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.

State your answer in

^^^^^^

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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