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
eflection 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 ?⁄ .
) (5pts) Using Ampere’s law, compute the free cu
ent 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
y 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 XXXXXXXXXXGHz).
a) (10 pts) Let ? = ? − ?? and plot the attenuation and phase constants for the
wave inside the lossy material.
) (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.