Chpter 7 Stedy Mgnetic Field september 2016 Microwve Lbortory Sogng University
Teching point Wht is the mgnetic field? Biot-Svrt s lw: Coulomb s lw of Mgnetic field Stedy current: current flow is independent of time Line current, Surfce current, Volume current Ampere s Circuitl Lw Curl 2016-09-06
Biot-Svrt Lw From experiments, it ws known tht dh due to current I flowing differentil vector length dl proportionl to IdL nd sinθ inversely proportionl to the squre of the distnce (1/R 2 ) Direction is norml to the plne contining dl nd R. In 1820, Biot nd Svrt formulted these observtions s follows IdL R R dh dh IdL R 2 4 R Since, current flows in closed pth, by superposition principle H dh IdL R 2 4 R 3
H due to infinitely long stright current I (1) Biot-Svrt Lw dh IdL R 2 4 R dh Idz ' ( z ' ) I dz ' 4 ( z ' ) z 2 2 3/ 2 2 2 3/ 2 4 ( z ' ) z dl 4 dz ' z R r r ' z R z ' 2 2 z ' z ' z H 2 2 3/ 2 I dz ' 4 ( z ' ) 2 2 3/ 2 I z ' 4 ( z ' ) I 2 dh I dz ' 4 ( z ' ) 2 2 2 1/ 2
Appliction of Biot-Svrt s lw Mgnetic field due to current loop: current element Idl = Idφ from symmetry considertion mgnetic field component z dir. use Biot-Svrt s lw; dh = Idφ 4π( 2 +h 2 ) ( 2 +h 2 ) H = I 2 2 ( 2 +h 2 ) 3 [ z] 2016-09-06
H due to infinitely long stright current I (2) The seprtion of the stremlines is proportionl to the rdius, or inversely proportionl to the mgnitude of H. H I 2 2 cf. n infinite line chrge D L V L 2 0 ln b 6
Ampere s circuitl lw If the symmetry is present in the problem, we cn use Ampere s circuitl lw insted of Biot-Svrt lw, H dl I The line integrl of H bout ny closed pth is exctly equl to the direct current enclosed by tht pth. This lw cn be derived from the Biot-Svrt lw. 7
Ex) H due to infinitely long stright current I H H ( ) I dl d H dl I H H ( ) Choose pth to ny section of which H is either perpendiculr or tngentil nd long which H is constnt. In this cse, circle of rdius ρ is pproprite for the pth. 2 2 H dl H d H d H 2 I I I H, H 2 2 0 0 8
Ex) H due to infinitely long stright current I An infinitely long, stright, solid, nonmgnetic conductor with circulr cross section of rdius crries stedy current I. Determine the mgnetic field intensity both inside nd outside the conductor. H dl I H H ( ) 2)0 I 1), H 2 2 H dl H d 2 H I 0 H, H 2 2 2 2 2 2 9
Ex) H of coxil cble (1) Infinitely long coxil cble crrying uniformly distributed current I in the center conductor nd I in the outer conductor. H dl I H H ( ) b c H I, 2 2π I, b 2π 10
Ex) H of coxil cble (2) c Externl field is zero. (shielding) : coxil cble crrying lrge currents would not produce ny noticeble effect in n djcent circuit. 11
Ex) H of infinite sheet current A sheet of current with uniform surfce current density of K = K y y (A/m) flowing in the positive y direction nd locted in the z= 0 plne. H H ( z) x x 12 H dl H x ( H )( x) K x x x y K y 1 H x H K N 2 2 : unit vector norml to the current sheet N If second sheet of current flowing in the opposite direction, K = -K y y, is plced t z = h, K N, 0 z h H 0, elsewhere
Solenoidl coil : H of Solenoidl Coil (1) H Cn control the -field inside Set up the constnt -field inside nd zero -field outside (idel cse) Cn be used electromgnet H H ( 전자석 ) N S Doorbell 13
H of Solenoidl Coil (2) 14
H of Solenoidl Coil (3) Mgnetic field of infinitely long solenoidl coil Inside : x component only nd constnt Outside : zero N windings per length L C H H C x x H dl I J ds S I 0 L x H H H x x HxL NI H x ni L NI ( n # of windings per unit length) x B H ni = 0 0 x L n 2 S 15
Toroidl Coil (1) Determine the mgnetic flux density inside closely wound toroidl coil with n ir core hving N turns of coil nd crrying current I. The toroid hs men rdius b, nd the rdius of ech turn is. Toroidl inductor H dl I H H( ) C H is constnt long ny circulr pth bout the xis of the toroid 16
Ex5-2) Toroidl Coil (2) Mgnetic field inside H NI,( b ) ( b ) 2π 0, elsewhere 17
Ex5-2) H field of Toroidl Coil (3) H field inside nd outside the toroid H dl = H 2πρ = NI H = NI 2πρ inside toroid I = 0 H = 0 outside toroid H NI,( b ) ( b ) 2π 0, elsewhere 18
Curl(1) Differentil form of Guss lw S D d S = Q = V ρ v dv S D d S = V ( D)dv D = ρ v Differentil form
Curl(2) Differentil form of Ampere s circuitl lw C H dl = S J d S = I integrl form Stokes theorem C H dl = Differentil form results from S ( H) d S S ( H) d S = S J d S H = J differentil form
Mgnetic flux density Define the mgnetic flux density B = μ 0 H (T) where μ 0 = 4π 10 7 (H/m) : permebility Mgnetic flux (c.f. D = ε 0 E ) S Φ = S B d S (Wb) B d S = 0 no mgnetic monopole B = 0
Mxwell s equtions Time-invrint cse: D = ρ v, S D d S = Q = V ρ v dv E = 0, C E dl = 0 H = J, C H dl = I = S J d S B = 0, S B d S = 0 constitutive reltions: D = ε 0 E B = μ 0 H
potentils E = 0, C E dl = 0 E = V electrosttic sclr potentil H = J, C H dl = I = S J d S mgnetosttic sclr potentil not defined insted mgnetic vector potentil defined B = 0 B = A : vector potentil
Exmple: mgnetic flux Mgnetic flux crossing the coxil line: Φ = H = I 2πρ ( < ρ < b) B = μ 0 H = μ 0I 2πρ S B d S = 0 d b μ 0 I 2πρ dρdz = μ 0I 2π ln b flux crossing < ρ < b nd 0 < z < d.