329fall08hw4sol

329fall08hw4sol - ECE-329 Fall 2008 Homework 4 Solution 1....

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Unformatted text preview: ECE-329 Fall 2008 Homework 4 Solution 1. Curl and divergence exercises. a) V = x x + y y Vector field x-axis y-axis- 2- 1 1 2- 2- 1 1 2 Curl V = x y z x y z x y = . Divergence V = x ( x ) + y ( y ) = 1 + 1 = 2 . b) V =- y x + x y Vector field x-axis y-axis- 2- 1 1 2- 2- 1 1 2 Curl V = x y z x y z- y x = 2 z. 1 ECE-329 Fall 2008 Divergence V = x (- y ) + y ( x ) = 0 . c) Curl and divergence properties. i. V 6 = implies the field strength varies across the direction of the field V . ii. V 6 = 0 implies the field strength varies along the direction of the field V . 2. Verifying vector calculus identities, ( f ) = and ( A ) = 0 . a) The gradient of a scalar field f is defined as f = f x x + f y y + f z z. Taking the curl, we obtain f = x y z x y z f x f y f z = y f z- z f y x + z f x- x f z y + x f y- y f x z = . b) The curl of a vector field A = A x x + A y y + A z z is defined as A = x y z x y z A x A y A z = A z y- A y z x + A x z- A z x y + A y x- A x y z. Taking the divergence, we obtain ( A ) = x A z y- A y z + y A x z- A z x + z A y x- A x y = 2 A z xy- 2 A y xz + 2 A x yz- 2 A z yx + 2 A y zx- 2 A x zy = 0 . 3. 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 1 = (2 x + 3 y ) x + (3 x- 2 y ) y Divergence F 1 = x (2 x + 3 y ) + y (3 x- 2 y ) = 2- 2 = 0 ....
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329fall08hw4sol - ECE-329 Fall 2008 Homework 4 Solution 1....

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