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No credit card required Problems 29
Problems
1.1. Some visible light detection systems are limited by background or
extraneous light from the surroundings. Usually 300K blackbody radi-
ation is not a problem. To verify this:
(a) Calculate what fraction of the power radiated at 300 K occurs at
wavelengths shorter than
1
jMn.
Use the approximation of Eq. (1.26).
(b) For wavelengths shorter than 1 jum, how many photons I second
are emitted by a 1-m^ source with unit emissivity at 300 K. Use a
photon energy corresponding to a wavelength of 1 fjm.
1.2 A long two-wire (single polarization mode) transmission line of
length, L, acts as a resonator with modes determined by nXjl - L, with
n an integer. Find the mode density in the frequency domain and then
find the thermal energy per unit length per unit frequency in terms of
kT and hv. What is the limiting form for low-frequency, that is, hv«
kT?
1.3 There is an atmospheric transmission band from 8 to 12 fjm that is
near the peak of the 300 K blackbody spectrum. Estimate the total power
radiated from a 1-m'^ unit emissivity surface in both watts and
photons/second, within this band. Approximate by calculating the power
spectral density at 10
/urn
and multiplying by the bandwidth. Compare
with a calculation based on Figure 1.13. Use the average energy of the
photons at 20 iJm.
1.4 The reflectivity of the earth averaged over the solar spectrum, some-
times called the earth albedo (literally, whiteness), is on the average
0.35. Using the solar constant and an earth temperature of
300K,
find
the effective emissivity e averaged over a 300K spectrum. Remember
that the earth receives solar energy from one direction but radiates its
thermal energy isotropically.
1.5 If the emissivity of the earth decreases by 10% of its current value,
find the rise in average temperature, assuming an unchanged albedo.
You may approximate by using differential forms dP and dT, 30 Chapter 1 Blackbody Radiation, Image Plane Intensity, and Units
1.6 The radii of the orbits of Mercury, Earth, and Pluto are respectively
36, 93, and 3700 million miles. With Earth at 300 K, find the temp-
erature of the other two planets assuming the solar reflectivity and
average emissivity are the same as those of earth.
1.7 Tv^o planes of unit emissivity are at temperatures T.^ and T^. They
are separated by a vacuum region as shown in the sketch. If
(T^
-T^) is
small and both are near 300 K, find a numerical value for the radiative
heat conductance, G^ =
('llA)dP/dT,
for heat flow between the planes.
^^^^^^
vacuum
1.8 One of the legendary causes of fire is that due to glasses (spectacles)
left on the window sill on a sunny day. Assume a 4 cm lens diameter
and a focal length of
1 m
(the optician would call this a one diopter , or
l/f=
1
m'^ correction). Let the sun shine normal to the lens and strike a
perfectly absorbing surface at the focal distance. If the surface can only
lose heat by radiation from the incident side, find its temperature.
1.9 You are performing medical thermography (remote sensing of skin
temperature) using an infrared camera operating in the wavelength band
of
8
to
12
jjm. Using the same approximation as Problem 1.3, calculate
the fractional change in the emitted power with temperature, T.
Specifically, find (l/P)(dP/dT).
1.10 Continuing from Problem 1.9, your camera has a
lO-crn^
lens of
focal length 10 cm and is observing the patient at a range of I m. A
detector in the image plane "sees" a skin area of
1
cm^. Assume the skin
has unit emissivity and calculate the number of photons received by the
detector in a
1-msec
exposure time.

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