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10 電磁 III How is an aurora so thin yet so tall and wide? Sections 1. 磁場 2. 電流與磁場 10-1 磁場 The electric field and the magnetic field Electromagnets and permanent magnets 10-1.1 The definition of B FB qv B F E (N/C) (V m) q0 The tracks in a bubble chamber The SI unit for B 1 tesla = 1T =1 N/A‧m=104 gauss 108 T 1.5 T 10-2 T 10-4 T 10-10 T 10-14 T Magnetic Field Lines Magnetic vs. electric dipoles A horseshoe and a Cshaped magnets 例 1 A 5.3 MeV proton v B = 1.2 mT 2 K / m 3.2 10 m/s 7 FB qvB sin 6.1 10 15 a FB / m 3.7 10 m/s 12 N 10-1.2 Crossed Fields: Discovery of the Electron A cathode ray tube Thomson’s procedure: 設定E = 0, B = 0, 並記錄光點位置 開啟電場 開啟磁場,並調至與電場相等 Calculation 2 qEL y , qE qvB 2 2mv 2 2 m B L v E/B, q 2 yE the charge-to-mass ratio of the electron 1.75881961011 C·kg-1 F QE ay m m 1 2 y a y t , L vxt 2 QEL2 y 2mvx2 10-1.3 Crossed Fields: The Hall Effect By the conduction electrons in copper: V Ed eE evd B J i vd ne neA Bi A n (l ) Vle d 例 2 A cube generator d = 1.5 cm, v = 4.0m/s, B = 0.05T eE evB V Ed V dvB V 3.0mV 10-1.4 A Circulating Charged Particle F ma mv / r 2 qvB mv / r 2 r mv / qB T 2r / v 2m / qB 頻率與軌跡 The frequency and angular frequency 1 qB qB f 2f T 2m m The magnetic bottle machine Helical Paths V∥ and V⊥ v11 v cos v v sin The pitch (螺距) of the helical path 2m p v11T v cos qB 極光橢圓圈 例 3 The Mass Spectrometer (質譜儀) 1 2 mv qV 2 v 2qV / m mv 1 r qB B x 2r 2mV q 質譜儀 x = 1.6254m, V = 1000.0V, B = 80.000mT 2 x 2r B 2 2mV q 2 B qx m 203.93u 8V Isotope Separation Centrifuge and diffusion chamber 10-1.5 Cyclotrons and Synchrotrons (迴旋加速器與同步加速器) Fermilab: 6.3km ring Synchrotrons The f f osc resonance condition: When proton energy > 50Mev: Out of resonance (relativistic effect) A huge magnet (4×106 m2) is needed for high energy (500Gev) protons The proton sychrotron at Fermilab can produces 1Tev proton CERN LHC The LHC is 27km long and sits 100m below the surface. 10-1.6 Magnetic Force on a Current-Carrying Wire Magnetic Force q it iL / vd FB qvd B sin iLB FB iL B For a wire segment: dFB idL B 例 4 A length of wire with a semicircular arc Calculation F1 F3 iLB dF iBdL iB ( Rd ) F2 0 dF sin iBR sin d iBR cos 0 0 2iBR F F1 F2 F3 2iB ( L R ) 線圈 10-1.7 Torque on A Current Loop F2 and F4 cancel F1 and F3 form a force couple F2 ibB sin( 90 ) ibB cos b b (iaB sin ) (iaB sin ) 2 2 iabB sin N ( NiA) B sin 例 5 A galvanometer for analog meters NiAB sin NiAB sin [( 250)(100 10-6 A )( 2.52 10-4 m 2 ) (0.23T )(sin 90 )] / 28 5.2 108 M m / degree 10-1.8 The Magnetic Dipole The magnetic dipole moments NiA B sin B (cf : p E ) The magnetic potential energy U ( ) p E U ( ) B 磁能 U ( B ) ( B ) 2 B 10-2 Magnetic Fields due to Currents Conventional rocket EM Rail Gun 10-2.1 Calculating the Magnetic Field due to a current dE dE 1 4 e 0 1 4 e 0 dq 2 r dq 3 r r 0 ids sin dB 4 r2 0 id s r dB 4 r3 The law of Biot and Savart Magnetic Field Due to a Current in a Long Straight Wire 0 4 10 7 T m/A permeabili ty 0 ids sin dB 2 4 r B dB 2 dB 0i 2 0 0 sin ds 2 r Integration r s R , sin = 2 R 2 s R 2 2 Rds 0i B 2 3/ 2 2 0 (s R ) 2 0i s 0i ] [ 2 = 2 1/ 2 0 2R 2R ( s R ) Magnetic Field Due to a Current in a Circular Arc of Wire 0 ids sin 90 0 ids dB = 2 4 R 4 R 2 0 iRd B dB 0 4 R2 0i 0i = d 4R 0 4R 例 6 What B does the current produce? ds r 0 B1 B2 0 0i ( / 2) 0i B3 = 4R 8R 10-2.2 Two Parallel Currents 0ia Ba 2d Fba ib L Ba 0 Liaib Fba ib LBa sin 90 2d 例 7 The Field Between Two wires B ( x) Ba ( x) Bb ( x) = 0i 2 ( d x) 0i 2 ( d x) 0id 2 ( d 2 x 2 ) 10-2.3 Ampere’s Law Comparing Gauss’ lawand Ampere’s law Ampere’s law B ds 0ienc The Magnetic Field Outside a Long Straight Wire with Current B ds B cos ds B ds B(2r ) B(2r ) 0i The Magnetic Field Inside a Long Straight Wire with Current B ds B (2r ) ienc r 2 i R 2 r 2 B ( 2r ) 0i R 2 例 7 A hollow conducting cylinder r ienc JdA cr ( 2rdr ) 2 a c( r a ) 4 4 2 B( 2r ) c( r a ) 4 2 4 10-2.4 Solenoids and Toroids Magnetic Field of a Solenoid (螺線管) Magnetic Field of a Toroid (螺線環) Magnetic Field of a Solenoid b c B ds a B ds b B ds d a B ds B ds c Bh 0inh d B 0in Magnetic Field of a Toroid 0iN B(2r ) 0iN B 2r 磁圍阻核融合反應器 Tokamak Fusion Test Reactor 10-2.5 A Current Carrying Coil as a Magnetic Dipole A current loop and a bar magnet Magnetic Field of a Coil B dB cos 0ids R 4r 2 r 0iR 2R 2 2 3/ 2 4 ( R z ) 0iR 2 2( R 2 z 2 ) 3 / 2 0i (at center of coil) 2R Coulomb’s Law •Using Gauss’s law to take advantage of special symmetry situations •Gaussian surfaces •高斯面上各點電場與 面內總電荷相關 Gauss’ Law •Flux enclosed charge e 0 e 0 E dA qenc Gauss’Law and Coulomb’Law •From G.L. to C.L. e 0 E dA e 0 EdA q e 0 E dA q e 0 E ( 4r ) q 2 1 q E 2 4e0 r 敬請期待 電磁 IV