Please solve the two problems provide. Make sure to include all steps and details leading to the answers

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Please solve the two problems provide. Make sure to include all steps and details leading to the answers


ECE 407 – ELECTROMAGNETIC COMPATIBILITY Advanced Electromagnetic Fields and Waves I 1. (50 pts) Let’s consider a homogeneous Tellegen medium. A Tellegen medium is bi-isotropic, and is usually described using the constitutive relations: �⃗⃗� = ?�⃗� + ?�⃗⃗� (1) �⃗� = ?�⃗� + ?�⃗⃗� (2) Here ?, ?, and ? are the constitutive parameters with ?2 < ??. these parameters do not depend on the position since the medium is homogenous. assume there no sources in the region of interest. a) (5 pts) show that (1) and (2) may be written as: �⃗⃗� = ?�⃗� + ?�⃗� (3) �⃗⃗� = −?�⃗� + ?�⃗� (4) here ?, ?, and ? are the new constitutive parameters. determine these in terms of the old parameters ?, ?, and ?. b) (5 pts) using (3) show that ∇ ∙ �⃗� = 0. c) (10 pts) derive the following wave (helmholtz) equation for �⃗� : ∇2�⃗� + ?2�⃗� = 0 where ?2 = ?2(?? − ?2). d) (10 pts) verify that the following plane wave is a solution of the helmholtz equation: �⃗� = �̂�?0? −??? e) (15 pts) use maxwell’s equations to derive the following expression for �⃗⃗� : �⃗⃗� = (−�̂� ? ? + �̂� ? ?? )?0? −??? is this wave tem? f) (5 pts) the phase fronts for this wave propagate in the z-direction. does the power flows in the z-direction as well? 2. (50 pts) a parallel-plate waveguide is formed by placing infinitely large pec plates in the planes ? = 0 and ? = ?. the parallel-plate waveguide is filled by a lossless material with permittivity ? and permeability ?. let consider the tmz modes propagating in the z-direction. the magnetic field can be expressed as a combination of the following two modes: �⃗⃗� + = �̂�?0? ??(−? sin ?−? cos?) �⃗⃗� − = �̂�?0? ??(? sin ?−? cos?) a) (5 pts) show by adding �⃗⃗� + and �⃗⃗� − that the total magnetic field is equal to �⃗⃗� = �̂�?0 cos(???) ? −?? b) (10 pts) derive �⃗� and then ?? by using the boundary conditions at the two metallic plates. c) (15 pts) derive ? and the cut-off frequencies of the modes, ??. d) (20 points) if the plates are not pec, but instead have a surface resistance ℛ derive the attenuation constant due to imperfect conductors, ??. (hint: use the perturbation method). .="" these="" parameters="" do="" not="" depend="" on="" the="" position="" since="" the="" medium="" is="" homogenous.="" assume="" there="" no="" sources="" in="" the="" region="" of="" interest.="" a)="" (5="" pts)="" show="" that="" (1)="" and="" (2)="" may="" be="" written="" as:="" �⃗⃗�="?�⃗�" +="" �⃗�="" (3)="" �⃗⃗�="−?�⃗�" +="" �⃗�="" (4)="" here="" ,="" ,="" and="" are="" the="" new="" constitutive="" parameters.="" determine="" these="" in="" terms="" of="" the="" old="" parameters="" ,="" ,="" and="" .="" b)="" (5="" pts)="" using="" (3)="" show="" that="" ∇="" ∙="" �⃗�="0." c)="" (10="" pts)="" derive="" the="" following="" wave="" (helmholtz)="" equation="" for="" �⃗�="" :="" ∇2�⃗�="" +="" 2�⃗�="0" where="" 2="?2(??" −="" 2).="" d)="" (10="" pts)="" verify="" that="" the="" following="" plane="" wave="" is="" a="" solution="" of="" the="" helmholtz="" equation:="" �⃗�="�̂�?0?" −???="" e)="" (15="" pts)="" use="" maxwell’s="" equations="" to="" derive="" the="" following="" expression="" for="" �⃗⃗�="" :="" �⃗⃗�="(−�̂�" +="" �̂�="" )?0?="" −???="" is="" this="" wave="" tem?="" f)="" (5="" pts)="" the="" phase="" fronts="" for="" this="" wave="" propagate="" in="" the="" z-direction.="" does="" the="" power="" flows="" in="" the="" z-direction="" as="" well?="" 2.="" (50="" pts)="" a="" parallel-plate="" waveguide="" is="" formed="" by="" placing="" infinitely="" large="" pec="" plates="" in="" the="" planes="" =="" 0="" and="" =="" .="" the="" parallel-plate="" waveguide="" is="" filled="" by="" a="" lossless="" material="" with="" permittivity="" and="" permeability="" .="" let="" consider="" the="" tmz="" modes="" propagating="" in="" the="" z-direction.="" the="" magnetic="" field="" can="" be="" expressed="" as="" a="" combination="" of="" the="" following="" two="" modes:="" �⃗⃗�="" +="�̂�?0?" (−?="" sin="" −?="" cos?)="" �⃗⃗�="" −="�̂�?0?" (?="" sin="" −?="" cos?)="" a)="" (5="" pts)="" show="" by="" adding="" �⃗⃗�="" +="" and="" �⃗⃗�="" −="" that="" the="" total="" magnetic="" field="" is="" equal="" to="" �⃗⃗�="�̂�?0" cos(???)="" −??="" b)="" (10="" pts)="" derive="" �⃗�="" and="" then="" by="" using="" the="" boundary="" conditions="" at="" the="" two="" metallic="" plates.="" c)="" (15="" pts)="" derive="" and="" the="" cut-off="" frequencies="" of="" the="" modes,="" .="" d)="" (20="" points)="" if="" the="" plates="" are="" not="" pec,="" but="" instead="" have="" a="" surface="" resistance="" ℛ="" derive="" the="" attenuation="" constant="" due="" to="" imperfect="" conductors,="" .="" (hint:="" use="" the="" perturbation="">
Answered 5 days AfterOct 02, 2024

Answer To: Please solve the two problems provide. Make sure to include all steps and details leading to the...

Bhaumik answered on Oct 07 2024
4 Votes
Advanced Electromagnetic Fields and Waves I
Solutions
Solution for Part 1:
a) Expressing the constitutive relations in terms of new parameters :
Given:
(1)
(2)
The following must be expressed:
(3)
(4)
The process is about identifying how to translate the parameters into
By comparing, we can set:
· (from the term in ),
· (from the term in ),
· (from the term in ).
Thus, we have the relations:
, , and
b) Showing
Using the modified form:
As there are no free charges in the region, Gauss’s law for the electric displacement field is:
Substituting into this equation:
Since the medium is homogeneous, the parameters and are constants, so the divergence of this expression becomes:
Now, using the facts in the absence of magnetic monopoles, we have:
Thus, the equation reduces to:
Since for a non-trivial medium, we conclude:
This shows that the electric field is solenoidal (divergence-free), consistent with the problem’s assumptions.
c) Deriving the wave (Helmholtz) equation for
From Maxwell’s equation, we know that:
Substituting into this:
Since time-harmonic fields are assumed (), this becomes:
Similarly, for the curl of H:
Now, let’s derive the wave equation,
Taking the curl of
Using the vector identity and knowing , this simplifies to:
To obtain Helmholtz equation for using the constitutive relations and and assuming time-harmonic fields (), we substitute Maxwell’s curl equations.
d) Verifying the plane wave solution:
Given:
First, calculate Since, the wave is propagating in the z-direction, we only need to take the second...
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