450fall09hw1_Sol - ECE 450 Fall 2009 1. Utilize boundary...

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ECE 450 Fall 2009 Homework 1 Solutions Due: Tuesday, Sep 1 st , 2009 1. Utilize boundary conditions: (1) ˆ The normal components of the the electric flux density across an interface are discontinuous ˆ 0 The normal components of the the s n n ρ +− ⎡⎤ ⋅−=→ ⎣⎦ DD BB magnetic flux density across an interface are continuous ˆ 0 The tangental components of the the electric field intensity across an n ×−=→ EE interface are continuous ˆ The tangental components of the the magnetic field intensity across an interface are discontinuous n ×−= s HH J dium 1 PEC ( σ = and E = 0) (i) Since me ׵ (therefore) 22 2 ˆˆ ˆ 0( ss nn z z z ρρ ε ++ + + − = → =⋅ =⋅ =⋅ = E E 0 ˆ 0 ) 2 z z 9 0 2 1 21 0 18 s C m π == Thus the answer is a) (ii) Again, since medium 1 PEC ( σ = and H = 0) ׵ 3 2 10 0 ) ( 4 A z z z m + ˆ ) x ×− = = × = × = = × HJ J H H 3 1 4 0 ˆ s A y m = ⎢⎥ J Thus the answer is d) (iii) Medium 1 is not a PEC, therefore E exists. ׵ [] 11 00 0 0 0 )3 ) · 3 ) ˆ ˆ (0 ) ) ˆ 4( 2( ( 4 2 2 s VV xz x z nz z z z mm ρε + ⋅−= = = = ⎤⎡ = = ⎦⎣ ⋅− = E E 9 2 1 10 18 s C m =− Thus the answer is b) 1
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(iv) Medium 1 is not a PEC, therefore H exists. ׵ 21 0 0 00 44 ˆ ˆ (0 ) ) , s i n c e 10 10 ˆ ) ) 10 10 ˆˆ ˆ () ( 2 2 nz z z zz x z yy μ μμ +− + −− ⎡⎤ ×−= = × = = = ⎣⎦ = − = = ⎛⎞ ˆ ) z x z × × −+ ⎜⎟ ⎝⎠ =− ss s HH J J H H B H BB J 3 310 ˆ 2 · 4 s A y m π = J Thus the answer is c) 2. a) * Apply the divergence to both sides on the point form of Ampere’s Law: t ∇× = + D HJ * ·· t ∇∇ × = ∇ + D , a vector identity states that the divergence of a curl of a vector is zero. Makes since physically since curl of a vector field is a measure of a field’s
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450fall09hw1_Sol - ECE 450 Fall 2009 1. Utilize boundary...

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