This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 450 Fall 2009 Homework 9 Solutions Due: Thursday , Oct 29 th , 2009 1. In medium 1, the dielectric (glass), 2 · 2 ˆ ˆ ˆ which can be expressed in more general terms as 2 i x z j j i i o x z e e π + − − − = = k r E E ¡ ¡ E e a. Since then from the 1 ˆ i i k k = k 2 2 x z j e π + − term, one may intuitively see that that ˆ ˆ ˆ 2 i x z k + = which satisfies ˆ ˆ · i i k k 1 = and thus 1 2 k π = ♠ Or just equate ( ) 1 1 2 2 x z x z j j k x k z e π + − − + e , which yields: → 2 2 1 1 1 1 x z k 1 and 2 2 2 2 2 x z k k k k π π = + π = = ⇒ = ♠ ( ) 1 1 1 9 1 1 1 1 2 2 3 rad 10 300 MHz m p p r k v k c k v f 8 10 1.26 2.25 π ω λ π ω = = = = = × ♠ = ε ˆ × = b. Since you know e from the incident field, you can determine i θ graphically given that incidence angle is in the plane of incidence and is perpendicular to the direction or propagation. As a matter of fact, since is discernable from the given incident field, determining i E ¡ ˆ i k i θ becomes easier. (See figure below). From the figure, it is easy to see that 1 2 1 2 ˆ cos 45 ˆ sin x i i z i k k θ θ θ = = = ° ♠ = = → 1 1 0 , r μ Critical angle: Snell’s Law Æ 1 1 sin n 2 sin n 2 θ θ = , with 2 90 θ = ° , we find c θ as 2 2 1 1 1 1 2.25 1 sin 41.8 2.2 s n 5 i r c r c n n θ θ − = = = ∴ = = ° ♠ ε ε Æ An evanescent wave will be produced in the half space x > 0 because the angle of incidence is greater than the critical angle. That is, 45 i 41.8 c θ θ = ° > = ° . This scenario leads to total internal reflection ( TIR ) of the incidence field impinging upon the interface. c. Starting from Snell’s Law: 1 2 2 1 2 1 2 2 2 2 2 1 2 1 2 1 2 2 sin sin Since, sin sin sin cos 1 sin cos 1 n n n n n n θ θ θ θ θ θ θ θ = = = − = − Since we are in a TIR condition, 2 2 1 1 2 2 sin 1 n n θ > , and thus we can write , μ ε ε ε i E ¡ ˆ k i θ → i θ ˆ x k ˆ k z 2 2 2 2 1 1 2 1 1 2 2 2 2 2 sin cos 1 1 1 sin n n j j n n θ θ θ ⎛ ⎞ = − − = ± − = ± ⎜ ⎟ ⎝ ⎠ α 2 2 2 sin 45 2.25 cos 1 0.354 cos 0.354 1 j j j θ θ ∴ = ± ° − = ± → = − ♠ Æ 2 θ is complex because as the wave travels from medium 1 to medium 2 in conjunction with the fact that...
View
Full
Document
This note was uploaded on 02/21/2010 for the course ECE 450 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff

Click to edit the document details