Fields, Charges, and Field Lines

Transcription

1 Felds, Charges, and Feld Lnes Electrc charges create electrc felds. (Gauss Law) Electrc feld lnes begn on + charges and end on - charges. Lke charges repel, oppostes attract. Start wth same dea for magnetc felds Magnetc feld lnes begn on North(+) poles and end on South(-) poles. Lke poles repel, oppostes attract. Important dfference: N & S poles alwas occur together! No magnetc monopoles (solated magnetc charges) Basc element of sources of magnetc feld s the magnetc dpole. Magnetc Felds and Electrc Charges Force due to electrc feld: F = q E But there are no equalent magnetc charges g: Do magnetc felds accelerate electrc charges? Yes! But onl on mong electrc charges. Epermentall: F = q B F = g B Unts of B are: Tesla = Ns mc F B = q 1 2 Magnetc Felds Unt Tesla named for nentor Ncola Tesla (AC generator, motor, ) Another common unt: Gauss = 1-4 Tesla Magnetc Felds n Nature: Shelded Lab: 1-14 T Intergalactc Space: 1-1 T Interstellar Space: 1-1 T Earth s Surface: 1µT efrgerator Magnet: 1mT Electromagnet: 1 T Superconductng MI Magnet: 2 T Superconductng Lab Magnet: 1-2 T Hgh Feld Magnet Labs: -6 T Neutron Stars: 1 MT Magnetars: 1 14 T and up Vector Cross Product and ght Hand ule Often we ll need to do c = a b Propertes of the Cross Product: 1) c s perpendcular to a and b. 2) ght-hand rule: If the fngers of our rght hand curl from a to b, c wll be along the drecton of our thumb. ) Magntude of c: c = ab sn(θ) Queston: What s b a? Aes and unt ectors must be rght handed : = 4 1

2 Charge Mong n a Unform Magnetc Feld Crcular Moton of Charged Partcles B s along drecton. A partcle wth mass m and charge q< moes wth eloct n the drecton. What s the drecton of the acceleraton caused b the B feld? Force s F=q B H rule: B n drecton q negate so F s n - drecton - B What happens to the magntude of? Is work done? No! F W=F d s alwas ero. Knetc energ s constant, but drecton of changes. 5 F = q B B = B Force s alwas perpendcular to eloct. Veloct s constant. Just lke unform crcular moton F = ma = m 2 r = m qb r = qb F T = 2πr = 2π m qb = 2πm qb ω = 2π T = qb m Notce ω does not depend on. 6 Combnng Electrc and Magnetc Felds Electrc Feld ges F E =qe Magnetc Feld ges F B =q B Can we construct a combnaton of, B and E to ge unform lnear moton? Yes!, E, and B all perpendcular. Tr along +, B along -, and E along +. s n E B drecton. + B E F E =q E =qe F B =q B =qb( ) =qb( ) F tot = F E + F B =q(e B) F tot = f E=B. Ths s a eloct selector! Onl =E/B goes straght. 7 A Mass Spectrometer Usng a eloct selector wth an acceleratng electrc feld we can construct a smple dece to measure mass (reall q/m). 1) Accelerate partcles through a oltage drop V. Then: K = qv = 1 2 m 2 2) Pass through a regon where E and B are crossed and E/B =. ) Partcles wll go straght onl f: = 2qV m = E B q m = E 2 2VB 2 (Thompson used a slghtl more complcated procedure to measure q/m for the electron) 8 2

3 Another Mass Spectrometer Force on a Current-Carrng Wre elease a poste on from the source S. When t enters the bo t has knetc energ K = qv = 1 2 m 2 It traels n a semcrcle and mpacts a dstance 2r awa. Usng our result for the radus of curature for a partcle n a B feld, we get = 2r = 2m qb = 2m 2qV qb m = 2 2mV B q Sole for m: m = B 2 q 2 8V For mong pont charges: F = q B How do we deal wth mong charges formng currents on a wre? Consder charge dq on a wre: dq = dq d dt = d dq dt = d Integrate d along a wre to get a ector along segment length: So: F = L B L For B nto page, force on wre s up. For 1 A n a 1 Tesla feld, force s 1 N per meter of length. 9 q = L 1 Eample-Force on a Wre What s the force on the wre shown for a B feld out of the page? For each straght segment: F = L B = LB( ) = LB θ L Semcrcle: F = d L B d L = ( sn(θ) + cos(θ) )dθ d L B = Bsn(θ)( )dθ + B cos(θ)( )dθ = Bsn(θ) dθ + B cos(θ) dθ π F = ( B sn(θ) + B cos(θ) )dθ π = B ( sn(θ) + cos(θ) )dθ [ ] π = B cos(θ) + sn(θ) = 2B L 11 Total: F = 2( LB ) 2B = B (2L + 2) Just lke a straght wre along spannng the same dstance. Force on a Current Loop What are the forces on the current loop? F = L B F 1 = ab( ) = ab F 2 = bb( ) = bb F = ab( ) = ab F 4 = bb( ) = bb F 1 b F 4 F 2 a F No net force. Consder torques about as. τ = r F Wth loop n plane all torques are ero too. 12 B = B

4 Lng Near Power Lnes Power lnes carr electrc power across long dstances at hgh oltages (5-5 kv) and currents of ~5A. What s the magnetc feld 1 m awa due to the current n a power lne? B = µ I Tm 5A = 4π 1 7 A 2π(1m) = 1 6 T = 1µT Compare ths to the tpcal Earth s feld of 1 µt. What are tpcal felds from currents n a home? Bot-Saart Law Integraton of the Bot-Saart* law ges B feld for a current dstrbuton: db = µ d s r 4π r Ths s an nerse-square law, just lke Coulomb s law. µ s the permeablt constant. µ 4π 1 7 Tm * (The re French, so the fnal t s are slent.) A 1 14 Magnetc Feld of a Long Straght Wre µ ds r µ sn( θ ) ds db = = 2 4π r 4π r r 2 = 2 + s 2 sn(θ) = sn(π θ) = µ B = 4π + µ = 2π 2 + s 2 ds = 2π 2 2 ( s ) ( + s ) ( + s ) 2 s= s= µ + µ = ds Electrc Currents and Magnetc Felds Electrc currents feel forces due to magnetc felds. The also create magnetc felds! Feld lnes encrcle the currents whch are ther source. ght hand rule: Thumb along, fngers curl n drecton of B B falls as 1/ for an nfnte wre

5 Forces Between Parallel Wres Consder the force on the bottom wre due to the top one. ght-hand rule: B s nto screen Wres attract f currents are parallel; epel f currents are antparallel. B = µ F = LB = µ 2 L For two 1A currents 1 m apart, force s N/m Ths s how the Ampere s actuall defned. 17 Sdes are of length r, currents are. What s the B feld (magntude and drecton) at the center of the square? Calculate B Long wre: a) and b) : B= snce opposte corners cancel. B = µ c) Each wre contrbutes same feld n drecton. felds cancel. µ B = 4 sn(45 ) µ = 4 1 d) Lower left and upper rght cancel; 2 others add n -,- drecton µ B = 2 ( + ) 2π r = µ ( + ) 2 πr 2 2π r 2 2π r 2 18 = 2µ πr Magnetc Feld of a Wre Loop To calculate the feld at the center of a wre loop of radus, ntegrate ds around the loop. Note that ds r s alwas the same (out of the screen) Each loop carres current, counterclockwse. Arc segments are at r, 2r, r. What s B at the dot? What s the feld? d B = µ d s r = µ ds 4π r 4π 2 = µ dφ = µ dφ 4π 2 4π B = 2π µ dφ 4π = µ 2π dφ 4π = µ 2 Feld of a Loop: B = µ 2 a) B = µ = µ 2 = µ 2 2 b) B = µ = µ 1 = µ c) B = µ = µ 1 = 1µ

6 Magnetc Flu We formulated the concept of electrc flu and used t n Gauss Law. Φ E = E d A We can also defne a correspondng quantt for the magnetc feld: Φ B = B d A Unt of magnetc flu s the Weber. 1 weber = 1 Wb = 1 T m 2 The law for magnetc flu correspondng to Gauss Law dffers snce there are no magnetc charges (monopoles): B d A = 21 Bot-Saart Law Integraton of the Bot-Saart* law ges B feld for a current dstrbuton: db = µ d s r 4π r Ths s an nerse-square law, just lke Coulomb s law. µ s the permeablt constant. µ 4π 1 7 Tm * (The re French, so the fnal t s are slent.) A 22 Magnetc Feld of a Long Straght Wre B falls as 1/ for an nfnte wre. B = µ Feld lnes encrcle the wre, and are alwas tangent to a crcle. Gauss Law: closed surface Ampere s Law E d A = Q enc ε o Ampere s Law: closed loop B d s = µ enc True for an closed loop! What s lne ntegral of B feld along a crcle wth the wre at the center? crcle B d s = µ o crcle ( ) ds = µ o = µ o Same for an radus crcle. Hmmm 2 enc s the sum of all currents passng through an surface enclosed b the loop. Sgns gen b rght hand rule. 24 6

7 Ampere s Law for an Infnte Wre We can start from Ampere s Law and get back the feld of an nfnte wre. B d s = µ enc closed loop B smmetr, B s ndependent of and of constant magntude at same r. Choose an Amperan contour : a crcle centered on the wre. B d s = µ enc = B tangental crcle B tangental = µ enc ( ) We take the ntegral (and sum the current) usng the rght hand rule. 25 Ampere s Law for a Thck Wre Now look at an nfnte wre, but spread unform current denst oer a thck wre. Outsde the wre(r>): same as before. Insde the wre (r<), enc s proportonal to fracton of area nsde r. B tangental = µ 2πr enc = πr2 π 2 B d s = 2πrB tangental = µ enc = µ r2 r B tangental = µ 2 2 B r 26 Magnetc Feld for Coaal Cable Man tpes of sgnals (ncludng cable TV) are transmtted on coaal cable. A thn core wre transmts the current core A clndrcal sheld carres a return current sheld =- core Insde: B n (2πr) = µ core What s B -nsde (between core and sheld)? -outsde the cable? B n = µ core / 2πr Outsde: B out (2πr) = µ ( core - sheld ) B out = You can t steal the sgnal wthout gettng nto the cable! Solenods If the cols are close together and the solenod s nfntel long, B nsde s unform and B outsde s ero! A col of current wound around a clnder s a common wa to make a regon contanng a farl unform magnetc feld

8 Ampere s Law and Solenods Torods Solenod of length L, N turns n = N/L turns per unt length If L>>r, no feld outsde. (Consder contours contanng the whole solenod) Insde, enc s N Take a fnte solenod and connect t to tself. The result s lke a solenod wthout ends: a torod. Insde, onl contrbuton to ntegral along loop s segment from a to b. B d s = B n h = µ enc = µ (nh) B n = nµ Note ths s ndependent of where the contour s placed n the solenod. The feld s perfectl unform f the solenod s nfntel long! B = µ N 2πr Feld s completel contaned. 29 8