HW2_solutions

HW2_solutions - EEE241 Fundamentals of Electromagnetics...

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EEE241 – Fundamentals of Electromagnetics – Spring 2007 Solutions to Homework 2 P2-21a:(10 points) Given ( )( ) 12 ,. 2 , 1 , 1 8 , 2 , 1 xy E a y a x evaluate E d from P to P =+ r rr l along the parabola 2 2 x y = Solution: 2 2 x y = ; 4 dx y dy = da d x a d y r l 2 1 2 2 2 23 1 1 .6 2 14 P P Eaya x Ed ydx xdy y dy y =+= == = ∫∫ r r r l r r l P2-23 (10 points) Given the scalar function sin sin z Vx y e ππ ⎤⎡ ⎛⎞ = ⎜⎟ ⎥⎢ ⎝⎠ ⎦⎣ Determine (a) the magnitude and direction of the maximum rate of increase of V at P(1,2,3). (b) the rate of increase of V at P in the direction of the origin (a) The magnitude of and direction of maximum rate of increase of a function V is given by its gradient V . cos sin sin cos sin sin 22 3 2 33 2 3 xyz zz z Va a a V ax y e a x y e a x y e π −− ∂∂∂ ∇= + + ⎡⎤ ⎛⎞⎛⎞ =+− ⎜⎟⎜⎟ ⎢⎥ ⎝⎠⎝⎠ ⎣⎦ rrr () 3 1, 2, 3 62 0.026 0.043 yz tP a e a e aa =−
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HW2_solutions - EEE241 Fundamentals of Electromagnetics...

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