3.1 Electrostatic Potential Energy and Potential Difference

Transcription

1 3. lectostatc Potental negy and Potental Dffeence RMMR fom mechancs: - The potental enegy can be defned fo a system only f consevatve foces act between ts consttuents. - Consevatve foces may depend only on the poston ; they do not depend on velocty o acceleaton. - Only the dffeence between two values of potental enegy (not the values by themselves) does have a physcal meanng. Ths allows to assgn zeo potental enegy to such a locaton that smplfes the calculatons. (xample; In geneal, one chooses U G = 0 fo gavtatonal enegy of an object on eath suface but f the object can move only nsde the lab one chooses U G = 0 at floo level. Note that one may choose U G = 0 at eath cente o even at nfnty f ths choce smplfes the soluton of consdeed poblem.) - The pncple of enegy consevaton; U G + K = W ext tells that, povded that K=0, the change of potental enegy U G s equal to the wok done by extenal foces. - Consde an object at est on floo whee U G (0) = 0. If you move t up at constant speed and leave on a table at heght y (fg.), the object gets a potental enegy U G (y) = mgy. The extenal foce (you foce) has done a postve wok W ext whch nceases the potental enegy of object (.e. of system object- eath). y () Fext mg Fg. Meanwhle, the ntenal consevatve foce F nt (.e. the weght), does the same amount of wok but ths one s a negatve wok. So, () -Fo an nfntesmal dsplacement dy, the ntenal consevatve foce F nt, does the nfntesmal wok (3) The sgn (-) n (3) tells that when a consevatve foce does postve wok a decease of potental enegy happens. Fom (3) we get the elaton between a consevatve foce and the potental enegy (4) x: ath-object sys; If Oy dected vetcally up U G (y) = mgy and (dected down) - The potental enegy of the object s actually the enegy of the system eath-object and t s due to the gavtatonal feld of the eath. One defnes the potental functon of the gavtatonal feld as follows; The gavtatonal potental G (y) at the locaton y s equal to the potental enegy of a Kg mass located at ths pont of the space, but ts unt s dffeent; n SI system t s [J/kg] nstead of [J]. Ths way: a) One defnes a functon G g * y whch s a chaactestc of gavtatonal feld of the eath. b) Fo a m Kg, one can fnd and ts change fom the elatons U U G and G G G (5) m m c) One can calculate the wok done by gavtatonal feld dung the shft of a mass m nsde t as Wnt UG m* G (6) Wth espect to the system eath - object

2 - t ths pont we have to emnd that the Coulomb s foce s a consevatve foce, too. So, we just adopt all the uppe pesented steps to electc feld and get to defnton of electc potental (x,y,z); a) It s numecally equal the potental enegy of a +C chage at locaton (x,y,z) of electostatc feld. b) Smlaly to (-6) one can get ts elaton to wok done by electc feld (W n ) on fo q =+C as : fom _ 0 _ enegy _ locaton U W nt q q (7) U W nt (8) q q Fom (7): The electc potental s equal to the - ntenal wok spent by feld when q = +C chage moves fom the pont wth 0-potental enegy to the consdeed space locaton; but ts SI unt s [J/C]. Fom (8); The change of electc potental between any two space locatons s equal to - amount of wok done by the feld when a chage +C s shfted between these two locatons; ts SI unt s [J/C]. - When the electc feld 3 shfts a postve chage along ts lnes, t acheves a postve wok, W nt > 0. The elaton (8) shows that = - = -( W nt /q) < 0, whch means that < ; the chage s shfted fom hghe to lowe potental. So, f a postve chage s placed nsde an electc feld, the electc feld tents to shft t vesus locatons wth lowe -values(fg.b). smla stuaton s met nsde gavtatonal feld; objects n fee fall decease the potental enegy by movng vesus eath(fg.a). If a negatve chage (q < 0) s placed nsde a feld, the electc foce wll move t aganst feld lnes. s ( S F ) the feld acheves el postve ntenal wok (W nt > 0), but n ths case = - = - ( W nt /q) > 0 and = - > 0 o >. Ths means that the electc feld pushes the negatve chages vesus hghe values (fg.c). Hgh Hgh Hgh G m +q -q Low Low Low Fg.a Fg. b Fg.c The electc feld wll push a + chage vesus lowe -values and a - vesus hghe -values. - When a chage q s shfted by s nsde an electc feld, the feld does the wok (see 8) whch bngs to * s (9) Fo an nfntesmal shft one gets * ds d (0) d So, d * ds s * ds () and s () ds The elatons (9 -) ae vey mpotant because they elate the two man paametes of the same electc feld and show a way how to calculate one of them f one knows the othe one. -The electc potental s a sot of potental enegy; so, ts value depends only on the locaton. Ths means that ts change depends only on the ntal and fnal postons and not on the path followed fo a shft fom to. Ths comment offes a gude fo smplfyng the calculaton of wok done by Lke unvesal gavtaton law, the Coulomb s law depends only on the postons and not on veloctes o acceleaton. 3 Intenal consevatve foce n the system electc feld - electc chage.

3 the feld when a chage s shfted nsde a feld, especally f t s a non unfom electc feld (fg.3). If a chage q moves nsde a non unfom feld the wok done by the feld can be calculated as follows: Fgue 3 W nt qds qd q( ) ette, keep n mnd that potental dffeence W ds nt q (3) 3. Potental and Potental negy nsde a Unfom Feld -Consde that an extenal foce shfts the chage q nsde the unfom electc feld fom to followng an egula path (Fg.4). s we know, the wok by electc foce ( Fel q ) s Wnt F ds q ds q ds q s el * * * (4) path path path Fg. 4 because the feld s constant (one can take t out ntegal) and the vecto sum of nfntesmal shfts s equal to s. s = - = - W nt /q one gets * S * S * d (5) Note that the esult s the same fo any poston of pont on lne C. Ths means that fo all these postons the potental s the constant. In othe wods: The lne C s an equpotental lne. Next, one may fx the ogn of an axe Ox at pont and select ts decton along the feld (fg.4). s =(0) at ogn, fo a pont on C wth coodnate x on Ox axs, eq. (5) could be wtten n fom: ( x) (0) * x _ and _ ( x) (0) * x (6) The elaton (6) tells that the potental deceases lnealy (Fg.4) along sense. One concludes that nsde an unfom feld: a) The potental value s the same on a plane pependcula to electc feld lnes. b) The potental deceases lnealy whle movng along the decton of electc feld. c) Dmensonal ule appled at (6) gves []=[]*[x] and [] volt = (N/C)*m= N*m/C = J/C - In the case of unfom feld we found that: TH LCTRIC FILD LINS R PRPNDICULR TO TH QUIPOTNTILS ND POINT DOWNHILL FROM TH HIGHR TO TH LOWR LUS OF POTNTIL. 3

4 ctually, ths s a geneal ule that apples fo all electc felds (not only fo unfom ones). In fact, fo a shft ds of chage q =+C along an equpotental lne (on whch Δ = 0), the wok done by electc feld s zeo because * ds Wnt 0. s * ds 0 means that ds and ds s algned on a equpotental lne, t comes out that the electc feld s pependcula to equpotental lnes. - Consde now a movng chaged patcle enteng nsde an electc feld. If othe foces ae neglgble, one may assume that W ext 0 and apply the enegy consevaton pncple fo the system feld-chaged patcle ( Δ = W ext = 0 ). s long as ths patcle s nsde the feld ts total enegy wll be constant and Δ = ΔK + ΔU = 0. Ths means that nsde the feld ΔK = - ΔU. Then, as ΔU = q* Δ one gets K q* (7) Dependng on sgn of ts chage, the knetc enegy of patcle may ncease o decease nsde the feld. If q > 0, and the chaged patcle entes the feld movng : a) o along the decton of potental decease ( downhll ), Δ < 0 and the feld wll ncease ts K. b) o along the decton of potental ncease ( uphll ), Δ > 0 and the feld wll decease ts K. If q < 0, and the chaged patcle entes the feld movng : a) o along the decton of potental decease ( downhll ), Δ < 0 and the feld wll decease ts K. b) o along the decton of potental ncease ( uphll ), Δ > 0 and the feld wll ncease ts K. These ules ae mpotant when dealng wth electons o potons acceleated by an electostatc feld. One has ntoduced a specal unt electonvolt (e) fo the enegy of mcoscopc patcles; t s wdely used n chemsty, atomc and nuclea physcs. The electon- volt s the knetc enegy ganed by a patcle wth chage q = +e when movng downhll a dffeence of potental. e 9 9 (.60*0 C)*.60*0 J (8) 3.3 The lectc Potental of Pont Chages a) One pont chage - The electc feld ceated by a pont chage +Q n the space aound t s a adal feld Q k ( ) (9) One may fnd the expesson fo the potental by applyng expesson (3) fo a path nsde the feld (fg.6). s, one get Fg.6 kq ds kq Q d ( ) d k d kq (0) 4

5 y conventon, one fxes the zeo value of potental at =,( ) = 0. Then one gets the potental at pont as ( ) ( ) ( ) kq () Fgue 7 pesents the pofle of ths functon n space. The dashed ccles pesent the equpotental lnes. They ae close to each othe nea the chage because the potental changes apdly n ths egon. The feld lnes ae nomal to equpotental lnes. Note that the magntude of feld s lage at locatons whee the equpotental lnes ae dense (nea the chage whch s the souce of feld ) and t deceases at locatons whee they ae ae (bg -values). Fg.7 b) Two pont chages - Consde that two pont chages Q and Q ae pesent at the same space. Let be _ and _ the electc felds coespondng to each of them at a gven pont of ths space. We can fnd the esultant electc feld at ths pont by applyng the supeposton pncple () s n the case of one patcle, we chose the statng pont of a path at nfnty ( = ( ) = 0) and expess the dffeence of potental by the wok done by net electc feld dung the shft fom to. ds Q Q ( ) ds ds ds k d k d... kq kq In the geneal case, when the net feld s due to seveal chages Q,Q,..Q the net potental s calculated kq kq though the expesson o moe geneally ( ) (3) Notes: a) s the dstance fom -chage to the locaton whee we calculate the potental value b) one has to nclude the chage sgns nsde the sum (3). -Fgue 8.a shows the potental evoluton(up) n the electc feld bult by a dpole and the equpotental lnes and the felds lnes (down). Fgue 8.b shows the potental evoluton (up) n the electc feld due to two equal postve chages, the equpotental lnes and the feld lnes n the plan that contans chages. 5

6 Fg. 8a Fg.8b 3.4 Geneal fomula elatng lectc Feld to lectc Potental - Let s consde anew the elaton In ths elaton, s s the component of feld along an abtay decton ds nsde the feld. Ths means that the component of along a gven decton s elated to the change of along same decton. y ntoducng a ectangula system of axs Oxyz, one gets ds dx dy j dz k and x y j z k (4) Fg.9 Then, d * ds ( dx dy dz) x y z (5) If one consdes a dsplacement n Ox decton, dy = dz = 0 and d dx. It comes out that x x and smlaly, x y, z _ const y and y x, z _ const z z x, y _ const Then, fom (4) one fnds out the expesson j k (6) x y z The expesson gad ( x, y, z) j k s known as gadent of functon (x,y,z). x y z In many textbooks, the expesson (6) s wtten n the fom gad( x, y, z) (7) 6

7 3.5 Contnuous Chage Dstbutons -The potental, due to a contnuous chage dstbuton, at a pont P nsde a feld can be found by: kdq a) xpessng the contbuton of each elementay chage dq as d b) Calculatng the total potental as sum of elementay contbutons ove the total dstbuton as Fg. 0 k egon _ of _ ch ag e dq (8) 3.6 Conductos - s explaned n secton.3 the electostatc feld s zeo nsde a conducto and pependcula to ts suface outsde close by. s * ds 0 fo a path connectng two ponts located nsde t o on ts suface, t comes out that the electc potental of a conducto s the same eveywhee on ts suface and nsde t. Note that ths s vald fo any chaged conducto and any neutal conducto placed nsde an statc electc feld. Fg. One mght emembe that that the pesence of a conducto n an extenal feld modfes the feld patten aound ts suface so that the feld lnes become pependcula to ts suface. Fg. shows the defomaton of equpotental planes (and feld lnes) of a unfom electc feld aound a sphecal unchaged conducto. Nea the sphee, the equpotental planes become sphecal sufaces. -s the chages ae dstbuted on the conducto suface and the feld nsde t s zeo, one may open a cavty nsde the conducto and use the conducto suface as a sheld fom extenal felds. Ths sheld effect happens n a Faaday cage whch s used to block extenal felds n the space nsde t. - If a conductng we connects two chaged conductng objects, the chage edstbutes so that the potental become the same on and nsde any pont of the set(ob_, ob_, we). The feld s zeo at any pont wthn them and on the suface. Outsde, t s dected pependcula to the suface. If one places the chage +Q on a metallc sphee, t wll be dstbuted unfomly on sphee suface and Q ths wll buld a constant chage densty on sphee suface. The chage densty on suface s 4R bgge f adus R s smalle. When the suface of the metallc object s not sphecal, the chage densty s hghe at the at shap ponts (smalle R); the electc feld outsde ( ~ ) s lage n font of those locatons. Ths effect s used to poduce electc felds wth hgh magntude. On the othe sde, ths effect may become poblematc because t onzes the a and may be the ogn of coona dschages. 7

8 RMMR: ()= -W feld (+C shfted_fom _nfty _to_) and = () - () = -W feld ((+C shfted_fom ). One should not foget that s not exactly a potental enegy; ts unt s [volt =J/C ] and not [J]. Fo a chage q the potental enegy s U = q *, U = q* and W feld = - U = -( q* ) kq ( ) kq (fo one chage) and ( ) (fo many chages) whle the change of electc potental between two ponts nsde an electc feld s. s always pependcula to equpotental sufaces (o lnes) and dected vesus the decease of potental values. x y j z k 8