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329fall10hw1sol

# 329fall10hw1sol - ECE 329 Fall 2010 Homework 1 Solution Due...

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Unformatted text preview: ECE 329 Fall 2010 Homework 1 - Solution Due: Aug. 31, 2010 1. a) A + B- 4 C = 4 b x + 2 b y- 3 b z- 4( b x- 2 b y + 3 b z ) = 10 b y- 15 b z | A + B- 4 C | = p 10 2 + 15 2 = 18 . 03 \ b) A + 2 B- C = (3 b x + b y- 2 b z ) + 2( b x + b y- b z )- ( b x- 2 b y + 3 b z ) = 4 b x + 5 b y- 7 b z | A + 2B- C | = p 4 2 + 5 2 + 7 2 = 9 . 49 ˆ u = A + 2B- C | A + 2B- C | = 4 b x + 5 b y- 7 b z 9 . 49 = 0 . 42 b x + 0 . 53 b y- . 74 b z c) A · B = (3 b x + b y- 2 b z ) · ( b x + b y- b z ) = 3 × 1 + 1 × 1 + (- 2) × (- 1) = 6 d) B × C = ( b x + b y- b z ) × ( b x- 2 b y + 3 b z ) = b x × ( b x- 2 b y + 3 b z ) + b y × ( b x- 2 b y + 3 b z )- b z × ( b x- 2 b y +3 b z ) = (- 2 b z- 3 b y ) + (- b z + 3 b x )- ( b y + 2 b x ) = b x- 4 b y- 3 b z e) A · ( B × C ) = (3 b x + b y- 2 b z ) · [( b x + b y- b z ) × ( b x- 2 b y + 3 b z )] = (3 b x + b y- 2 b z ) · ( b x- 4 b y- 3 b z ) = 3 × 1 + 1 × (- 4) + (- 2) × (- 3) = 5 f) The unit of the triple product is m 3 . The magnitude of the cross product, | B × C | , can be thought of as the area of a parallelogram formed by B and C (see Fig. 1). The area of a parallelogram with sides length | B | and | C | and interior angle θ is | B || C | sin θ , which is also the magnitude of the cross product | B × C | . Taking the dot product with A introduces a third dimension of length | A | cos α , where a is the angle between A and B × C . Thus, the magnitude of the triple product is the same as the volume of the parallelepiped illustrated in Fig. 1. g) The triple product has units of m 2 · T / s where 1T = 1 V · s m 2 so the triple product has units of 1V = 1 J C . Thus, the triple product describes the work done per Columb of charge that moves through a magnetic eld B between two points separated by vector A at velocity C . 1 ECE 329 Fall 2010 B C θ A B×C α Figure 1: Parallelepiped formed by triple product A · ( B × C ) Figure 2: Charges and electric elds for Problem 2 2. We know Q 1 = 8 πε , so Q 2 =- Q 1 / 2 =- 4 πε . The electric eld at the point P 3 is the superposition...
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329fall10hw1sol - ECE 329 Fall 2010 Homework 1 Solution Due...

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