan nse: