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329sp09hw4sol - ECE-329 Spring 2009 Homework 4 — Solution...

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Unformatted text preview: ECE-329 Spring 2009 Homework 4 — Solution February 18, 2009 1. Curl and divergence exercises. a) F = x ˆ x + y ˆ y x-axis y-axis- 2- 1 1 2- 2- 1 1 2 • Curl ∇ × F = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z x y = . • Divergence ∇ · F = ∂ ∂x ( x ) + ∂ ∂y ( y ) = 1 + 1 = 2 . b) F = y ˆ x- x ˆ y x-axis y-axis- 2- 1 1 2- 2- 1 1 2 • Curl ∇ × F = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z y- x =- 2 ˆ z. 1 ECE-329 Spring 2009 • Divergence ∇ · F = ∂ ∂x ( y ) + ∂ ∂y (- x ) = 0 . c) Curl and divergence properties. i. ∇ × F 6 = implies the field strength varies across the direction of the field F . ii. ∇ · F 6 = 0 implies the field strength varies along the direction of the field F . 2. The magnetic field B in regions where the current density J and the displacement current ∂ D ∂t are both zero satisfies ∇ · B = 0 (Gauss’ law) and ∇ × B = (Ampere’s law). Below, we will verify if the following vector fields can be realized as magnetic fields in such regions. a) F a = (2 x + 3 y ) ˆ x + (3 x- 2 y ) ˆ y • Divergence ∇ · F a = ∂ ∂x (2 x + 3 y ) + ∂ ∂y (3 x- 2 y ) = 2- 2 = 0 . • Curl ∇ × F a = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z (2 x + 3 y ) (3 x- 2 y ) =- ∂ ∂z (3 x- 2 y ) ˆ x + ∂ ∂z (2 x + 3 y ) ˆ y + ∂ ∂x (3 x- 2 y )- ∂ ∂y (2 x + 3 y ) ˆ z = (3- 3) ˆ z = . Since ∇ · F a = 0 and ∇ × F a = , the vector field F a can represent a magnetic field. b) F b = x 2 y ˆ x + xy 2 ˆ y • Divergence ∇ · F b = ∂ ∂x ( x 2 y ) + ∂ ∂y ( xy 2 ) = 2 xy + 2 xy = 4 xy. • Curl ∇ × F b = ˆ x ˆ y ˆ z ∂ ∂x ∂ ∂y ∂ ∂z x 2 y xy 2 =- ∂ ∂z ( xy 2 ) ˆ x + ∂ ∂z ( x 2 y ) ˆ y + ∂ ∂x ( xy 2 )- ∂ ∂y ( x 2 y ) ˆ z = ( y 2- x 2 ) ˆ z....
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329sp09hw4sol - ECE-329 Spring 2009 Homework 4 — Solution...

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