Workshop01VectorCalc

# Workshop01VectorCalc - ECE 3030 Electromagnetic Fields and...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 WORKSHOP 1 INSTRUCTOR NOTES: Vector Calculus Reminder: Help students as they need it. Date: Tuesday 9/3 1. Problem 2.9: Let ~ B = 2ˆ x + 3ˆ z . (f) The magnitude of | B | = 2 2 + 3 2 = 13. (i) The unit vector= ~ B | ~ B | = 1 13 (2ˆx + 3ˆ z). 2. Problem 2.17 A vector ~ F is given by ~ F = 3 r sin φ ˆ r + r 3 cos φ ˆ φ . a. Evaluate H ~ F · d around the contour ABCDA shown in Figure 1, where a = 1 m and b = 2 m. b. Find the expression for ~ ∇ × ~ F. c. Evaluate R ( ~ ∇ × ~ F) · ~ d a over the sur- face enclosed by the ABCDA contour. Be careful with the direction of ~ d a . Check your answers to (a) and (c) with Stokes’ Theorem. y x a b A B C D Figure 1: Contour ABCDA sur- rounds the shaded area. Solution: Matching results from a contour integral to a surface integral demonstrates Stokes Theorem. In general, ~ d = ˆxd x + ˆyd y + ˆ zd z , and in cylindrical coordinates, this becomes ~ d = ˆ rd r + ˆ φr d r . a. Path D A: r = 1, ~ d = d φ ˆ φ = 1d φ ˆ φ with d r = 0.

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