Steady magnetic field 2013 mk

Nguyễn Công Phương Engineering Electromagnetics The Steady Magnetic Field Contents I Introduction II Vector Analysis III Coulomb’s Law & Electric Field Intensity IV Electric Flux Density, Gauss’ Law & Divergence V Energy & Potential VI Current & Conductors VII Dielectrics & Capacitance VIII Poisson’s & Laplace’s Equations IX The Steady Magnetic Field X Magnetic Forces & Inductance XI Time – Varying Fields & Maxwell’s Equations XII The Uniform Plane Wave XIII Plane Wave Reflection & Dispersion XIV.Guided Waves & Radiation The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn The Steady Magnetic Field (1) Biot – Savart Law Ampere’s Circuital Law Curl Stokes’ Theorem Magnetic Flux & Magnetic Flux Density Magnetic Potential Derivation of the Steady – Magnetic – Field Law The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn The Steady Magnetic Field (2) • The source of the steady magnetic field may be: – Permanent magnet – Electric field changing linearly with time – Direct current • Consider the field produced by a differential dc element in free space only The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn Biot – Savart Law (1) dH = IdL × a R 4π R = Id L × R 4π R3 R12 dL1 I1 P aR12 H: magnetic field intensity (A/m) The direction of H is determined by the right-hand rule d H2 = dH = I1dL1 × a R12 IdL × a R 4π R 2 4π R12 →H= ∫ IdL × a R 4π R The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn Biot – Savart Law (2) I = Kb K b I = ∫ KdN I IdL = KdS H= ∫ IdL × a R 4π R =∫ S K × a R dS 4π R The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn z dL Biot – Savart Law (3) d H2 = R12 = ρ a ρ − z ' a z → a R12 = → dH = I → H2 = 4π 3/2 ρ dz ' aϕ ∫−∞ ( ρ + z '2 )3/2 = = 4π ρaρ x y I ∞ Idz ' a z × (ρ a ρ − z ' a z ) −∞ 4π ( ρ + z '2 )3/2 → H2 = ∫ I ρ aϕ R12 ρ + z '2 4π ( ρ + z ' ) ∞ z’az 4π R12 dz ' a z dL1 = ρaρ − z ' a z Idz ' a z × (ρ a ρ − z ' a z ) IdL1 × a R12 aR a z × a ρ = aϕ ; a z × a z = ∞ dz ' ∫−∞ ( ρ + z '2 )3/2 z '=∞ I ρ aϕ z' 4π ρ ρ + z '2 = z '=−∞ The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn I 2πρ aϕ z Biot – Savart Law (4) H= az z aφ I 2πρ dL aR z’az R12 aϕ ρaρ y α2 α ρ y x I aρ ρ x z z y I φ H= I 4πρ (sin α − sin α1 )aϕ x The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn H= I 2πρ Biot – Savart Law (5) aϕ The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn Biot – Savart Law (6) H= ∫ IdL × a R 4π R2 K × aR dS H=∫ S 4π R H=∫ V J × a R dV 4π R The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 10 Magnetic Potential (4) µ0 IdL dA = 4π R dL = dza z → dA = R = ρ + z2 → dA z = µ0 Idza z 4π ρ + z H= ∇× A µ0 Idz 4π ρ + z z IdL = Idzaz R = ρ + z2 P(ρ, φ, z) y az x φ dAϕ = dA ρ = → dH = µ0 ρ ∇ × dA  ∂dAz = −  µ0  ∂ρ µ0 → dH = z   aϕ  Idz ρ aϕ 2 3/ 4π ( ρ + z ) The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 59 Magnetic Potential (5) à0 IdL dA = R ã In the case of current flow throughout a volume with a density J, then: IdL = Jdv →A=∫ V µ0 Jdv 4π R The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 60 Magnetic Potential (6) Ex A very long, straight conductor lies along the z axis, carrying a uniform current I in the z direction Find the magnetic potential difference between two points in space? a Vm,ab = − ∫ H.dL b dL = d ρ aρ + ρ dϕ aϕ + dza z → Vm, ab = − a I dϕ = I (ϕb − ϕ a ) ∫b 2π 2π I H= aϕ 2πρ The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 61 z L Magnetic Potential (7) Ex IdL Find the vector magnetic potential in the plane bisecting a straight piece of thin wire of finite length 2L in free space µ0 IdL dA = 4π R R I ρ P( ρ ,0, 0) −L IdL = Idz ' a z R = ρ + ( z ') → A ( ρ ,0,0) = L ∫ z ' =− L 4π µ0 Idz ' ρ + ( z ')2 az L2 + ρ + L µ0 I = ln az 4π L2 + ρ − L The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 62 The Steady Magnetic Field Biot – Savart Law Ampere’s Circuital Law Curl Stokes’ Theorem Magnetic Flux & Magnetic Flux Density Magnetic Potential Derivation of the Steady – Magnetic – Field Law The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 63 (1) • Use formulae/definitions: H= ∫ IdL × a R 4π R B = à0 H B =ìA ã to show that A= V A=∫ V µ0 Jdv 4π R µ0J dv →H= 4π R ∫ IdL × a R 4π R2 The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 64 (2) A=∫ V µ0J dv →H= 4π R ∫ IdL × a R 4π R2 Current element at (x1, y1, z1), A at (x2, y2, z2) B = à0 H B ì A = H = B = ì A à0 à0 H2 = ì A2 à0 = à0 ì V → H2 = 4π µ0J1dv1 = 4π R12 4π µ0 J1dv1 V 4π R 12 → A2 = ∫  J1  J1dv1 ∫V ∇ × R12 = 4π ∫V  ∇ × R12  dv1 ∇× ( SV ) = (∇S ) × V + S (∇ × V )    ∫V  ∇2 R12  × J1 + R12 ( ∇2 × J1 ) dv1   The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 65 (3) A=∫ V µ0J dv →H= 4π R ∫ IdL × a R 4π R2 Current element at (x1, y1, z1), A at (x2, y2, z2)    ∫V  ∇ R12  × J1 + R12 ( ∇ × J1 ) dv1   ∇ × J1 =   1  → H2 =  ∇  × J1  dv1 ∫ V 4π  R12   R12 a R12 2 R12 = ( x2 − x1 ) + ( y2 − y1 ) + ( z2 − z1 ) → ∇ =− =− R12 R12 R12 → H2 = 4π → H2 = 4π ∫V a R12 × J1 R12 dv1 = 4π ∫V J1 × a R12 R12 dv1 The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 66 (4) A=∫ V µ0J dv →H= 4π R ∫ IdL × a R 4π R2 Current element at (x1, y1, z1), A at (x2, y2, z2) → H2 = 4π ∫V J1 × a R12 R12 dv1 J1dv1 = I1dL1 → H2 = ∫ I1dL1 × a R12 4π R12 The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 67 (5) ∇×H = J ìH = J B = à0 H B = ì A ì H = ì B à0 = à0 ì ì A ì ì A = ∇ (∇ A) − ∇ A ∇ A = ∇ Axa x + ∇ Ay a y + ∇ Az a z → ∇× H = ∇ (∇ A) − ∇ A µ0 The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 68 (6) ∇×H = J → ∇× H = ∇ (∇ A) − ∇ A µ0 A2 = ∫ V µ0 J1dv1 4π R12 ∇.(S A ) = A.(∇ S ) + S (∇.A ) µ0 → ∇ A = 4π     ∫V J1  ∇ R12  + R12 (∇2 J1 )  dv1   The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 69 (7) ìH = J à0 A = 4π     ∫V J1  ∇ R12  + R12 (∇2 J1 )  dv1   ∫V R12 (∇ J1 )dv1 = R12 ∇1 = = −∇ R12 R12 R12 µ0 → ∇2 A = 4π    ∫V −J1  ∇1 R12  dv1   The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 70 (8) ìH = J à0   → ∇2 A = −J1  ∇1 dv1    ∫ 4π V   R12   ∇.(S A ) = A.(∇ S ) + S (∇.A ) µ0 → ∇2 A = 4π  J1  ∫V  R12 ( ∇1 J1 ) − ∇1 ( R12 )  dv1 The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 71 (9) ∇×H = J µ0  J1  → ∇ A = ∇1 J1 ) − ∇1 ( )  dv1 (  ∫ 4π V  R12 R12  ∂ρ v ∇1 J = − =0 ∂t ∫ µ0 → ∇ A = − 4π S J.dS = ∫ ∇.Jdv V ∫S J1 dS1 = R12 The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 72 (10) ∇×H = J ∇ (∇ A) − ∇ A ì H = à0 à0 J x dv .A = Ax = ∫ V 4π R ∇2 Ax = − µ0 J x  ρv dv V =∫ → ∇ A = − µ J → ∇ A = − J µ  y y V 4πε R  ∇ Az = −µ0 J z ρv  ∇ V =− ε0 → ∇× H = J The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 73 ... Derivation of the Steady – Magnetic – Field Law The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn The Steady Magnetic Field (2) • The source of the steady magnetic field may be: – Permanent... Magnetic Potential Derivation of the Steady – Magnetic – Field Law The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn 14 Ampere''s Circuital Law (1) ∫ H.dL = I I The Steady Magnetic Field. .. The Steady Magnetic Field - sites.google.com/site/ncpdhbkhn The Steady Magnetic Field (1) Biot – Savart Law Ampere’s Circuital Law Curl Stokes’ Theorem Magnetic Flux & Magnetic Flux Density Magnetic