ELEC 3909
PROBLEM SET 2
1. In space just outside the earth’s atmosphere, sunlight has a
power density of 1.5 kWm-2. An aluminized mylar sheet
25 μm thick is oriented perpendicular to the sunlight. The
mylar has a density of 0.8 gcm-3. If the sheet starts from
rest and moves in response to the pressure of the sunlight,
how long will it take it to travel from the earth to the moon,
a distance of 380,000 km? Ignore gravitational forces.
(Inspired by the Arthur C. Clarke short story The Wind
From the Sun )
2. A Carleton graduate student is planning to build antennas
for 24 GHz operation on high resistivity (ρ = 500 Ωcm)
silicon substrates. What is the characteristic length for
TEM wave attenuation in the silicon at this frequency?
Take εr = 11.9 for silicon.
3. During the cold war, it was once suggested that submarines
carrying ICBMs should be stationed in Lake Superior.
(The idea was that the location of the submarines would
never be accurately known, so they could not be the targets
of a "first strike" attack). This problem considers the
possibility of communicating with such submarines using
radio waves. Suppose that εr = 81, μr = 1 and σ = 10-4 Sm-1
in the lake water. For a TEM radio wave of frequency of
1 MHz propagating vertically downward into the lake,
estimate the ratio of the power at a depth of 30 m to the
power just beneath the surface.
4. Communicating with submarines at sea presents an even
more difficult problem than communicating through lake
water. The US Navy has constructed an Extremely Low
Frequency (ELF) radio system operating at a frequency of
76 Hz for this task. The antenna for this system is located
in northern Michigan, and is approximately 30 km long.
For a 76 Hz TEM radio wave propagating vertically
downward into the water, estimate the ratio of the power at
a depth of 20 m to the power just beneath the surface.
Take σ = 4 Sm-1, εr = 81, and μr = 1 for the seawater
5. Consider a material which is both a weak conductor with
conductivity σ and a lossy dielectric with complex
permittivity εc = εe-jφ. Assume that σ << εω and φ is close
to 0. Show that a possible solution to Maxwell’s equations
in such a material has the form

E  ( E0 e z e j (t z ) ,0,0)
where   

2


 2
and    
6. A TEM wave with electric field

E  ( E0 e  z /  e j (t  z /  ) ,0,0)
is propagating into a metal with conductivity σ. Consider
a thin layer inside the metal of thickness Δz. Determine
the change ΔSav in the magnitude Sav of the time-average
Poynting vector on crossing this layer. Show that ΔSav is
equal to the resistive heating I2R per unit area resulting
from induced current in the layer averaged over a cycle.
May 2013
7. Consider a material for which σ = εω exactly. Show that
there is a solution

E  ( E0 e j (t  z ) e z , 0 , 0)
to the wave equation in this material provided
   21 / 4  sin( / 8)
8. Radio waves at a frequency of 100 MHz have been used to
measure the thickness of the ice caps over Greenland. A
pulse of radio waves is transmitted into the ice, and the
return pulse reflected off the rock beneath the ice is
detected. The time of flight of the pulse determines the ice
depth. Glacial ice has εr = 3.17 and tanδ = 0.002 at this
frequency. Assume rock has εr = 6.
a) If a radio pulse is transmitted through a 100 m thick ice
sheet, what fraction of the incident power returns to the
surface?
b) What is the travel time for the pulse?
9.
A linearly polarized TEM radio wave in air with power
density of 1 mWcm-2 strikes a perfect conductor at normal
incidence. Determine the peak density of the surface
current (in Am-1) induced in the conductor
10. Consider the standing wave produced when a linearly
polarized TEM wave with electric field

E  ( E0 e j (t  kz ) ,0,0)
reflects from a perfect conductor, as described in section
3.1 of the notes. Find an expression for the instantaneous
Poynting vector S(t) at a distance λ/8 from the conductor
surface. Explain why S changes direction as time
advances. Also find the time average Poynting vector Sav.
Explain your result for Sav.
11. Circularly polarized light with electric field

E  ( E0 , E0 e j / 2 ,0) e j (t  kz )
reflects from a perfectly conducting metal mirror. Write
equations for the electric and magnetic fields in the
resultant standing wave. Sketch the electric and magnetic
field patterns.
12. A 100 MHz TEM radio wave strikes an aluminum foil
sheet 50 μm thick at normal incidence. The aluminum
has a resistivity of 2.7 μΩcm, and has ε r = μr = 1. In the
following, ignore multiple reflections, but justify this
assumption.
a) Estimate the fraction of the power in the incident wave
that penetrates through the sheet.
b) Estimate the fraction of the incident power which is
absorbed in the aluminium.