Chapter 23: Electric Potential

Transcription

1 Chapte 23: Electc Potental Electc Potental Enegy It tuns out (won t show ths) that the tostatc foce, qq 1 2 F ˆ = k, s consevatve. 2 Recall, fo any consevatve foce, t s always possble to wte the wok done by the foce as Δ U, fo some potental enegy, U. So, fo the tostatc foce, t must be tue that the wok W done by the tostatc foce can always be wtten: W fo some tc potental enegy, What s the functon U? = Δ U, (1) U. 1

2 Ch. 23: Electc Potental Electc Potental Enegy of Two Pont Chages Consde a postve test chage q held above a souce chage, Q, whch s negatve. The tostatc foce that Q exets on q s: qq qq F ˆ ˆ = k = k 2 2 Ths foce ponts towad the (negatve) souce chage Q. Now magne gabbng q and movng t upwad, away fom Q. To do ths, you apply a foce to q that I wll call F appled. You could jek q that s much lage n magntude apdly upwad, exetng a foce F appled than F. But f you dd, you d be acceleatng q, and the wok that you do would go nto nceasng ts knetc enegy as well as ts potental enegy. By contast, f you magne lftng q quas-statcally, meanng so slowly that at evey nstant, q s essentally n statc equlbum, then the acceleaton of q wll be abtaly small. In ths dealzed stuaton, the wok that you do goes nto changng just the potental enegy, not the knetc enegy. 2

3 Ch. 23: Electc Potental If q s ased quas-statcally, then Fappled = F qq F ˆ appled = k 2 and the wok done by F appled s: f f f qq 1 W ˆ 2 ( ˆ) 2 ( ˆ ˆ appled = Fappled d = k d = kq Q d ) But ˆ ˆ= 1, so: f f 1 1 kqq kqq Wappled = kqq d kq 2 = Q = f Because q was magned to be ased quas-statcally, Wappled = Δ U kqq kqq f = U U ( ) f 3

4 Ch. 23: Electc Potental f Lookng at ( ), t seems temptng to dentfy kqq f as U and kqq as U. Ths s exactly what we do. In geneal, then, the potental enegy of any two pont chages q 1 and q 2 sepaated by a dstance s: kq1q2 U = (2) 4

5 Ch. 23: Electc Potental Electc Potental Enegy of N Pont Chages Imagne a collecton of N pont chages, q 1, q 2,, qn. Let the dstance between the th chage, q, and the j th chage, q j, be called j. Then the potental enegy of ths pa of chages s: kqq j U = j Then the total potental enegy of the system of N chages s: kqq j Utotal =, (3) < j j n whch the sum s pefomed ove and j fom 1 to N, but ncludng only tems fo whch < j to avod ovecountng o countng the nteacton of any chage wth tself. It s mpotant to ealze that ths enegy s enegy assocated wth the ente system of N chages, not any sngle chage. 5

6 Ch. 23: Electc Potental Electc Potental The tc potental, V, s defned to be the potental enegy pe unt test chage : U V (4) q Note: Unt (SI): JC Volt, V (n hono of Alessando Volta, nvento of the voltac ple battey) Voltage The voltage between two ponts s the dffeence n potental, Δ V, between them. Fom (4), t follows mmedately that: ΔU Δ V = (5) q 6

7 Ch. 23: Electc Potental Potental due to Pont Chages Fo a souce chage Q and a test chage q, we saw eale that the potental enegy was: kqq U = Fom (4), then, the tc potental ( potental, fo shot) at the locaton of q s: kqq U V = = q q kq V = (6) Note: V s just a popety of the souce chage and the dstance that you ae fom the souce chage; q has been dvded out. V s a scala, not a vecto! 7

8 Ch. 23: Electc Potental Potental due to Collecton of N Pont Chages If we have N souce chages Q,, 1 QN poducng a potental at some pont P, the net potental at P s found by just addng the ndvdual potentals due to each souce chage: V V = V + V + + V net kq 1 2 kq N kq 1 2 N net = (7) 1 2 N Notes: To get the net potental, we just add ndvdual potentals lke numbes (scalas). Thee s no such thng as the x (o y) component of the potental. Thee s no chage at the pont P. In fact, f you ted to calculate the potental at the locaton of a postve pont chage Q, you d get V = kq, whch blows up (.e., nceases wthout bound) as. 8

9 Ch. 23: Electc Potental Potental due to Contnuous Dstbuton of Chage Imagne a 3-D blob havng total chage Q dstbuted contnuously thoughout the volume of the blob. Ths chaged blob ceates some potental at a pont P outsde the blob. How do we wte down the potental V that ths blob poduces at P? In pncple, you could magne the chage to be a collecton of pont chages (tons, e.g.) and thnk about calculatng the net potental by summng up all the kq tems, as n (7). But f the total chage Q s even modeately szed (1 μ C, fo example), then the numbe of tons and theefoe the numbe of 12 tems n (7) wll be on the ode of 1! 9

10 Ch. 23: Electc Potental Instead, we magne an nfntesmal element of chage dq, so small that we can appoxmate t as a pont chage. Then the nfntesmal contbuton to the total potental at P fom just ths nfntesmal chage dq s, fom (6): kdq dv = To get the total potental at P, we sum up (ntegate) all such nfntesmal contbutons dv fom all the lttle bts of chage dq n the whole blob: kdq V = (8) The ntegal must be taken ove the ente chage dstbuton (length, aea, o volume). 1

11 Ch. 23: Electc Potental Fndng V fom E Fo some contnuous chage dstbutons, t s ease to get the tc feld E fst and then get V fom E, nstead of dong the ntegal n (8). The cases fo whch ths s a useful tck ae pecsely those fo whch you can get E easly fom Gauss s law, namely, cases n whch the chage dstbuton has: sphecal symmety cylndcal symmety plana symmety 11

12 Ch. 23: Electc Potental To see how to do ths, just ecall that the wok done by F defnton of the wok done by a vaable foce: f W = F d But F = qe, so: f W = q E d And, because F s a consevatve foce, W = Δ U, so: f Δ U = q E d Now Δ V =Δ U q, so: f Δ V = E d s, fom the Eq. (9) tells us how to calculate the dffeence n potental (the voltage) f we know the tc feld. To get the potental at a pont, we need to defne some efeence level at whch V s chosen to be zeo. (9) 12

13 Ch. 23: Electc Potental We aleady made a choce of efeence level when we defned V due to a sngle pont chage: kq V = Ths defnton chooses V to be zeo at =. Adoptng the same efeence level fo V n (9), then, we can ewte (9): f f f Δ V = E d + E d = E d + E d = E d E d f Vf V = E d E d So we defne f Vf = E d and V = E d O, fo any geneal : V = E d (1) 13

14 Ch. 23: Electc Potental Fndng E fom V Consde a egon of space n whch E ponts n the + x decton, so that E = E x,,. If we go along a path fom a pont a at = x,, to a pont b at = x,,, the change n potental s, fom (9): Now, we know that f f f f xf Δ V = E d = E,, dx, dy, dz = E dx x x x Δ V = dv, n whch dv means the nfntesmal change n the potental along some nfntesmal bt of the path fom x to x+ dx. Compang the two expessons fo Δ V mmedately above, we get: dv = Exdx Ths looks lke the defnton of the total dffeental of V : dv dv dx, dx fom whch we fnd: x f x 14

15 Ch. 23: Electc Potental dv Ex = dx Now consde the moe geneal case of a egon of space n whch E changes n magntude and decton along a path n 3-D fom a pont a at = x, y, z to a pont b at = x, y, z. Ths means that the f f f f components E x, E y, and z Ex x, y, z, Ey ( x, y, z ), and Ez ( x, y, z ). Smlaly, the potental s a functon of x, y, V x, y, z. The change n potental fom a to b s, once agan: and z : ( ) E ae functons of x, y, and z : ( ) f Δ V = E d The nfntesmal change n potental along an nfntesmal bt of ths path, fom to + d, s evdently: dv = E d = E,,,, ( ) ( ) ( ) x Ey Ez dx dy dz = Ex dx + Ey dy + Ez dz Ths looks lke the genealzaton of the total dffeental to 3-D: V V V dv dx + dy + dz, x y z 15

16 Ch. 23: Electc Potental n whch V x, V y, and V z ae the patal devatves of V wth espect to x, y, and z, espectvely. These ae the devatves of V wth espect to one vaable, holdng the othe vaables fxed (.e., teatng the othe vaables as constants.) Compang the two equatons mmedately above, we see that: V Ex = (11) x V Ey = (12) y V Ez = (13) z These thee equatons ae often wtten moe compactly as a sngle vecto equaton by ntoducng the gadent opeato, del : ˆ ˆ j kˆ + + =,, x y z x y z Wth ths opeato, Eqs. (11) though (13) can be wtten as: E = V Ths says that E s the negatve of the gadent of V. 16

17 Ch. 23: Electc Potental Popetes of Conductos n Electostatc Equlbum (evsted) When we talked about Gauss s law n Chapte 22, we dscussed some popetes of conductos n tostatc equlbum ( E = nsde, etc.) Now thee ae two moe popetes we can add to the eale lst: V s unfom eveywhee on the suface of a conducto n tostatc equlbum. V nsde a conducto n tostatc equlbum s unfom and equal to V on the suface of the conducto. These two popetes mean that any conducto n tostatc equlbum s one bg blob of equpotental. 17