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EE5831 Homework
Homework should be submitted in a single PDF file to Canvas.
Q1.
The E-field of a uniform plane wave propagating in a dielectric nonmagnetic (μ = μ0 ) medium is given by
E(z, t) = 2 cos(108 t - z / 、3 ) (V/m)
(i) Determine the frequency and wavelength of the wave.
(ii) What is the dielectric constant of the medium?
(iii) Find an expression for the corresponding instantaneous H-field.
Q2.
Show that the linearly polarized wave E = ( 
+ 
)E0 eikz can be decomposed into a linear superposition of a left hand circularly polarized (LHCP) and a right hand circularly polarized (RHCP) wave.
Q3.
An electromagnetic wave with the following electric field
is propagating in free space.
(a) What is the wave vector k ?
(b) Write down the phasor of the electric field E .
(c) Write down the instantaneous magnetic field H (t) .
(d) Specify the type of polarization of the wave.
Q4.
Two Hertzian dipole antennas of length l are located at the origin with current densities:
J1 = I1lδ(x)δ(y)δ(z) and J2 = 
I2lδ(x)δ(y)δ(z)
as shown in Fig. Q4. The radiation field at observation point ( r, θ, φ) ( kr 
1 , where k is the
wavenumber) due to the current density J1 is given by
where wis the angular frequency, μ0 is the permeability of free space.
Fig. Q4
Note that the observation point is in the far field ( kr 》 1 ) in all the following questions.
For parts (a) and (b), we assume I2 = 0.
(a). What is the polarization of the wave if the observation point is in the x axis? What is the
polarization of the wave if the observation is in the direction θ= and φ= ?
(b). What is the magnetic field at an arbitrary observation point ( r, θ, φ)?
In parts (c) and (d), the current I2 is not equal to zero.
(c). Find the total electric field at the observation point in the positive x axis.
. If the polarization of the wave at the observation direction θ= andφ= is right-hand circular polarization, find the ratio of I1 / I2 .
Q5.
For two-dimensional problems, calculate the elements in the dyadic Green’s function
in Cartesian coordinate system, where g . Note that since there is no variation with respect to z, i.e., = 0 , 4 items 
of the
dyadic Green’s function are zero. The zz component is just the scalar Geen’s function
, whereas the 2 × 2 matrix in the upper left corner is not straightforward.
Derive the four elements in the upper left corner 2 × 2 matrix. Hint: Applying chain rule for derivative in
and using the recurrence formulas of the Hankel function
Q6.
A uniform plane wave in air with Ei (z) = 
10 ei6z is incident normally on an interface at z = 0 with a medium having a dielectric constant of 2.56 and relative permeability μr = 1 . Find the following:
(i) Instantaneous expressions for Er (z, t) , Hr (z, t) , Et (z, t) and Ht (z, t) .
(ii) The expression for the time-average Poynting vectors in air Sav1 and in the dielectric medium S . av2
Q7.
Fig. Q7 depicts a beaker containing a block of glass on the bottom and water over it. The glass block contains a small air bubble at an unknown depth below the water surface. When viewed from above at an angle of 60o, the air bubble appears at a depth of 6.81 cm. What is the true depth of the air bubble?
Fig. Q7 Apparent position of the air bubble
Q8.
Consider a plane wave incident from the medium 1 ( μ1,ε1 ) onto the medium 2 ( μ2,ε2 ) . For an oblique incidence, the reflection coefficients for the S and P waves are generally different from
each other. Consider a special case where the reflection coefficient is equal to zero. When it occurs, the incidence angle is defined as the Brewster angle θB .
(1) What are the values of Brewster angle θB for the S and P waves, respectively?
(2) Repeat part (1) for nonmagnetic materials, μ1 = μ2 = μ0 .
Q9.
Consider a two-dimensional TM scattering problem. The incident electric field is polarized in the z-direction, which is parallel with a PEC cylinder that has a circular cross section. The incident wave is a plane wave propagating along the positive x-axis. The magnitude of electric field is 1. The wavelength is 1 meter, and the radius of the cylinder is 0.15 meters. Let the center of the cylinder to be the origin. 30 receivers are placed uniformly on a circle of 5 meters radius, concentrically with the cylinder.
Implementing the MoM in numerical software, such as Matlab, to calculate the magnitude and phase of scattered field received at the 30 receivers. Two plots are required to be presented, the vertical axes are the magnitude and phase of scattered field respectively, and the horizontal axis is the angles of the receivers.
Please provide your source code that is used to generate the results. Place all your results for this question in a single PDF file, including source code, two plots, and other optional materials such as derivations and/or discussions.
Q10.
Two z-oriented dipole antennas are located at x= - λ and x=λ, respectively. We know that I1l = I2 l. In the xz plane, what is the smallest angle θ where the far field is null?
Fig. Q10
Q11.
For a radiation problem, the solution to the equation (▽2 + k2 )g(r) = -δ(r) is given by g(r) = ,
where both the plus and the minus signs are mathematically correct. Determine the sign in the exponent and provide justifications.