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Unformatted text preview: ECE 329 Spring 2010 Homework 3  Solution Due: Feb. 9, 2010 1. E =∇ V = ˆ x ∂ ∂x + ˆ y ∂ ∂y + ˆ z ∂ ∂z ( xz 2 ) = z 2 ˆ x 2 xz ˆ z, ρ = ∇ · D = ∇ · ( E ) = ( ∇ · E ) = ˆ x ∂ ∂x + ˆ y ∂ ∂y + ˆ z ∂ ∂z · ( ˆ x z 2 ˆ z 2 xz ) = 2 x. 2. According to the relationship between electric eld and potential, we have V (0 , , 0) V ( x,y,z ) = ˆ ( x,y,z ) (0 , , 0) E · d l = ˆ ( x,y,z ) (0 , , 0) ( x ˆ x + y ˆ y 4ˆ z ) · ( ˆ xdx + ˆ ydy + ˆ zdz ) = ˆ ( x,y,z ) (0 , , 0) ( x dx + y dy 4 dz ) . Since ∇ × E = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z x y 4 = 0 , the integration above is pathindepent and we can simply choose three segments along x, y and z directions, respectively. Here, we choose the integral path as (0 , , 0) to ( x, , 0) , ( x, , 0) to ( x,y, 0) , and ( x,y, 0) to ( x,y,z ) . Accordingly, V (0 , , 0) V ( x,y,z ) = " ˆ ( x, , 0) (0 , , 0) + ˆ ( x,y, 0) ( x, , 0) + ˆ ( x,y,z ) ( x,y, 0) # ( x dx + y dy 4 dz ) = ˆ ( x, , 0) (0 , , 0) x dx + ˆ ( x,y, 0) ( x, , 0) y dy ˆ ( x,y,z ) ( x,y, 0) 4 dz = x 2 2 + y 2 2 4 z....
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 Spring '10
 Goddart
 Trigraph, Electric charge, D2z z, xy V/m

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