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 x + y y x-axis y-axis- 2- 1 1 2- 2- 1 1 2 Curl F = x y z x y z x y = . Divergence F = x ( x ) + y ( y ) = 1 + 1 = 2 . b) F = y x- x y x-axis y-axis- 2- 1 1 2- 2- 1 1 2 Curl F = x y z x y z y- x =- 2 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 = (Amperes 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 ) x + (3 x- 2 y ) y Divergence F a = x (2 x + 3 y ) + y (3 x- 2 y ) = 2- 2 = 0 . Curl F a = x y z x y z (2 x + 3 y ) (3 x- 2 y ) =- z (3 x- 2 y ) x + z (2 x + 3 y ) y + x (3 x- 2 y )- y (2 x + 3 y ) z = (3- 3) z = . Since F a = 0 and F a = , the vector field F a can represent a magnetic field. b) F b = x 2 y x + xy 2 y Divergence F b = x ( x 2 y ) + y ( xy 2 ) = 2 xy + 2 xy = 4 xy. Curl F b = x y z x y z x 2 y xy 2 =- z ( xy 2 ) x + z ( x 2 y ) y + x ( xy 2 )- y ( x 2 y ) z = ( y 2- x 2 ) z....
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This note was uploaded on 11/16/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

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

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