Time varying field 2013 mk

Nguyễn Công Phương Engineering Electromagnetics Time – Varying Fields & Maxwell’s Equations 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 Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Time – Varying Fields & Maxwell’s Equations Faraday’s Law Displacement Current Maxwell’s Equations in Point Form Maxwell’s Equations in Integral Form The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (1) dΦ emf = − V dt Emf is nonzero if one of any: • A time-changing flux linking a stationary closed path • Relative motion between a steady flux and a closed path • A combination of the two Minus sign ? Lenz’s law Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn dΦ emf = − dt Faraday's Law (2) d emf = ∫ E.dL → emf = ∫ E.dL = − dt ∫S B.dS B = B (t ) Φ = ∫ B.d S S ∂B → emf = ∫ E.dL = − ∫ dS S ∂t Stokes' theorem: ∫ E.dL = ∫ ( ∇ × E).dS S ∂B ∂B dS → ∫ (∇ × E).dS = −∫ dS → (∇ × E).dS = − S S ∂t ∂t ∂B →∇×E = − ∂t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (3) ∂B ∇×E = − ∂t ∂B emf = ∫ E.dL = − ∫ d S S ∂t ∂B = (steady) ∂t ∫ E.dL = ∇×E = Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (4) z B y v x=d x Φ = ∫ B.dS = Byd S dΦ emf = − dt dy → emf = − B d = − Bvd dt Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (5) z F = Qv × B B F → = v×B Q Em = v × B emf = y v x=d x ∫ Em dL = ∫ ( v × B).dL = ∫d vBdx = − Bvd Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (6) ∂B emf = ∫ E.dL = − ∫ dS + ∫ ( v × B).dL S ∂t B Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (7) Ex A single turn loop is situated in air, with a uniform magnetic field normal to its plane The area of the loop is 10 m2 If the rate of change of flux density is Wb/m2/s, what is the emf appearing at the terminals of the loop? dΦ emf = − N dt Φ = B.S dB → emf = − S = × 10 = 50 V dt Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 10 Time – Varying Fields & Maxwell’s Equations Faraday’s Law Displacement Current Maxwell’s Equations in Point Form Maxwell’s Equations in Integral Form The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 11 Displacement Current (1) ∇ × H = J → ∇.∇ × H = ∇.J ∂ ρ ∇ ∇× H = → v = (unreasonable) ∂ρv ∂t ∇.J = − ∂t ∇× H = J + G → = ∇.J + ∇.G ∂ρ v → ∇.G = ∂ρ v ∇.J = − ∂ t ∂t ∇.D = ρ v ∂ ∂D ∂D → ∇.G = (∇.D) = ∇ →G = ∂t ∂t ∂t ∇×H = J +G ∂D → ∇×H = J + ∂t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 12 Displacement Current (2) ∂D ∇×H = J + ∂t → ∇× H = J + J d ∂D Define J d = ∂t ∂D In nonconducting medium J = → ∇ × H = ∂t ∂D I d = ∫ J d dS = ∫ dS S S ∂t ∂D ∫S (∇× H).dS = ∫S J.dS + ∫S ∂t dS ∂D → ∫ H.dL = I + I d = I + ∫ dS S ∂t ∫ H.dL = ∫S (∇ × H ).dS Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 13 Displacement Current (3) C emf = V0 cos ωt I k → I = −ωCV0 sin ωt = −ω ∫ k εS d V0 sin ωt B H.dL = I k  V0  D = ε E = ε  cos ωt  d  → I = −ω ε S V sin ωt d ∂D ∂D d Id = ∫ dS = S S ∂t ∂t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 14 Time – Varying Fields & Maxwell’s Equations Faraday’s Law Displacement Current Maxwell’s Equations in Point Form Maxwell’s Equations in Integral Form The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 15 Maxwell’s Equations in Point Form (1) ∂B ∇×E = − ∂t ∂D ∇×H = J + ∂t ∇ D = ρv ∇.B = Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 16 Ex Maxwell’s Equations in Point Form (2) Given an electric field E = Acosω(t – z/c)ay Determine the time-dependent MFI H in free space? B H ìE = = à0 t t ∂E y ω  z ∇× E = − a x = − A sin ω  t −  ∂z c  c ωA  z →H= sin ω  t −  a x ∫ c µ0  c  z = cos ω  t −  a x c µ0  c A Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 17 Time – Varying Fields & Maxwell’s Equations Faraday’s Law Displacement Current Maxwell’s Equations in Point Form Maxwell’s Equations in Integral Form The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 18 Maxwell’s Equations in Integral Form ∂B ∇×E = − ∂t ∂B ∫ E.dL = −∫S ∂t dS Et1 = Et ∂D ∇×H = J + ∂t ∂D ∫ H.dL = I + ∫S ∂t dS H t1 = H t ∇ D = ρv ∫ S D.dS = ∫V ρv dv ∇.B = ∫ S B.dS = DN − DN = ρ S BN = BN Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 19 Time – Varying Fields & Maxwell’s Equations Faraday’s Law Displacement Current Maxwell’s Equations in Point Form Maxwell’s Equations in Integral Form The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 20 The Retarded Potentials (1) E = −∇V → ∇ × E = −∇ × (∇V ) → ∇ × E = ∂B = ∇ × (∇ V ) ∂B → =0 ∇ ×E = − ∂t ∂t (unreasonable) E = −∇V + N → ∇ × E = −∇ × (∇V ) + ∇ × N ∂B ∇ × (∇V ) = → ∇× N = − ∂B ∂t ∇×E = − B = ∇× A ∂t ∂ ∂A ∂A → ∇ × N = − (∇ × A ) → ∇ × N = −∇ × →N =− ∂t ∂t ∂t ∂A → E = −∇V − ∂t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 21 The Retarded Potentials (2) B = ∇×A ∂A E = −∇V − ∂t ∂D ∇× H = J + ∂t ∇ D = ρv 1  ∂V ∂ A  −   ∇ ×∇ × A = J + ε  −∇ ∂t ∂t  µ  → ∂    −∇ ∇ V − ∇ A = ε ρ   v   ∂t   Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 22 The Retarded Potentials (3) 1  ∂V ∂ A  −   ∇ ×∇ × A = J + ε  −∇ ∂t ∂t  µ   ∂    −∇ ∇ V − ∇ A = ε ρ   v   ∂t     ∂ V ∂ A +  ∇(∇.A ) − ∇ A = µ J − µε  ∇  ∂t   ∂t → ρv ∂  ∇ V + ∂t (∇.A ) = − ε Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 23 The Retarded Potentials (4)   ∂ V ∂ A +  ∇(∇.A ) − ∇ A = µ J − µε  ∇  ∂t ∂t    ρv ∂  ∇ V + ∂t (∇.A ) = − ε ∂V Define ∇.A = − µε ∂t  ∇ A = −µ J + µε → ∇ 2V = − ρv + µε  ε ∂2 A ∂t ∂ 2V ∂t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 24 The Retarded Potentials (5) V =∫ V ρv dv 4πε R R t′ = t − v →V = ∫ V [ ρv ] dv 4πε R   R  Ex : ρ v = e cos ωt → [ ρv ] = e cos ω  t −   v    −r A=∫ V −r µJ µ[J ] dv → A = ∫ dv V 4π R 4π R Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 25 ... 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 Time – Varying. .. ∂t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (3) ∂B ∇×E = − ∂t ∂B emf = ∫ E.dL = − ∫ d S S ∂t ∂B = (steady) ∂t ∫ E.dL = ∇×E = Time – Varying Fields... ∫d vBdx = − Bvd Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn Faraday's Law (6) ∂B emf = ∫ E.dL = − ∫ dS + ∫ ( v × B).dL S ∂t B Time – Varying Fields & Maxwell’s