Download Mid-Term Exam

Mid-Term Exam, Electromagnetic Theory II
Dr. Christopher S. Baird, Spring 2015
University of Massachusetts Lowell
Part I: Multiple Choice (30 Points)
Circle the one best answer to each question.
1. What is the most general, fundamental expression for the Poynting vector?
(a) S=
√
1 ϵ ∣ ∣2 ̂
E k
2 μ
(b) S=E×H
1
(c) S= ℜ( E×H *)
2
(d) S=σ E
2. What is the difference between “loss” and “attenuation” when talking about electromagnetic waves
traveling in materials?
(a) There is no difference. They are exactly the same thing.
(b) “Loss” describes the energy that is reflected by the material back towards the incident direction,
while “attentuation” describes the narrowing of the beam as it is refracted.
(c) “Loss” describes energy absorbed by the material, while “attenuation” describes the decrease in
wave strength across space, which may be caused by loss, but can be caused by other things.
(d) “Loss” describes the portion of the wave scattered sideways and “attenuation” describes the portion
of the wave scattered forwards.
3. How do we decrease the electromagnetic skin depth of a good conductor?
(a) increase the conductivity
(b) decrease the conductivity
(c) increase the temperature
(d) decrease the temperature
4. An electromagnetic plane wave is incident on the planar interface between two dielectric materials at
Brewster's angle. In which case will the plane wave be totally transmitted through the interface?
(a) The incident wave is polarized parallel to the plane of incidence
(b) The incident wave is polarized perpendicular to the plane of incidence
(c) The incident wave is going from high-index material to low-index material
(d) The incident wave is going from low-index material to high-index material
(e) The incidence wave is always totally transmitted at Brewster's angle.
5. When can an incident electromagnetic wave travel through an ideal plasma?
(a) always
(b) never
(c) above the plasma frequency
(d) below the plasma frequency
6. For a perfectly circularly-polarized plane monochromatic EM wave traveling in free space, how
does the electric field magnitude, |E|, vary through space?
(a) The E field magnitude gradually decays through space.
(b) The E field magnitude oscillates back and forth along one line.
(c) The E field magnitude oscillates according to a spinning sine wave.
(d) The E field magnitude is constant across all points in space, although its direction spirals around.
7. When do electromagnetic waves exist that are exactly monochromatic?
(a) never
(b) only in waveguides
(c) only in free space
(d) only in plasmas
8. What is a good description of electromagnetic wave dispersion?
(a) When a wave packet keeps its shape as it travels.
(b) When a polarized electromagnetic wave becomes unpolarized.
(c) When an electromagnetic wave is scattered in many different directions.
(d) When different frequencies in a wave packet experience different dielectric constants.
9. When does the group velocity of a wave packet exceed the speed of light in vacuum c?
(a) never
(b) for all frequencies below the plasma frequency
(c) for all frequencies above the plasma frequency
(d) only when the group velocity loses meaning because the wave packet is too distorted
10. What do the Kramers-Kronig relations tell us?
(a) That the wavenumber is always real-valued
(b) That the real part and imaginary part of the dielectric constant depend on each other
(c) That the real part and imaginary part of the dielectric constant do not depend on each other
(d) That the plasma frequency is always less than the resonant frequency.
11. When are the magnetic field and electric field of a transverse EM plane wave in phase with each
other?
(a) Never
(b) Always
(c) Only in materials where the permittivity and permeability are real-valued.
(d) Only in materials with a lot of loss.
12. What best describes the role played by the imaginary part of the wavenumber of an EM wave?
(a) dispersion
(b) wave refraction
(c) wave attenuation
(d) energy loss/absorption
13. According to the Fresnel equations, what best describes the sum of the reflection coefficient R and
transmission coefficient T?
(a) R + T = 1
(b) R + T < 1
(c) R + T > 1
(d) R + T = 0
14. What best describes refraction?
(a) reflected waves have an angle of reflection that is equal to the angle of incidence
(b) transmitted waves change speed and bend when entering a different material
(c) wave packets spread out longitudinally as they travel in certain materials
(d) two coherent waves can cancel each other out.
15. In general, how does the reflectivity of a dielectric material's surface change as we increase the
angle of incidence (i.e. the angle between the incident wave vector and the surface normal increases)?
(a) R stays constant
(b) R decreases
(c) R increases
(d) R becomes infinite
Part II: Diagram Problem (25 Points)
A large volume is filled with a linear, dispersive, lossy dielectric material. For simplicity, this material
follows the simple harmonic model, only has one resonant oscillation at frequency ω0, and a plasma
frequency of ωp. The material is significantly lossy.
1) On the appropriate diagram on the next page, sketch the real part of the relative permittivity εr of this
material.
2) On the appropriate diagram on the next page, sketch the imaginary part of the relative permittivity εr
of this material.
3) A wave packet travels in this material, centered at one of the various frequencies shown on the
diagram (ω1, ω2, ω3, ω4). For each of these possible center frequencies, describe using words what the
wave experiences by filling in each box in the table on the next page.
Re(εr)
1
ω0
ω
ωp
Im(εr)
ω
ω0
ω1
Dispersion
(low, high, or anomalous)
ω1
ω2
ω3
ω4
ω2
ω3
Absorption
(low or high)
ω4
Attenuation
(low or high)
Type of Wave
(standing or traveling)
Part III: Work Problems (45 Points)
Show all your work and explain each major step to receive full credit.
Problem 1
Consider an ideal plasma. For wave frequencies below the plasma frequency, find the following things.
Write you answers as explicitly as possible.
(a) The dielectric constant, real and imaginary parts.
(b) The wave number of a plane wave in the plasma, real and imaginary parts.
(c) The electric field of a plane monochromatic wave in the plasma.
Problem 2
A telescope that detects only waves of frequency ω is aimed at several points in a distant nebula and
polarization filters are used to determine the Stokes parameters at each point. The Stokes parameters
are found to be:
s 0=
Rr
,
R
s1 =
R−r
,
R
s 2=0 ,
s 3=2

r
R
where R is the total radius of the nebula and r is the distance between the observation point in the
nebula and the nebula's center.
(a) Determine the polarization vector of the waves from the nebula that this telescope is detecting and
write out the electric field as explicitly and detailed as possible.
(b) What type of polarization of light is emitted at the nebula's center?
(c) What type of polarization of light is emitted at the nebula's edge?