PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Potental Dsclame: These lectue notes ae not meant to eplace the couse textbook. The content may be ncomplete. Some topcs may be unclea. These notes ae only meant to be a study ad and a supplement to you own notes. Please epot any naccuaces to the poesso. Wok and Potental Enegy Applyng a oce ove a dstance eues wok: W = Fd Fand dae constant W 12 = F ds othewse to move the object om ntal pont to nal pont The wok done by a oce on an object to move t om pont to pont s opposte to the change n the potental enegy: ( ) W = Δ U = U U In othe wods, the wok expended by the oce s postve, the potental enegy o the object s loweed. Fo example, an apple s dopped om the banch o a tee, the oce o gavty does wok to move (acceleate actually) the apple om the banch to the gound. The apple now has less gavtatonal potental enegy. These concepts ae ndependent o the type o oce. So the same pncpal also apples to the electc eld actng on an electc chage. We dene the electc potental as the potental enegy o a postve test chage dvded by the chage 0 o the test chage. U V = 0 It s by denton a scala uantty, not a vecto lke the electc eld. The SI unt o electc potental s the Volt (V) whch s 1 Joule/Coulomb. The unts o the electc eld, whch ae N/C, can also be wtten as V/m (dscussed late). Changes n the electc potental smlaly elate to changes n the potental enegy: ΔU Δ V = 0 D. Acosta Page 1 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental So we can compute the change n potental enegy o an object wth chage cossng an electc potental deence: Δ U = Δ V Ths motvates anothe unt o potental enegy, snce oten we ae nteested n the potental enegy o a patcle lke the electon cossng an electc potental deence. Consde an electon cossng a potental deence o 1 volt: ( )( ) 19 19 1.6 10 C 1 V 1.6 10 J = 1 ev Δ U = Δ V = eδ V = = Ths s a tny numbe, whch we can dene as one electon-volt (abbevated ev ). It s a basc unt used to measue the tny eneges o subatomc patcles lke the electon. You can easly convet back to the SI unt Joules by just multplyng by the chage o the electon, e. A common conventon s to set the electc potental at nnty (.e. nntely a away om any electc chages) to be zeo. Then the electc potental at some pont just ees to the change n electc potental n movng the chage om nnty to pont. Δ V = V V V The wok done by the electc eld n movng an electc chage om nnty to pont s gven by: ( ) W = Δ U = Δ V = V V = V whee the last step s done by ou conventon. But keep n mnd that t s only the deences n electc potental that have any meanng. A constant oset n electc potental o potental enegy does not aect anythng. Electc Potental om Electc Feld Consde the wok done by the electc eld n movng a chage 0 a dstance ds: dw = F ds= E ds 0 The total wok done by the eld n movng the chage a macoscopc dstance om ntal pont to nal pont s gven by a lne ntegal along the path: W = 0 E d s Ths wok s elated to the negatve change n potental enegy o electc potental: D. Acosta Page 2 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental W 0 ( ) = Δ V = V V Δ V = V V = E d s= E d s The last step changes the decton o the ntegaton and eveses the sgn o the ntegal. Eupotental Suaces Eupotental suaces ae suaces (not necessaly physcal suaces) whch ae at eual electc potental. Thus, between any 2 ponts on the suace ΔV=0. Ths mples that no wok can be done by the electc eld to move an object along the suace, and thus we must have E ds = 0 Theeoe, eupotental suaces ae always pependcula to the decton o the electc eld (the eld lnes). Feld lnes Eupotental lnes The potental lnes ndcate suaces at the same electc potental, and the spacng s a measue o the ate o chage o the potental. The lnes themselves have no physcal meanng. Potental o a Pont Chage Let s calculate the electc potental at a pont a dstance away om a postve chage. That s, let us calculate the electc potental deence when movng a test chage om nnty to a pont a dstance away om the pmay chage. Δ V = V V = E d s D. Acosta Page 3 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Let us choose a adal path. Then E ds= E ds snce the eld ponts n the opposte decton o the path. Howeve, we choose ntegatng vaable d, then ds = d snce ponts adally outwad lke the eld. We thus have: Δ V = E ds= Ed d d 1 2 2 = K = K = K = K Snce the electc potental s chosen (and shown hee) to be zeo at nnty, we can just wte o the electc potental a dstance away om a pont chage : ( ) V = K It looks smla to the expesson o the magntude o the electc eld, except that t alls o as 1/ athe than 1/ 2. We also could ntegated n the opposte sense: Δ V = V V = E d s Then E ds= E d Δ V = V = E ds= Ed d d 1 = K = K = K = K 2 2 V = K Potental o Many Pont Chages By the supeposton pncpal, the electc potental asng om many pont chages s just: V = K whee s the chage o the th chage, and s the dstance om the chage to some pont P whee we wsh to know the total electc potental. The advantage o ths calculaton s that you only have to lnealy add the electc potental asng om each pont chage, athe than addng each vecto component sepaately as n the case o the electc eld. D. Acosta Page 4 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Dpole y + + P Let s see how to calculate the electc potental at pont P due to an electc dpole. By the supeposton pncple, the total potental s: V = V+ V = K + + V = K + d θ x whee + s the dstance om the postve chage to pont P, and - the dstance om the negatve chage. Now o lage dstances, electc dpole. d dcosθ, whee d s the sepaaton o the + dcosθ pcosθ V = K = K 2 2 whee = + and p d s the electc dpole moment. Potental o Contnuous Chage Dstbutons Potental between 2 Paallel Plates Let s calculate the electc potental deence between 2 lage paallel conductng plates sepaated by a dstance d, wth the uppe plate (denoted + ) at hghe electc potental than the lowe. + Fom what we leaned by Gauss s Law and conductos, we know that the electc eld σ asng om a conducto wth a chage densty σ s E =. It s thus a constant between ε 0 the two plates n ths example. The electc potental deence s gven by a lne ntegal: E ds D. Acosta Page 5 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental + Δ V = V+ V = E d s E d s = E ds Δ V = E d opposte dectons Anothe way to vew ths esult s that we apply an electc potental deence between two conductng plates (lage compaed to the sepaaton d), the magntude o the electc eld between them s: E ΔV = d Ths motvates the altenate unts o electc eld o V/m. See moe examples n the textbook on contnuous chage dstbutons! Electc Feld om Electc Potental We have seen n the pevous example o the electc potental between two paallel plates, that E ΔV = Δ s whee Δs s the spacng between the plates, whee the path s paallel to the eld decton (and pependcula to eupotental suaces). In act, the eld ponts n decton opposte to nceasng electc potental deence along path s: ΔV E= s ˆ Δ s Now n the nntesmal lmt, dv E= s ˆ ds whch apples to the eld calculated n any egon, unom o not. Wtng ths nto the usual Catesan coodnates: E = V whee s the gadent opeato. It s a shot-hand o: D. Acosta Page 6 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental V Ex = x V Ey = y V Ez = z So the electc eld s elated to the negatve ate o change o the electc potental. Ths s a specc manestaton o a moe geneal elaton that a oce s elated to the ate o change o the coespondng potental enegy: F = U ( n one dmenson: du F = ) dx Fo the case o the electc eld, F= E and U = V, so E= V E = V D. Acosta Page 7 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Conductos and Electc Potental Recall that the valence electons n a conducto ae ee to move, but that n electostatc eulbum they have no net velocty. Anothe conseuence o ths s that: ΔV = 0 acoss a conducto I not, electons would move om hghe to lowe potental, and thus not be n statc eulbum. Ths mples that the suace o the conducto, no matte what shape, s also an eupotental suace. We leaned aleady that the electc eld s pependcula to the suace o a conducto (othewse chages would acceleate along the suace), and eupotental lnes ae always pependcula to the electc eld lnes. Eupotental lne Suace o conducto Example: 1 1 2 2 Let s consde as an example 2 conductng sphees connected by a thn conductng we. One sphee has a smalle adus ( 1 ) than the othe ( 2 ); and the chages on the two sphees ae 1 and 2 espectvely. By the above agument, all suaces ae at the same electc potental. Let s ase the ente system to potental V wth espect to a pont nntely a away. The two sphees must have the same potental, so by euatng the potental enegy o each chaged sphee (whch s the same as that o a pont chage at the cente o the sphee) we get: 1 2 K = K 1 2 1 1 = 2 2 D. Acosta Page 8 9/12/2006
PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Now let s detemne the suace chage denstes. Snce σ =, the last euaton can 2 4π be wtten: σ 4π σ 2 1 1 1 = 2 24π2 2 σ1 2 = σ 2 1.e. the suace chage densty s nvesely popotonal to the adus o the sphee. Now the magntude electc eld at the suace o the sphee s: E 2 1 σ 4π σ = K = = 4πε ε 2 2 0 0 Thus, the eld stength s popotonal to the suace chage densty, whch s nvesely popotonal to the adus o the sphee. Fo a lage enough chage 1 and small enough adus 1, the beakdown electc eld 6 stength n a could be exceeded ( 3 10 V/m) and a dschage (lghtenng bolt) occu. Ths s the bass o a lghtenng od. Let the lage sphee epesent a lage suace, such as the Eath, and the small sphee a small naow pont such as a od. I the two suaces accue a lage chage, such as dung a lghtenng stom, the electc eld s stongest at the naow od and a beakdown s most lkely to occu thee. Do not stand nea tall naow objects (lke tees!) n an electcal stom! D. Acosta Page 9 9/12/2006