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Chapter 10. Time-Varying Fields and Maxwell’s Equations 1. Faraday’s Law • • • A time-varying magnetic field produces an electromotive force (emf) which may establish a current in a suitable closed circuit. An electromotive force is merely a voltage that arises from conductors moving in a magnetic field or from changing magnetic fields. Faraday’s Law: emf d V dt • 1. 2. 3. The minus sign is an indication that the emf is in such a direction as to produce a current whose flux, if added to the original flux, would reduce the magnitude of the emf.(Lenz’s Law) A time-changing flux linking a stationary closed path Relative motion between a steady flux and a closed path A combination of the two. 목원대학교 정보통신공학전공 9-1 • If the closed path is taken by an N-turn filamentary conductor emf N • Define: emf E dL d dt emf E dL d B dS dt S • The fingers of our right hand indicate the direction of the closed path, and our thumb indicates the direction of dS. • A flux density B in the direction of dS and increasing with time thus produces an average values of E which is opposite to the positive direction about the closed path • Stationary Path B dS and S t emf E dL E dS B dS t E dL E dS (Stokes' Theorem) S E B t Electrosta tics : E dL 0 and E 0 목원대학교 정보통신공학전공 9-2 Example) Within th e cylindrica l region b, B Bo e kt a z Choosing the circular path a, a b, in the z 0 plane 1 emf 2aE kBo e kt a 2 E kBo e kt a 2 ( E ) 1 Ez kBo e kt 1 kBo e kt 2 E 2 1 E kBo e kt a 2 A filamentar y conductor of resistance R would have a current flowing in the negative a and this current establish a flux withi n the circular loop in the negative a z direction. [Time - constant flux and a moving closed path] Byd d dy emf B d Bvd dt dt E dL EL Bvd * Lenz' s law 목원대학교 정보통신공학전공 9-3 F Qv B F vB Q The force per unit charge : the motional electric field intensity E m Em v B If the moving conductor were lifted off the rails, this electric field intensity would force electrons to one end of the bar (the far end) until the static field due to these charges just balanced the field induced by the motion of the bar. The resultant tangential electric field intensity would then be zero along the length of the bar. emf E m dL v B dL vBdx Bvd 0 d In the case of a conductor moving in a uniform constant magnetic field emf E dL E m dL v B dL If the magnetic flux density is changing with time , B dS v B dL S t transfor mer emf motional emf emf E dL emf - d dt 목원대학교 정보통신공학전공 9-4 2. Displacement Current B t Steady magnetic field H J E v t v D 0 J G Thus G ( D) t t t H 0 J but the eq. of continuity : J H J G G D t H J D t D : displaceme nt current density t Conduction current density, the motion of charge (zero net charge density) : J E Convection current density, the motion of volume charge density : J v v H J Jd Jd In a nonconduct ing medium, H 목원대학교 정보통신공학전공 D (J 0) t 9-5 D dS S t I d J d dS S The time varying version of Ampere' s cirsuital law : Applying Stokes' theorem : D H d S J d S S S S dS t D dS S t H dL I I d I Circuit : a filamentar y loop and a parallel - plate capacitor A magnetic field varying sinusoidal ly with ti me emf Vo cos t Negligible resistance and inductance : I C H k k dV S CVo sin t Vo sin t dt d dL I k V cos t Within the capacitor : D E o d D S Id S Vo sin t t d Displaceme nt current is associated with time - varying electric fields and therefore exists in all imperfect conductors carrying a time - varying conduction current. 목원대학교 정보통신공학전공 9-6 3. Maxwell’s Equations in Point Form B t (If a changing magnetic field is present, E may have circulatio n.) D H J t D v (Charge density is a source of electric flux density) E B 0 (no magnetic charges) D E o E P B H o (H M ) J E conduction current density J v v convection current density For linear materials, P e o E M m H Lorentz force equation ( the force per unit volum e) f v (E v B) 목원대학교 정보통신공학전공 9-7 4. Maxwell’s Equations in Integral Form B t D H J t D v B dS (Faraday' s law) S t D H d L I S t dS (Ampere' s circuital law) E E dL B 0 D dS B dS 0 S vol S v dv (Gauss' s law for the electric field) (Gauss' s law for the magnetic field) [Boundary conditions ] Et1 Et 2 H t1 H t 2 (K 0) DN 1 DN 2 S BN 1 BN 2 In a perfect conductor, E 0 H 0 J 0 If region 2 is a perfect conductor, E t1 0 H t1 K D N 1 S 목원대학교 정보통신공학전공 BN 1 0 9-8 5. The Retarded Potentials v dv 2 V v (static), E V (static) vol 4R Jdv Vector magnetic potential : A (dc), 2 A J (dc), B A (dc) vol 4R Scalar Electric potential : V In time - varying fields, B A ( B 0) B E E V N ( V 0) t B A A E 0 N N N A N t t t t E V - A t D H J t V 2 A A - A J 2 t t D v V - A v 2V A v t t V Define A t v 2A 2V 2 2 A J 2 and V 2 t t 9-9 2 목원대학교 정보통신공학전공 V 2 A A J t t 1