ECE-305-Lecture18 - Engineering Electro-Magnetics...

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Engineering Electro-Magnetics Engineering Electro-Magnetics ECE-305, Lecture 18 Dennis McCaughey, Ph.D. 30 March, 2010
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 2 Home work Home work Problems 10.4, 10.10, 10.13, 10.21, and 10.22 Due 4/6
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 3 Summary of Maxwell’s Equations for Summary of Maxwell’s Equations for Static Fields Static Fields 0 0 v ρ ∇• = ∇× = ∇× = ∇• = D E H J B 0 0 v s v s s d Q dv d d I d d ρ = = = = = = D S E L H L J S B S Ñ Ñ Ñ Ñ Either set may be obtained from the other by the relevant application of either the Divergence Theorem or Stoke’s Theorem
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 4 Faraday’s Law Faraday’s Law A time varying magnetic field produces an electromotive force (emf) which may induce a current in a suitable closed circuit. The time variation may arise from a time varying magnetic field or a moving path in a stationary magnetic field. The closed path may not be a closed conducting path, e.g.. It may include a capacitor, or it may be a purely imaginary line in space A non-zero value of may be the resultof: 1. A time varying magnetic flux linkin a stationary closed path 2.Relative motion between a steady flux and a close path d emf volts dt d dt Φ = - Φ e 3. A combination of the two
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 5 Lenz’s Law Lenz’s Law The minus sign indicates 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. This is known as Lenz’s Law If the closed path is taken as an N-turn filamentary conductor, and consider the turns as coincident then: d emf N volts dt Φ = -
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 6 Differences from Electrostatics Differences from Electrostatics It is the voltage about a specific closed path. If any part of the path changes the emf will change The right hand rule is used to indicate the direction of the path in (1) Thumb indicates the direction of d S . A flux density B in the direction of d S and increasing with time thus produces an average value of E which is opposite to the positive direction about the closed path ( 29 ( 29 0 for electrostatics 1 s s emf d d d emf d d dt = = Φ = = = - E L B S E L B S Ñ Ñ
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 7 Case 1: Stationary Path, Time Varying Case 1: Stationary Path, Time Varying Magnetic Field Magnetic Field ( 29 ( 29 s Applying Stoke's Theorem The integrals are taken over identical surfaces which are perfectly general and may be chosen as differentials and s s emf d d t dS d t d d t t = = - ∇× = - × × = - = E L B S E B S E E B S B S Ñ
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03/30/2010 Dennis McCaughey, ECE305, Spring 2010 8 Example Example 0 2 0 2 0 Beginning with in a cylindrical region circular path: , , in the 0 plane from symmetry must be constant 2 2 , s kt z kt s kt d d t B e a b a a b z d E a d kB e a t emf aE kB e a a φ φ φ φ φ ρ ρ π π π π ρ = - = < = < = =
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