Homework05WorkWaves - ECE 3030: Electromagnetic Fields and...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 Homework 5 SOLUTIONS for GRADER: Virtual Work and Plane Waves Reminder: Note items to be kept for Workshop or Homework! Date: Tuesday 10/6 Problem 8.15 Find the force using virtual work concepts. a. Basic equations: U = 1 2 CV 2 = 1 2 CV 2 , Q = CV = ε 0 A x V , C = ε 0 A ) x Force = - ∂U ∂x Q = const. + V ∂Q ∂x V = const. = ∂x [ - U + V Q ] = ∂x - 1 2 CV 2 + V CV = ∂x + 1 2 CV 2 V = const = 1 2 ∂C ∂x V 2 Now ∂C ∂x = ∂x ε 0 A x = - C x , so Force = - 1 2 C x V 2 , which is in the direc- tion of decreasing x and the top panel is pulled down. Problem 9.3 Finding the Laplacian and divergence of a wave electric field. b : Take the Laplacian and identify the terms in the result: ~ ∇ · ~ E( ~ r) = ± 2 ∂x 2 + 2 ∂y 2 +
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