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329fall08hw1sol

# 329fall08hw1sol - ECE-329 Fall 2008 Homework 1(Solution...

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ECE-329 Fall 2008 Homework 1 (Solution) September 3, 2008 1. Given vectors ~ A = 3ˆ x + ˆ y - z, ~ B = ˆ x + ˆ y - ˆ z, ~ C = ˆ x - y + 3ˆ z, we can calculate the following. (a) The vector magnitude of ~ A + ~ B - 4 ~ C = 10ˆ y - 15ˆ z is ~ A + ~ B - 4 ~ C = p 10 2 + 15 2 = 325 . (b) The unit vector of ~ A + 2 ~ B - ~ C = 4ˆ x + 5ˆ y - z is ˆ u = x + 5ˆ y - z 4 2 + 5 2 + 7 2 = x + 5ˆ y - z 90 . (c) The dot product ~ A · ~ B = (3ˆ x + ˆ y - z ) · x + ˆ y - ˆ z ) = 3 + 1 + 2 = 6 . (d) The cross product ~ B × ~ C = ˆ x ˆ y ˆ z 1 1 - 1 1 - 2 3 = (3 - 2) ˆ x - (3 + 1) ˆ y + ( - 2 - 1) ˆ z = ˆ x - y - z. (e) The scalar triple product ~ A · ~ B × ~ C = (3ˆ x + ˆ y - z ) · x - y - z ) = 3 - 4 + 6 = 5 . (f) If vectors ~ A, ~ B, and ~ C have units of meters, then the units of ~ A · ~ B × ~ C would be m 3 . ~ A · ~ B × ~ C is the volume of a parallelepiped formed by these vectors. (g) If vector ~ C has velocity units (m / s), ~ B has flux density units (Tesla), and ~ A has distance units (m), then the units of ~ A · ~ B × ~ C would be m s Tesla · m = Joules Coulomb , this unit describes energy per unit charge. 2. Charge Q 1 = 8 π o C is placed at ~ r 1 = ( - 1 , 0 , 0) and generates ~ E 1 ( x, y, z ) = Q 1 4 π o ~ r - ~ r 1 | ~ r - ~ r 1 | 3 = 2 ( x + 1) ˆ x + y ˆ y + z ˆ z (( x + 1) 2 + y 2 + z 2 ) 3 / 2 V m . 1

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ECE-329 Fall 2008 Charge Q 2 = - 4 π o C is placed at ~ r 2 = (1 , 0 , 0) and generates ~ E 2 ( x, y, z ) == Q 2 4 π o ~ r - ~ r 2 | ~ r - ~ r 2 | 3 = - ( x - 1) ˆ x + y ˆ y + z ˆ z (( x - 1) 2 + y 2 + z 2 ) 3 / 2 V m .
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329fall08hw1sol - ECE-329 Fall 2008 Homework 1(Solution...

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