ELEG413lec2 - ELEG 648 Lecture#2 Mark Mirotznik Ph.D Associate Professor The University of Delaware Tel(302)831-4221 Email Maxwell’s Equations in

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Unformatted text preview: ELEG 648 Lecture #2 Mark Mirotznik, Ph.D. Associate Professor The University of Delaware Tel: (302)831-4221 Email: Maxwell’s Equations in Differential Form m i c B D J J t D H M t B E ρ ρ = ⋅ ∇ = ⋅ ∇ + + ∂ ∂ = × ∇- ∂ ∂- = × ∇ Faraday’s Law Ampere’s Law Gauss’s Law Gauss’s Magnetic Law Faraday’s Law s d B t l d E t B E c s ⋅ ∂ ∂- = ⋅ ∂ ∂- = × ∇ ∫ ∫ ∫ S C t B ∂ ∂ E 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 Gauss’s Law ∫ ∫ ∫ ∫ ∫ = = ⋅ = ⋅ ∇ v tot s Q dv s d D D ρ ρ tot Q D Gauss’s Magnetic Law ∫ ∫ = ⋅ = ⋅ ∇ s s d B B B “all the flow of B entering the volume V must leave the volume” 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) POWER and ENERGY Ji 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 × ∇ ⋅- × ∇ ⋅ = × ⋅ ∇ ) ( ) ( ) 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 )] [ ) ( ) 6 ( = ⋅ + ⋅ + ∂ ∂ ⋅ + ∂ ∂ ⋅ + ⋅ × ∫ ∫ ∫ ∫ ∫ v i s dv J E E E t E E t H H ds H E eq σ ε μ POWER and ENERGY (continued) )] [ ) ( ) 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 σ σ ε ε μ μ = ⋅ ∂ ∂ = ∂ ∂ = ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ = ∂ ∂ ⋅ ] [ ] [ ) ( ) 7 ( 2 = + ⋅ + ∂ ∂ + ∂ ∂ + ⋅ × ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ v v i v e m s dv E dv J E dv t w t w ds H E eq σ [ ] ] [ ] [ ) ( ) 8 ( 2 = + ⋅ + + ∂ ∂ + ⋅ × ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ v v i v m e s dv E dv J E dv w w t ds H E eq σ , ] [ ] 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...
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This note was uploaded on 12/13/2011 for the course ELEG 413 taught by Professor Mirotznik during the Spring '11 term at University of Delaware.

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ELEG413lec2 - ELEG 648 Lecture#2 Mark Mirotznik Ph.D Associate Professor The University of Delaware Tel(302)831-4221 Email Maxwell’s Equations in

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