please answer all parts of the two problems provided
Make sure to include all steps and details to the answers
ECE 407 – ELECTROMAGNETIC COMPATIBILITY ECE 835 Advanced Electromagnetic Fields and Waves I Homework Set 2 Fall 2024 Due: 10/7/24 100 points total 1. (60 pts) A layer of lossy material is placed against a perfectly conducting metallic plate. The front of the lossy material is located in the plane ? = 0, and the conducting plate is located in the plane ? = ∆. A plane wave of radian frequency =2f (f is the frequency), traveling in free space, is normally impinging over the lossy material from ? < 0.="" assume="" that="" the="" lossy="" material="" and="" the="" plate="" are="" in="" infinite="" in="" the="" x="" and="" y-directions.="" the="" lossy="" material="" has="" a="" complex="" permittivity="" .="" the="" permittivity="" and="" permeability="" are="" considered="" frequency="" independent.="" the="" electric="" field="" in="" each="" region="" may="" be="" written="" as="" �⃗�="" (?)="?0�̂�(?" −??0?="" +="" 0?),="">< 0="" �⃗�="" (?)="?0�̂�(??" −???="" +="" +???),="" 0=""><>< ∆ where ?0 = ?√?0?0 and ? = ?√?0?. ?0 is the complex phasor amplitude of the electric field incident on the lossy material, and r is the complex dimensionless reflection coefficient for the wall. a) (10 pts) using faraday’s law, compute the magnetic fields in each region. for the derivation consider the intrinsic impedances of the media, ?0 = √?0 ?0⁄ and ? = √?0 ?⁄ . b) (5pts) using ampere’s law, compute the free current density, ? , inside the lossy material. c) (5pts) using gauss’law, compute the free charge, ?, inside the lossy material. d) (20 pts) apply the boundary conditions on the tangential electric and magnetic fields at the two interfaces to determine the reflection coefficient r. (hint: three equations are required to solve the problem). e) (10 pts) replace the perfectly conducting metallic plate with a perfect magnetic plate. apply the boundary conditions on the tangential electric and magnetic fields at the two interfaces to determine the reflection coefficient r. comment on the difference with the previous case considering a perfect electric conductor. f) (10 pts) derive the time-average power per unit area reflected and dissipated by the lossy material as function of ?0 and r. comment if any of the power impinging over the lossy material is transmitted beyond the electric wall. (hint: consider poynting’s theorem). 2. (40 pts) the permittivity of the lossy material is ? = 9?0 − ? ? ?⁄ with ? = 0.1 s/m. the lossy material has a thickness δ = 2 mm and we consider the problem in the s-band (2 - 4 ghz). a) (10 pts) let ? = ? − ?? and plot the attenuation and phase constants for the wave inside the lossy material. b) (5 pts) plot the skin depth and relate it to the size of the lossy media. c) (5 pts) plot the phase velocity and relate it to the speed of light. d) (10 pts) plot the magnitude |r| in db of the reflection coefficient. e) (10 pts) if the incident electric field has an amplitude of 10 v/m, plot the power per unit area dissipated and reflected by the lossy material. ∆="" where="" 0="?√?0?0" and="" =="" √?0?.="" 0="" is="" the="" complex="" phasor="" amplitude="" of="" the="" electric="" field="" incident="" on="" the="" lossy="" material,="" and="" r="" is="" the="" complex="" dimensionless="" reflection="" coefficient="" for="" the="" wall.="" a)="" (10="" pts)="" using="" faraday’s="" law,="" compute="" the="" magnetic="" fields="" in="" each="" region.="" for="" the="" derivation="" consider="" the="" intrinsic="" impedances="" of="" the="" media,="" 0="√?0" 0⁄="" and="" =="" √?0="" ⁄="" .="" b)="" (5pts)="" using="" ampere’s="" law,="" compute="" the="" free="" current="" density,="" ,="" inside="" the="" lossy="" material.="" c)="" (5pts)="" using="" gauss’law,="" compute="" the="" free="" charge,="" ,="" inside="" the="" lossy="" material.="" d)="" (20="" pts)="" apply="" the="" boundary="" conditions="" on="" the="" tangential="" electric="" and="" magnetic="" fields="" at="" the="" two="" interfaces="" to="" determine="" the="" reflection="" coefficient="" r.="" (hint:="" three="" equations="" are="" required="" to="" solve="" the="" problem).="" e)="" (10="" pts)="" replace="" the="" perfectly="" conducting="" metallic="" plate="" with="" a="" perfect="" magnetic="" plate.="" apply="" the="" boundary="" conditions="" on="" the="" tangential="" electric="" and="" magnetic="" fields="" at="" the="" two="" interfaces="" to="" determine="" the="" reflection="" coefficient="" r.="" comment="" on="" the="" difference="" with="" the="" previous="" case="" considering="" a="" perfect="" electric="" conductor.="" f)="" (10="" pts)="" derive="" the="" time-average="" power="" per="" unit="" area="" reflected="" and="" dissipated="" by="" the="" lossy="" material="" as="" function="" of="" 0="" and="" r.="" comment="" if="" any="" of="" the="" power="" impinging="" over="" the="" lossy="" material="" is="" transmitted="" beyond="" the="" electric="" wall.="" (hint:="" consider="" poynting’s="" theorem).="" 2.="" (40="" pts)="" the="" permittivity="" of="" the="" lossy="" material="" is="" =="" 9?0="" −="" ⁄="" with="" =="" 0.1="" s/m.="" the="" lossy="" material="" has="" a="" thickness="" δ="2" mm="" and="" we="" consider="" the="" problem="" in="" the="" s-band="" (2="" -="" 4="" ghz).="" a)="" (10="" pts)="" let="" =="" −="" and="" plot="" the="" attenuation="" and="" phase="" constants="" for="" the="" wave="" inside="" the="" lossy="" material.="" b)="" (5="" pts)="" plot="" the="" skin="" depth="" and="" relate="" it="" to="" the="" size="" of="" the="" lossy="" media.="" c)="" (5="" pts)="" plot="" the="" phase="" velocity="" and="" relate="" it="" to="" the="" speed="" of="" light.="" d)="" (10="" pts)="" plot="" the="" magnitude="" |r|="" in="" db="" of="" the="" reflection="" coefficient.="" e)="" (10="" pts)="" if="" the="" incident="" electric="" field="" has="" an="" amplitude="" of="" 10="" v/m,="" plot="" the="" power="" per="" unit="" area="" dissipated="" and="" reflected="" by="" the="" lossy="">