FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1
Recll tht we e inteested to clculte the electic field of some chge distiution. Howeve, sometimes we e given the electic field insted nd we e inteested to find wht kind of chge distiution give ise to tht pticul electic field. The given electic field is not just ny vecto field, it hs to stisfy the govening eqution, nmely : its divegence stisfy E ρ ε nd its culs should e zeo : E In fct these two condition e the necessy condition tht hs to e fulfilled y vecto field if it epesent n electic field. We will exploit these popety to find simple wy to clculte the electic field. Alexnde A. Isknd Electomgnetism 3 Clculting electic field is sometimes cumesome, ecuse of its vectoil ntue. Howeve, eclling the cul-lessless popety of the electic field E we cn define scl function, clled the electic potentil s E Let us tke pth integtion ti of the electic field gives d l Alexnde A. Isknd Electomgnetism
Which y gdient fundmentl theoem we hve dl E dl ( ) ( ) Whee is some efeence point, usully it is chosen such tht the potentil is zeo. Hence we hve the definition of electic potentil in tems of electic field s follows ) E dl Exmple.6 Alexnde A. Isknd Electomgnetism 5 In clculting the electic potentil, side fom the pevious integl fomultion, we cn lso find the potentil y solving the ppopite diffeentil eqution ρ ρ E E ε ε with ppopite oundy condition (this will e the topic of next chpte) The ove eqution is known s the Poisson s eqution. Fo egion whee thee e no chge distiution we otin the Lplce eqution Alexnde A. Isknd Electomgnetism 6 3
We cn clculte the electic potentil fo discete (loclized) chge distiution, y pplying the line supeposition pinciple. Fist conside point chge 1 ) πε q 1 q d πε Thus fo collection of point chges, we hve q 1 n 1 q ) i πε q i 1 i qi Genelizing fo continuous chge distiution 1 ρ( ) ) dτ πε Exmple.7 Alexnde A. Isknd Electomgnetism 1 πε q i 7 P Wht is the dvntge of clculting Electic Potentil? It s often esie to clculte, nd tke its gdient to find. E, thn to clculte E diectly. Resons: is scl; no vecto ddition to get it. Thee e mny situtions in ntue in which cn e egded s constnt ove egion in spce ne whee one would like to know. E. The solution of fo spce etween the constnt- ( equipotentil ) loctions nd the efeence point the pocess of which is clled oundy-vlue polem cn e shown to e unique. Finding y oundy-vlue solution, nd then clculting E, is in these cses usully much esie thn clculting E diectly. Although we stt fom electic potentil (one scl quntity) nd used it to detemine the electic field ( vectoil quntity with thee scl quntity), howeve these thees scl quntity is ctully elted y the govening eqution of the electic field itself. Alexnde A. Isknd Electomgnetism 8
Note the diffeence of the nme potentil in contst with potentil enegy: Electicl Potentil is popety p of the field (depends on loction). Electicl Potentil Enegy is popety of chged oject in n electic field. The choice of efeence point is ctully ity since in most pplictions, wht is impotnt is ctully the potentil diffeence (hence the vlue t the efeence point cncels out). ( ) ( ) Howeve, it is customy to choose the efeence point to e the point whee the potentil is zeo. Alexnde A. Isknd Electomgnetism 9 Summy of Electosttics The eltions etween chge density ρ, electic field nd electic potentil in electosttics cn e summized in the following digm : E 1 ρ( ) ) dτ πε ρ ρ 1 ρ E ˆ dτ πε ρ E E ε ε E ) E dl Alexnde A. Isknd Electomgnetism 1 E 5