Chaptr 33 Gauss s Law 33 Gauss s Law Whn askd t find th lctric flux thrugh a clsd surfac du t a spcifid nn-trivial charg distributin, flks all t ftn try th immnsly cmplicatd apprach f finding th lctric fild vrywhr n th surfac and ding th intgral f dt vr th surfac instad f just dividing th ttal charg that th surfac nclss by. Cncptually spaking, Gauss s Law stats that th numbr f lctric fild lins pking utward thrugh an imaginary clsd surfac is prprtinal t th charg nclsd by th surfac. A clsd surfac is n that divids th univrs up int tw parts: insid th surfac, and, utsid th surfac. An xampl wuld b a sap bubbl fr which th sap film itslf is f ngligibl thicknss. I m talking abut a sphridal sap bubbl flating in air. Imagin n in th shap f a tin can, a clsd jar with its lid n, r a clsd bx. Ths wuld als b clsd surfacs. T b clsd, a surfac has t ncmpass a vlum f mpty spac. A surfac in th shap f a flat sht f papr wuld nt b a clsd surfac. In th cntxt f Gauss s law, an imaginary clsd surfac is ftn rfrrd t as a Gaussian surfac. In cncptual trms, if yu us Gauss s Law t dtrmin hw much charg is in sm imaginary clsd surfac by cunting th numbr f lctric fild lins pking utward thrugh th surfac, yu hav t cnsidr inward-pking lctric fild lins as ngativ utward-pking fild lins. Als, if a givn lctric fild lin pks thrugh th surfac at mr than n lcatin, yu hav t cunt ach and vry pntratin f th surfac as anthr fild lin pking thrugh th surfac, adding + t th tally if it pks utward thrugh th surfac, and t th tally if it pks inward thrugh th surfac. S fr instanc, in a situatin lik: Clsd Surfac w hav 4 lctric fild lins pking inward thrugh th surfac which, tgthr, cunt as 4 utward fild lins, plus, w hav 4 lctric fild lins pking utward thrugh th surfac which tgthr cunt as +4 utward fild lins fr a ttal f 0 utward-pking lctric fild lins thrugh th clsd surfac. By Gauss s Law, that mans that th nt charg insid th Gaussian surfac is zr. 296
Chaptr 33 Gauss s Law Th fllwing diagram might mak ur cncptual statmnt f Gauss s Law sm lik plain ld cmmn sns t yu: Th clsd surfac has th shap f an gg shll. Thr ar 32 lctric fild lins pking utward thrugh th Gaussian surfac (and zr pking inward thrugh it) maning thr must (accrding t Gauss s Law) b a nt psitiv charg insid th clsd surfac. Indd, frm yur undrstanding that lctric fild lins bgin, ithr at psitiv chargs r infinity, and nd, ithr at ngativ chargs r infinity, yu culd prbably dduc ur cncptual frm f Gauss s Law. If th nt numbr f lctric fild lins pking ut thrugh a clsd surfac is gratr than zr, thn yu must hav mr lins bginning insid th surfac than yu hav nding insid th surfac, and, sinc fild lins bgin at psitiv charg, that must man that thr is mr psitiv charg insid th surfac than thr is ngativ charg. Our cncptual ida f th nt numbr f lctric fild lins pking utward thrugh a Gaussian surfac crrspnds t th nt utward lctric flux Φ thrugh th surfac. T writ an xprssin fr th infinitsimal amunt f utward flux dφ thrugh an infinitsimal ara lmnt, w first dfin an ara lmnt vctr whs magnitud is, f curs, just th ara f th lmnt; and; whs dirctin is prpndicular t th ara lmnt, and, utward. (Rcall that a clsd surfac sparats th univrs int tw parts, an insid part and an utsid part. Thus, at any pint n th surfac, that is t say at th lcatin f any infinitsimal ara lmnt n th surfac, th dirctin utward, away frm th insid part, is unambiguus.) utsid insid In trms f that ara lmnt, and, th lctric fild at th lcatin f th ara lmnt, w can writ th infinitsimal amunt f lctric flux dφ thrugh th ara lmnt as: 297
Chaptr 33 Gauss s Law d Φ θ Rcall that th dt prduct can b xprssd as cs θ. Fr a givn and a givn amunt f ara, this yilds a maximum valu fr th cas f θ 0 (whn is paralll t maning that is prpndicular t th surfac); zr whn θ 90 (whn is prpndicular t maning that is paralll t th surfac); and; a ngativ valu whn θ is gratr than 90 (with 80 bing th gratst valu f θ pssibl, th angl at which is again prpndicular t th surfac, but, in this cas, int th surfac.) Nw, th flux is th quantity that w can think f cncptually as th numbr f fild lins. S, in trms f th flux, Gauss s Law stats that th nt utward flux thrugh a clsd surfac is prprtinal t th amunt f charg nclsd by that surfac. Indd, th cnstant f prprtinality has bn stablishd t b whr (psiln zr) is th univrsal cnstant knwn as th lctric prmittivity f fr spac. (Yu v sn bfr. At th tim, w statd that th Culmb cnstant k is ftn xprssd as. Indd, th idntity k appars 4π 4π n yur frmula sht.) In quatin frm, Gauss s Law rads: QNCLOSD (33-) Th circl n th intgral sign, cmbind with th fact that th infinitsimal in th intgrand is an ara lmnt, mans that th intgral is vr a clsd surfac. Th quantity n th lft is th sum f th prduct fr ach and vry ara lmnt making up th clsd surfac. It is th ttal utward lctric flux thrugh th surfac. Using this dfinitin in Gauss s Law allws us t writ Gauss s Law in th frm: Φ Q (33-2) NCLOSD Φ (33-3) 298
Chaptr 33 Gauss s Law Hw Yu Will b Using Gauss s Law Gauss s Law is an intgral quatin. Such an intgral quatin can als b xprssd as a diffrntial quatin. W wn t b using th diffrntial frm, but, bcaus f its xistnc, th Gauss s Law quatin QNCLOSD is rfrrd t as th intgral frm f Gauss s Law. Th intgral frm f Gauss s Law can b usd fr svral diffrnt purpss. In th curs fr which this bk is writtn, yu will b using it in a limitd mannr cnsistnt with th mathmatical prrquisits and c-rquisits fr th curs. Hr s hw: QNCLOSD ) Gauss s Law in th frm Φ maks it asy t calculat th nt utward flux thrugh a clsd surfac that nclss a knwn amunt f charg Q NCLOSD. Just divid th 2 2 C amunt f charg Q NCLOSD by (givn n yur frmula sht as 8. 85 0 ) and 2 N m yu hav th flux thrugh th clsd surfac. 2) Givn th lctric fild at all pints n a clsd surfac, n can us th intgral frm f Gauss s Law t calculat th charg insid th clsd surfac. This can b usd as a chck fr a cas in which th lctric fild du t a givn distributin f charg has bn calculatd by a mans thr than Gauss s Law. Yu will nly b xpctd t d this in cass in which n can trat th clsd surfac as bing mad f n r mr finit (nt vanishingly small) surfac pics n which th lctric fild is cnstant vr th ntir surfac pic s that th flux can b calculatd algbraically as A r A csθ. Aftr ding s fr ach f th finit surfac pics making up th clsd surfac, yu add th rsults and yu hav th flux Φ thrugh th surfac. T gt th charg nclsd by th surfac, yu just plug that int QNCLOSD Φ and slv fr Q NCLOSD. If yu ar using th mthd as a chck, yu just cmpar yur rsult with th amunt f charg knwn t b nclsd by th surfac. 3) In cass invlving a symmtric charg distributin, Gauss s Law can b usd t calculat th lctric fild du t th charg distributin. In such cass, th right chic f th Gaussian surfac maks a cnstant at all pints n ach f svral surfac pics, and in sm cass, zr n thr surfac pics. In such cass th flux can b xprssd as A and n can QNCLOSD simply slv A fr and us n s cncptual undrstanding f th lctric fild t gt th dirctin f. Th rmaindr f this chaptr and all f th nxt will b usd t prvid xampls f th kinds f charg distributins t which yu will b xpctd t b abl t apply this mthd. 299
Chaptr 33 Gauss s Law Using Gauss s Law t Calculat th lctric Fild in th Cas f a Charg Distributin Having Sphrical Symmtry A sphrically-symmtric charg distributin has a wll-dfind cntr. Furthrmr, if yu rtat a sphrically-symmtric charg distributin thrugh any angl, abut any axis that passs thrugh th cntr, yu wind up with th xact sam charg distributin. A unifrm ball f charg is an xampl f a sphrically-symmtric charg distributin. Bfr w cnsidr that n, hwvr, lt s tak up th cas f th simplst charg distributin f thm all, a pint charg. W us th symmtry f th charg distributin t find ut as much as w can abut th lctric fild and thn w us Gauss s Law t d th rst. Nw, whn w rtat th charg distributin, w rtat th lctric fild with it. And, if a rtatin f th charg distributin lavs yu with th sam xact charg distributin, thn, it must als lav yu with th sam lctric fild. W first prv that th lctric fild du t a pint charg can hav n tangntial cmpnnt by assuming that it ds hav a tangntial cmpnnt and shwing that this lads t a cntradictin. Hr s ur pint charg q, and an assumd tangntial cmpnnt f th lctric fild at a pint P which, frm ur prspctiv is t th right f th pint charg. t q P (Nt that a radial dirctin is any dirctin away frm th pint charg, and, a tangntial dirctin is prpndicular t th radial dirctin.) 300
Chaptr 33 Gauss s Law Nw lt s dcid n a rtatin axis fr tsting whthr th lctric fild is symmtric with rspct t rtatin. Almst any will d. I chs n that passs thrugh bth th pint charg, and, pint P. t Axis f Rtatin q P Nw, if I rtat th charg, and its assciatd lctric fild, thrugh an angl f 80 abut that axis, I gt: Axis f Rtatin q P t This is diffrnt frm th lctric fild that w startd with. It is dwnward instad f upward. Hnc th lctric fild cannt hav th tangntial cmpnnt dpictd at pint P. Nt that th argumnt ds nt dpnd n hw far pint P is frm th pint charg; indd, I nvr spcifid th distanc. S, n pint t th right f ur pint charg can hav an upward cmpnnt t its lctric fild. In fact, if I assum th lctric fild at any pint P in spac thr than th pint at which th charg is, t hav a tangntial cmpnnt, thn, I can adpt a viwpint frm which pint P appars t b t th right f th charg, and, th lctric fild appars t b upward. Frm that viwpint, I can mak th sam rtatin argumnt prsntd abv t prv that th tangntial cmpnnt cannt xist. Thus, basd n th sphrical symmtry f th charg distributin, th lctric fild du t a pint charg has t b strictly radial. Thus, at ach pint in spac, th lctric fild must b ithr dirctly tward th pint charg r dirctly away frm it. Furthrmr, again frm symmtry, if th lctric fild is dirctly away frm th pint charg at n pint in spac, thn it has t b dirctly away frm th pint charg at vry pint in spac. Likwis, fr th cas in which it is dirctly tward th pint charg at n pint in spac, th lctric fild has t b dirctly tward th pint charg at vry pint in spac. W v bild it dwn t a 50/50 chic. Lt s assum that th lctric fild is dirctd away frm th pint charg at vry pint in spac and us Gauss s Law t calculat th magnitud f th lctric fild. If th magnitud is psitiv, thn th lctric fild is indd dirctd away frm th pint charg. If th magnitud turns ut t b ngativ, thn th lctric fild is actually dirctd tward th pint charg. 30
Chaptr 33 Gauss s Law At this pint w nd t chs a Gaussian surfac. T furthr xplit th symmtry f th charg distributin, w chs a Gaussian surfac with sphrical symmtry. Mr spcifically, w chs a sphrical shll f radius r, cntrd n th pint charg. Gaussian Surfac (An imaginary sphr f radius r, cntrd n th pint charg.) At vry pint n th shll, th lctric fild, bing radial, has t b prpndicular t th sphrical shll. This mans that fr vry ara lmnt, th lctric fild is paralll t ur utward-dirctd ara lmnt vctr. This mans that th in Gauss s Law, QNCLOSD valuats t. S, fr th cas at hand, Gauss s Law taks n th frm: Q NCLOSD Furthrmr, th magnitud f th lctric fild has t hav th sam valu at vry pint n th shll. If it wr diffrnt at a pint P n th sphrical shll than it is at a pint P n th sphrical shll, thn w culd rtat th charg distributin abut an axis thrugh th pint charg in such a mannr as t bring th riginal lctric fild at pint P t psitin P. But this wuld rprsnt a chang in th lctric fild at pint P, du t th rtatin, in vilatin f th fact that a pint charg has sphrical symmtry. Hnc, th lctric fild at any pint P n th Gaussian surfac must hav th sam magnitud as th lctric fild at pint P, which is what I st ut t prv. Th fact that is a cnstant, in th intgral, mans that w can factr it ut f th intgral. S, fr th cas at hand, Gauss s Law taks n th frm: Q NCLOSD 302
Chaptr 33 Gauss s Law Gaussian Surfac (An imaginary sphr f radius r, cntrd n th pint charg.) NCLOSD On th prcding pag w arrivd at. Nw, th intgral f vr th Gaussian surfac is th sum f all th ara lmnts making up th Gaussian surfac. That mans that it is just th ttal ara f th Gaussian surfac. Th Gaussian surfac, bing a sphr f radius r, has ara 4πr 2. S nw, Gauss s Law fr th cas at hand lks lik: 2 QNCLOSD 4π r Okay, w v lft that right sid aln fr lng nugh. W r talking abut a pint charg q and ur Gaussian surfac is a sphr cntrd n that pint charg q, s, th charg nclsd, Q NCLOSD is bviusly q. This yilds: 2 q 4π r Slving fr givs us: Q 4π This is psitiv whn th charg q is psitiv, maning that th lctric fild is dirctd utward, as pr ur assumptin. It is ngativ whn q is ngativ. S, whn th charg q is ngativ, th lctric fild is dirctd inward, tward th chargd particl. This xprssin is, f curs, just Culmb s Law fr th lctric fild. It may lk mr familiar t yu if w writ it in trms f th Culmb cnstant k in which cas ur rsult fr th utward lctric fild appars 4π as: It s clar that, by mans f ur first xampl f Gauss s Law, w hav drivd smthing that yu alrady knw, th lctric fild du t a pint charg. kq 2 r q 2 r 303