ELEG413lec2 - ELEG 648 Lecture #2 Mark Mirotznik, Ph.D....

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ELEG 648 Lecture #2 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Tel: (302)831-4221 Email: mirotzni@ece.udel.edu
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Maxwell’s Equations in Differential Form m i c B D J J t D H M t B E ρ = = + + = × - - = × Faraday’s Law Ampere’s Law Gauss’s Law Gauss’s Magnetic Law
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Faraday’s Law s d B t l d E t B E c s - = - = × ∫ ∫ S C t B E
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Ampere’s Law ∫ ∫ ∫ ∫ + = + = × s c s s d J s d D t l d H t D J H t D J J H H
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Gauss’s Law ∫ ∫ ∫ ∫ ∫ = = = v tot s Q dv s d D D ρ tot Q D
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Gauss’s Magnetic Law 0 0 ∫ ∫ = = s s d B B B “all the flow of B entering the volume V must leave the volume”
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Current Continuity Equation t q J - = J “Charge must be conserved”
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CONSTITUTIVE RELATIONS E J H B E D c & σ μ ε = = = ε= ε r ε o = permittivity (F/m) ε o = 8.854 x 10 -12 (F/m) μ= μ r μ o = permeability (H/m) μ o = 4 π x 10 -7 (H/m) σ = conductivity (S/m)
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POWER and ENERGY J i ε, μ, σ E, H V S n i c d i d J J J J E t E H eq M t H E eq + + = + + = × - = - = × σ ε μ ) 2 ( ) 1 ( ) 2 ( ) 1 ( eq E eq H - ) ( ) 3 ( i c d d J J J E M H H E E H eq + + - - = × - × take Using the vector identity ) ( ) ( ) ( B A A B B A × - × = × 0 ) ( ) ( ) 4 ( = + + + + × i c d d J J J E M H H E eq Integrate eq4 over the volume V in the figure ∫ ∫ ∫ ∫ ∫ ∫ + + + - = × v i c d d v dv J J J E M H dv H E eq )] ( [ ) ( ) 5 ( Applying the divergence theorem 0 )] [ ) ( ) 6 ( = + + + + × ∫ ∫ ∫ ∫ ∫ v i s dv J E E E t E E t H H ds H E eq
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POWER and ENERGY (continued) 0 )] [ ) ( ) 6 ( = + + + + × ∫ ∫ ∫ ∫ ∫ v i s dv J E E E t E E t H H ds H E eq σ ε μ 2 2 2 , 2 1 , 2 1 E E E w t E t t E E w t H t t H H e m = = = = = 0 ] [ ] [ ) ( ) 7 ( 2 = + + + + × ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ v v i v e m s dv E dv J E dv t w t w ds H E eq [ ] 0 ] [ ] [ ) ( ) 8 ( 2 = + + + + × ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ v v i v m e s dv E dv J E dv w w t ds H E eq 0 , 0 ] [ ] 2 1 [ , ] 2 1 [ ) ( 2 2 2 = = = = × = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ = = v d v i i v e v m s s dv E P dv J E P dv E W dv H W ds H E P d i e m s P P W t W t P + + + = Stored magnetic power (W) Stored electric power (W) Supplied power (W) Dissipated power (W) What is this term?
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POWER and ENERGY (continued) 0 , 0 ] [ ] 2 1 [ , ] 2 1 [ ) ( 2 2 2 = = = = × = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫
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ELEG413lec2 - ELEG 648 Lecture #2 Mark Mirotznik, Ph.D....

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