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329fall08hw5sol

# 329fall08hw5sol - ECE-329 Fall 2008 Homework 5 Solution 1...

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ECE-329 Fall 2008 Homework 5 — Solution October 23, 2008 1. Let us consider the following four plane waves in free space, E 1 = 2 cos( ωt - βz x V m E 2 = E o (cos( ωt - βz x - sin( ωt - βz y ) V m H 3 = cos( ωt + βz + π 3 x + sin( ωt + βz - π 6 y A m H 4 = cos( ωt - βx z + sin( ωt - βx y A m . a) The electric and magnetic fields ( E and H ) of uniform plane waves are perpendicular to each other and to the direction of propagation, therefore, it can be verified that such fields satisfy the following relation E = η H × ˆ β, where ˆ β is the unit vector parallel to the propagation direction and η is the intrinsic impedance. Using this relation, we can find the expressions for H or E that accompany the waves given above, ˆ β 1 = ˆ z H 1 = 2 η o cos( ωt - βz y A m ˆ β 2 = ˆ z H 2 = E o η o (cos( ωt - βz y + sin( ωt - βz x ) A m ˆ β 3 = - ˆ z E 3 = η o ( cos( ωt + βz + π 3 y - sin( ωt + βz - π 6 x ) V m ˆ β 4 = ˆ x E 4 = η o (cos( ωt - βx y - sin( ωt - βx z ) V m . b) The instantaneous power flow density is given by the Poynting vector P = E × H . Therefore, the instantaneous power that crosses some surface S is given by P = ´ S P · d S . In the case of uniform plane waves, this expression simplifies to P = A ˆ n · P , where ˆ n is the vector normal to the flat area A . Below, we are considering

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329fall08hw5sol - ECE-329 Fall 2008 Homework 5 Solution 1...

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