Chapter One: The Electric Field

Transcription

1 ) Analytial Physis quations fom nd dition Pull Jodan Boyd-Gab Chapt On: Th lti Fild (.) Coulomb s Law Two stationay lti hags pl o attat on anoth with a fo gin by: F q q (.9) Potntial ngy of a Systm of Chags Fato of ½ is to aount fo ounting ah pai twi. Supposition of th potntial ngis of ah pai of patils (Total of N hags). U N j k j q k q jk j (.) Gauss s Law Th lti flux though a sufa nlosing som hag is qual to fou pi tims th nlosd hag, gadlss of th gomty. da 4πq Sufa n (.6) lti fild fom a lin of Chag Wi with lina hag distibution lambda xtnding infinitly in ith dition a distan away points in th adial dition outwad with magnitud: F λ ˆ (.38) Th ngy of an lti Fild - Th potntial ngy U of a systm of hags whih is th total wok quid to assmbl th systm, an b alulatd fom th lti fild itslf by simply assigning an amount of ngy to y olum lmnt and intgating o all spa wh th is lti fild. U nti Fild d

2 Chapt Two: Th lti Potntial (.6) lti Potntial Dfinition, indpndnt of path: φ p f d s p i φ (.) lti Potntial of Unifomly Chagd Disk loatd on th xz plan with unifom sufa hag dnsity sigma. φ(, ± y,) πσ + ( y a m y) (.39) Gauss s Diffntial Fom lation btwn hag dnsity and lti fild. 4πρ (.56) Constaints of ltostati fild - Vto opato is known as Laplaian this follows fom impossibility of onstuting a onfinmnt of a hag using just ltostati filds. Also, th ul of th lti fild must b zo ywh th is no hag. (3.) Capaitan of onduto φ ds losd loop Chapt Th: lti Filds Aound Condutos Q C φ (3.6) Capaitan of aious gomtis: () Aa of plats is A, spaation of plats is s () adius of inn ylind is a, out is b, lngth L (3) adius of inn sph is a, out is b C paalll plats A 4πs Connti ylinds L b log a Connti sphs ab b a V [stat olts ] s [m] lti fild btwn paalll plat apaitos: 4πσ (3.5) ngy in a apaito Can b xpssd in tms of apaitan and oltag o lti fild and olum. U Cφ V 8π Th lti fild is always zo in a onduto and ppndiula to th sufa whn it las th onduto. Capaitans add insly in sis and ditly in paalll.

3 Chapt Fou: lti Cunts (4.5) Cunt Dnsity Assum som numb of ais of hag in a unt a ltons moing at an aag loity. Th unt dnsity is gin as: J N u (4.6) Poptis of th Cunt An lti unt going though an aa is dfind as th sufa intgal. Th dign of th unt dnsity is also latd to th hag dnsity baus of hag onsation. Also not that in () th dign is zo whn th is no tim dpndn. ρ I J da J t s (4.) sistan of Matial Th sistan of a matial an b xpssd in tms of Ohm s law (), a onstant lti fild (), ondutiity (3), and sistiity (4). In ah as th oss stional aa is A, th lngth is L, th unt dnsity is J, and th oltag is V. V L L L I ρ AJ Aσ A (4.) lati Populations Th atio of populations of ngy lls at thmal quilibium is latd to th ngy diffn, th tmpatu and Boltzman s onstant (k.38-6 g/klin). p kt p (4.) Adding sistos sistos add ditly in sis and insly in paalll N N sis i N + i i (4.3) Ciuit laws Ths laws an b usd to simplify omplx iuits.. Th unt though ah lmnt must qual th oltag aoss that lmnt diidd by th sistan of th lmnt.. At a nod of th ntwok, a point wh th o mo onnt wis mt, th algbai sum of th unts into th nod must b zo fom hag onsation. 3. Th sum of th potntial diffns takn in od aound a loop of th ntwok is zo. (4.4) Pow dissipatd by a unt Whn unt flows though a sisto, som pow is lost to hat. V P I VI (4.33) C Ciuits Whn a swith is thown allowing a hagd apaito to dissipat its ngy, th systm is haatizd by a tim onstant tau in th following xponntial quation. τ C i t t τ τ ( ) CV I ( t) Q t V

4 Chapt Fi: Th Filds of Moing Chags (5.) Th fo on a moing hag A tst hag moing in lti and magnti filds xpins a fo (wh, th spd of light, is.998 m/s in gs). q F q + B (5.) Th fild fom a moing hag Lt thta pim dnot th angl btwn this adius to and th loity of th hag Q, whih is moing in th positi x pim dition in th pimd fam. Q ' 3/ ' ( sin θ ) Chapt Six: Th Magnti Fild (6.3) Magnti fild fom wi Th fild dition is gin by th ight hand ul and is dition in th thta dition. B (6.7) Th fo on wis with paalll unts Th attati / pulsi fo on a lngth l of th sond wi fom anoth. Th sign is sd if th unts a anti-paalll. II F I θˆ (6.) Amp s Law (6.) Magnti Vto Potntial 4πJ ( A) 4π I Bds C J ( x, y, z) d B A A( x, y, z) (6.38) Fo on any hag aying wi W an intgat o th nti lngth of th wi in od find th magnti fild. Idl ˆ db (6.4) Filds of ings and Coils Th following a did fom Biot-Saat: unt aying ing of adius b on axis (), insid solnoid (), wh n is tuns p unit lngth. π b I z ) B z ( b + 3/ B z 4 π In (6.46) Magnti Chang at a Cunt Sht A magnti fild is hangd whn you mo fom on sid of a unt sht to th oth basd on how muh unt is going though th sht. 4π J B

5 (6.6) Tansfomation of Filds Th following gi th tansfomation to a fam moing at loity bta in a dition ointd with th assoiatd filds in th following mann. To tansfom fom a moing fam, swith th pims and th plus/minus signs. ' γ ( + B ) B' γ ( B ) pp pp ' pp pp B' B (6.64) Th Hall fft Suppos th a m mobil hag ais p ubi ntimts and dnot th hag of ah by q. Thn th unt dnsity is nq. If w now substitut in fo th aag loity, w an lat this tanss fild to th ditly masuabl quantitis J and B: J B t nq pp Chapt Sn: ltomagnti Indution (7.7) Flux and Indud MF A oil with N tuns that has a hang in flux xpins an indud oltag. Th thid fomulation is an quialnt fomulation. Φ B da d Φ dt pp B t (7.38) Mutual Indutan and ipoity Two iuits in a fixd onfiguation gnat som magnti fild and also ha som aa though whih a hanging magnti fild an flux. Th mutual indutan and th aompanying quialn fo any onfiguation - of on indutan of on iuit to th oth a gin blow. di dt M M M (7.54) Slf Indution - Whn th unt is hanging, th is a hang in th flux though a iuit itslf, and th is an ltomoti fo that is indud what th sou. This is alld th slf indutan and is gin by th following xpssions: gnal fom (), tooidal oil of tangula oss stion, inn adius a, hight h, and out adius b (), and a solnoid of adius a (3). di N h b 4π N a L L tous ln L solnoid dt a (7.66) Tansint Bhaio of L Ciuits th unt in iuit with a sisto and induto disonntd fom a batty that has atd a unt within th iuit is gin by an xponntial xpssion with tim onstant /L. τ t τ / L I( t) I (7.7) ngy in Magnti Fild Th induto is th analog to th lti ngy stod in a apaito. U LI B d 8π

6 Chapt ight: Altnating Cunt Ciuits (8.) sonant Ciuit a LC osillats in a dampd mann aoding to th following quations. ωt V( t) ( Aosωt+ Bsin ωt) ω LC 4L ω I( t) ACω Aosωt+ Bsin ωt ( ) t (8.5) sonant Fquny in LC iuit in a iuit without a sisto th iuit osillats with a sonant fquny. ω LC (8.9) Din L iuit A sinusoidal oltag psilon is gin to a iuit ontaining a sisto and an induto xpins th following unt: os φ + ω L I I ( t) I os( ωt φ) + ω L (8.4) Din LC iuit Similaly, w an xpss th bhaio of an LC iuit in th sam way. I ( t) os( ωt φ) tanφ ω C ωc + ωl ωc (8.57) Ntwoks in Altnating Ciuits W an du th omponnts in an AC iuit into thi omplx impdans and thn add th omplx alus as sistos obying th iuit laws fo DC iuits. Th impdan of ommon lmnts: Z Z L i i L ω Z C C ω (8.6) Pow and ngy Sin th of os-squad o many yls is ½ and w ha th oltag popotional to V/, th following xpssions xplain how ngy is dissipatd by an AC iuit. V V P ms I Chapt Nin: Maxwll s quations (9.5) Maxwll s quations Th lationship btwn th lti, magnti, hag dnsity ho, and unt dnsity J is gin by th following quations. B t 4π B + J t 4πρ B (9.7) Poynting Vto Th pow dnsity S is alignd with x B, in th dition of popagation. S 4π 8π

7 (9.33) M Wa Tansfomation If a fam is moing in th x-dition with spt to a fam of an obsd M wa popagating in x, thn th lti and magnti filds in y and z will b: / y B / z + / B z ( + Chapt Tn: lti Filds in Matt lim / / y B ) (.) Th Dipol Momnt n whn th nt hag of a distibution is zo, it an still at an lti fild baus of th distibution of intnal hags. This dipol distibution is indpndnt of wh th oodinats a hos (always points fom q to +q). p ρd q i i (.5) lti and Potntial Filds A dipol distibution ats both a potntial and xpssd lti fild. Sin th lti fild is somwhat omplx, it is bokn into adial and angula omponnts, wh thta is masud fom th alignmnt of th dipol to. ˆ p φ( ) p 3 i osθ θ p sin θ (.8) Toqu on Dipol A dipol will want to align itslf with an lti fild. Thus, th toqu xtd on a dipol and its ngy in a fild is gin by: N p U p (.3) Fo on Dipol A dipol will not xpin a fo in a onstant lti fild. It will, how, xpin a fo in a non-onstant fild. Not: w ha a to opato woking on, not p tims th gadint of. F ( p ) (.57) lti Susptibility Th polaization of matt an b haatizd by a polaization dnsity, whih is th dipol momnt p unit olum. This P is also th podut of th susptibility tims th lti fild. P P Nα pv χ N α (.58) Dilti Constant A dilti in an lti fild auss th lti fild in th gion to das by a fato th onstant, whil th ffti apaitan inass by th onstant. 3 z + 4πχ C C (.6) Bound-Chag Cunt A hanging polaization of a substan atually ats a momnt of hag within a polaizd substan. dp J dt (.7) M Wa in Dilti If th is an M wa within a dilti, it mos slow than if it w in a auum. Th following onstaints apply: ω B k

8 Chapt ln: Magnti Filds in Matt (.9) Magnti Dipol Momnt A magnti dipol momnt is a to whos dition is nomal to a loop of unt i.. alignd with th ditd aa to a. I m a (.) Magnti Dipol Potntial W again dfin a potntial funtion fo th fild podud by a magnti dipol. m ˆ A (.5) Fild, Fo, Toqu Th following a th sam algbaially as fo a dipol onfiguation. N m U m F ( m ) m m osθ θ 3 θ sin 3 (.3) Atom Momnts Th hang in th magnti momnt fo an atom of mass M fom an applid magnti fild is: q m B 4M (.39) Magnti Susptibility An applid lti fild ats a magnti polaization, whih in tun ats a unt dnsity aound th dg of th magntizd substan. M Vm χ B J M M m (.54) Th H-Fild A fild is atd abstatly that is podud fom unts und ou ontol whih satisfis Amp s Law: 4π H di I C (.57) Tu B-Fild W want to know th tu magnti fild within a substan, how, so w us th lationship dsibd in.39 (plaing B with H) and thn w an di an xpssion in tms of th susptibility as wll as th pmability, mu. B H + 4π M (+ 4πχ ) H µ H m