450fall09hw3 - ECE 450 Homework 3 Due Tue 5 PM 1 Consider a...

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Unformatted text preview: ECE 450 Homework 3 Due: Tue, Sept 15, 2009, 5 PM 1. Consider a circular wire loop of radius a carrying a time-harmonic current I = I o cos ( ωt ) and radiating in a homogeneous medium of electric permittivity ε and magnetic permeability μ . The loop is placed on the xy-plane of the reference coordinate system, with its center at the origin. The current flow in the loop is in the counterclockwise direction when looking down on xy-plane in- ˆ z direction. It can be shown 1 that if the loop is small enough (i.e., in the limit as a → ) the retarded vector potential phasor is given by ˜ A ( r, θ ) = ˆ φ μI o πa 2 4 πr e- jkr jk + 1 r sin θ where k = ω √ μ . a) Calculate the magnetic and electric field phasors ˜ H and ˜ E of the small loop antenna starting with ˜ B = μ ˜ H = ∇× ˜ A , and also using ∇× ˜ H = ˜ J + jω ˜ E where ˜ J = 0 . b) Using the expressions derived above in the limit as r λ , show that the radiation fields of the loop antenna are given by...
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