Bài giảng Engineering electromagnetic - Chapter XI: Time - Varying Fields & Maxwell’s Equations - Nguyễn Công Phương

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 2 Time – Varying Fields & Maxwell’s Equations 1. Faraday’s Law 2. Displacement Current 3. Maxwell’s Equations in Point Form 4. Maxwell’s Equations in Integral Form 5. The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 3 Faraday's Law (1) d 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 4 Faraday's Law (2) d emf  dt d emf E.dd L  B. S emf   E.d L dt S BB ()t  B.d S S B emf E.dd L  . S S t Stokes' theorem: EL.().dd E S S B B ().ESdd .S().ESdd  .S SSt t B E  t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 5 Faraday's Law (3) B E  t E.d L  0 B  emf E.dd L . S S t E 0 B  0 (steady) t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 6 Faraday's Law (4) B z y v x  d x  B.d S  Byd S dy emf B dBvd d emf  dt dt Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 7 Faraday's Law (5) B z FvBQ F y vB Q v EvBm  x  d x 0 emfE.Ldd ( v B.L ) vBdx Bvd m d Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 8 Faraday's Law (6) B emfE.dd L . S ( v B ) . d L S t  B Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 9 Ex. Faraday's Law (7) 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 5 Wb/m2/s, what is the emf appearing at the terminals of the loop? d emf N dB dt emf S 51050V  dt B.S Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 10 Time – Varying Fields & Maxwell’s Equations 1. Faraday’s Law 2. Displacement Current 3. Maxwell’s Equations in Point Form 4. Maxwell’s Equations in Integral Form 5. The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 11 Displacement Current (1) HJ  .H  .J  .H 0 v 0 (unreasonable)  t .J v t HJG   0 .J .G   v .J v .G  t t .D v  D D .G (.)  D . G D t t t HJ   t HJG Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 12 Displacement Current (2) D HJ   t HJJ   d D Define Jd  t D In nonconducting medium J = 0 H  t D I JS..dd S ddSSt D D ()..HSdd  JS  . d SHL..dIIId d S SSSt S t HL.().dd H S S Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 13 Displacement Current (3) C I emf Vt0 cos k ICVt 0 sin  S B  Vtsin d 0 H.dI L  k k V DE 0 cos t   S d IVt sin D D d d 0 IdS. S d S tt Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 14 Time – Varying Fields & Maxwell’s Equations 1. Faraday’s Law 2. Displacement Current 3. Maxwell’s Equations in Point Form 4. Maxwell’s Equations in Integral Form 5. 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 HJ   t .D  v .B  0 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? BH E   tt0 Ey  z Ea x Atsin   z cc Az Ha sin t  x cc0  Az cost ax cc0  Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 17 Time – Varying Fields & Maxwell’s Equations 1. Faraday’s Law 2. Displacement Current 3. Maxwell’s Equations in Point Form 4. Maxwell’s Equations in Integral Form 5. The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 18 Maxwell’s Equations in Integral Form B B E  EL..dd S Ett12 E t S t D D HJ   HL..dI d S HHtt12 t S t .D  DS.ddv  DD   v SVv NN12 S .B 0 BS.0d  BBNN12 S Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 19 Time – Varying Fields & Maxwell’s Equations 1. Faraday’s Law 2. Displacement Current 3. Maxwell’s Equations in Point Form 4. Maxwell’s Equations in Integral Form 5. The Retarded Potentials Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 20 The Retarded Potentials (1) E V E () V E 0 B 0()  V B 0 E  t t (unreasonable) ENV  EN () V  B ()0V  N  B t E  t BA   A A NA ()  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) BA  A E V  t D HJ   t .D  v  1 V 2A  AJ      2  tt    . V  .A   v  t Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 22 The Retarded Potentials (3)  1 V 2A  AJ      2  tt   . V  .A   v  t 2  2 V A ().A  A  J     tt2     2   V () .A v  t  Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 23 The Retarded Potentials (4) 2  2 V A ().A  A  J     tt2    2   V () .A v  t  V Define .A  t 2  2  A AJ2  t    2V 2V v    t 2 Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 24 The Retarded Potentials (5)  dv V  v V 4 R v dv V R V tt'   4 R v r r  R  Ex :v  et cos v etcos     v  J []J A  dv A dv V 4 R V 4 R Time – Varying Fields & Maxwell’s Equations - sites.google.com/site/ncpdhbkhn 25