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Chapter 10 Time-Varying Fields and Maxwell’s Equations Two New Concepts: The electric field produced by a changing magnetic field (Faraday) The magnetic field produced by a changing electric field (Maxwell) Faraday's Law Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc. Faraday's law is a fundamental relationship which comes from Maxwell's equations. It serves as a succinct summary of the ways a voltage may be generated by a changing magnetic environment. The induced emf in a coil is equal to the negative of the rate of change of magnetic flux times the number of turns in the coil. It involves the interaction of charge with magnetic field. Faraday's Law From GSU Webpage Faraday's Law Lenz’s Law When an emf is generated by a change in magnetic flux according to Faraday's Law, the polarity of the induced emf is such that it produces a current whose magnetic field opposes the change which produces it. The induced magnetic field inside any loop of wire always acts to keep the magnetic flux in the loop constant. In the examples below, if the B field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant. From GSU Webpage Faraday's Law d emf dt A nonzero value of emf may result from of the situations: 1 – A time-changing flux linking a stationary closed path 2 – Relative motion between a steady flux and a closed path 3 – A combination of the two Faraday's Law emf N emf d dt E_dot_ d L S d dt B_dot_ d S d emf E_dot_ d L B_dot_ d S dt S Applying Stokes' Theorem ( Del E)_dot_ d S S S d B_dot_ d S dt Removing the intregals - assuming same surface ( Del E)_dot_dS Del E d B dt d B _dot_dS dt Faraday's Law emf B y d d dt d y d dt B B v d Consider example using concept of motional emf F Qv B F v B Q Em v B The force per unit charge is called the motional electric field intensity Subject every portion of the moving conductor Faraday's Law Displacement Current Faraday's experimental law led to Maxwell's equation d B dt Ampere's circuital law for staedy magnetic fields Del E Del H J It shows its inadequacy for time-varying Del_dot_Del H 0 Del_dot_J The divergence of the curl is zero, and so the Del of J. However, the equation of continuity Del_dot_J d v dt Then it shows that d v dt Del H J can be true only if 0 This is an unrealistic limitation and Del H must be amended before we can accept it for time-varying fields J Displacement Current Suppose we add a term G Del H JG Taking the divergence 0 Del_dot_J Del_dot_G d v dt Replacing .v for Del_dot_D Del_dot_G G d D dt Del H J d D dt Displacement Current H_dot_ d L emf I d I D _dot_ d S dt S Vo cos t C Vo sin t Displacement current is associated with time-varying fields and exists in all imperfect Conductors carrying a time-varying conduction current. Maxwell’s Equations In Point Form Maxwell’s Equations In Integral Form The Retarded Potentials V A v 4 R dv Knowledge of the distribution of charge density and current density throughout space will enable us to determine V and A vol ( J) dv 4 R vol Every t appearing in v or (J) jas been replaced by a retarded time t1 v t R v 1 D10.7 while 8 4 cos 10 t A point charge of 8 4 cos 10 t C is located at P+(0, 0, 1.5) is at P-(0, 0, -1.5), both in free space Find V and P(r=450, , =0) 0 Pplus 0 1.5 0 Pminus 0 1.5 R1 P Pplus 7 12 0 8.854 10 cos 10 t 6 Vplus 76.083 0 0 P 0 450 R1 448.5 0 4 10 ro1 4 10 r 450 8 R1 v v 1 0 0 Vplus 9 t 15 10 ro1 12 4 8.854187917 10 R1 D10.7 R1 P Pminus 7 0 4 10 R1 451.5 12 0 8.854 10 cos 10 t 6 ro1 4 10 8 Vminus 75.524 V Vplus Vminus V 151.607 R1 v v 1 9 t 15 10 0 0 Vminus ro1 R1 12 4 8.854187917 10