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ENE 325 Electromagnetic Fields and Waves Lecture 10 Time-Varying Fields and Maxwell’s Equations

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Magnetic boundary conditions B n1 = B n2 and Inductance and mutual inductance self inductance L is defined as the ratio of flux linkage to the current generating the flux, henrys or Wb/A. mutual inductance M, where M 12 = M 21. Review

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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.

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Faraday’s law V where emf = electromotive force that may establish a current in a suitable closed circuit and is a voltage that arises from conductors moving in static or changing magnetic fields. is arisen from 1. the change of flux in a closed path 2. the moving closed path in a stationary magnetic field 3. both 1 and 2 For the N number of loops, v.

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emf in the closed loop is not zero a) direction of the induced current b) emf

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Changing flux in a stationary path (transformer emf) From apply Stokes’ theorem, So we have 1 st Maxwell’s equation

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Transformer To transform AC voltages and currents between a pair of windings in magnetic circuits With Faraday’s law, we have Since the term is the same for both voltages, so we get

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Ex1 Assume, prove the 1 st Maxwell’s equation.

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Changing flux in a moving closed path (1) a conductor moves in a uniform magnetic field. The sign of emf determines the direction of the induced current.

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Changing flux in a moving closed path (2) Examine in a different point of view So we get V.

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Changing flux in a moving closed path (3) Combing both effects yields

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Ex2 Let mT, find 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.

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Ex3 A moving conductor is located on the conducting rail as shown at time t = 0, a) find emf when the conductor is at rest at x = 0.05 m and T.

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b) find emf when the conductor is moving with the speed m/s.

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Displacement current (1) The next Maxwell’s equation can be found in terms of time- changing electric field From a steady magnetic field, From the equation of continuity, therefore this is impossible!

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Displacement current (2) Another term must be added to make the equation valid. 2 nd Maxwell’s equation In a non-conductive medium,

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Displacement current (3) We can show the displacement current as The more general Ampere’s circuital law:

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Where is the displacement current from? Consider a simple current loop, let emf = V o cos t

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Ex4 Determine the magnitude of for the following situations: a) a) in the air near the antenna that radiates V/m.

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b) a pair of 100 cm 2 area plates separated by a 1.0 mm thick layer of lossy dielectric characterized by r = 50 and = 1.0 S/m given the voltage across plates V(t) = 1.0cos(2 10 3 t) V.

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