* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Lecture 10 - web page for staff
Survey
Document related concepts
Neutron magnetic moment wikipedia , lookup
Equation of state wikipedia , lookup
Path integral formulation wikipedia , lookup
History of electromagnetic theory wikipedia , lookup
Magnetic field wikipedia , lookup
Field (physics) wikipedia , lookup
Magnetic monopole wikipedia , lookup
Time in physics wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Electromagnetism wikipedia , lookup
Superconductivity wikipedia , lookup
Electromagnet wikipedia , lookup
Transcript
ENE 325 Electromagnetic Fields and Waves Lecture 10 Time-Varying Fields and Maxwell’s Equations Review Magnetic boundary conditions Bn1 = Bn2 H 1 H 2 a n12 K . and Inductance and mutual inductance self inductance L is defined as the ratio of flux linkage to the current generating the flux, Ntotal henrys or Wb/A. L I N 212 mutual inductance M , M 12 I1 where M12 = M21. N M 21 1 21 I2 Time-Varying fields and Maxwell’s equations Concept The electric field E is produced by the change in the magnetic field B. The magnetic field B is produced by the change in the electric field E. Faraday’s law d emf dt V where emf = electromotive force that may establish a current in a suitable closed circuit and d is a voltage that arises dt from conductors moving in static or changing magnetic fields. d is arisen from dt 1. the change of flux in a closed path 2. the moving closed path in a stationary magnetic field 3. both 1 and 2 d For the N number of loops, emf N v. dt emf in the closed loop is not zero d emf E d L B d S 0 dt S a) direction of the induced current b) emf Changing flux in a stationary path (transformer emf) From emf E d L d B d S, dt S apply Stokes’ theorem, B dS ( E ) d S S S t B ( E ) d S dS t So we have B ( E ) . t 1st Maxwell’s equation Transformer To transform AC voltages and currents between a pair of windings in magnetic circuits N1i1 N2i2 With Faraday’s law, we have d d v1 N1 , v2 N 2 . dt dt Since the term d is the same for both voltages, so we get dt N v2 2 v1. N1 Ex1 Assume B Boekt a z, prove the 1st Maxwell’s equation. z r Changing flux in a moving closed path (1) a conductor moves in a uniform magnetic field. yw B d S Bo a z dxdya z Bo yw. 00 d dy emf Bo w Bo vw. dt dt The sign of emf determines the direction of the induced current. Changing flux in a moving closed path (2) Examine in a different point of view z y B va y w x So we get emf (v B) d L V. Changing flux in a moving closed path (3) Combing both effects yields emf d B E d L d S (v B) d L. dt S t Ex2 Let B 6cos10 t sin 0.01xa z mT, find 6 a) flux passing through the surface z = 0, 0 < x < 20 m, and 0 < y < 3 m at t = 1 S. b) value of closed line integral around the surface specified above at t = 1 S. Ex3 A moving conductor is located on the conducting rail as shown at time t = 0, z y B 0.05 x a) find emf when the conductor is at rest at x = 0.05 m and B 0.3sin104 ta z T. b) find emf when the conductor is moving with the speed v 150a x m/s. Displacement current (1) The next Maxwell’s equation can be found in terms of timechanging electric field From a steady magnetic field, H J H J 0. From the equation of continuity, rv J t rv this is impossible! therefore 0. t Displacement current (2) Another term must be added to make the equation valid. H J J d . In a non-conductive medium, J 0. 2nd Maxwell’s equation Displacement current (3) We can show the displacement current as D Id J d d S d S. s s t The more general Ampere’s circuital law: H d L I Id Where is the displacement current from? Consider a simple current loop, let emf = Vocost Ex4 Determine the magnitude of J d for the following situations: a) in the air near the antenna that radiates E 80cos(6.277 108 t 2.092 y)a z V/m. b) a pair of 100 cm2 area plates separated by a 1.0 mm thick layer of lossy dielectric characterized by r = 50 and = 1.010-4 S/m given the voltage across plates V(t) = 1.0cos(2103t) V.