8.7 Problem Set

Problem 1 Show that the energy density per unit wavelength is given by: $$E(\lambda)=\frac{8\pi hc}{\lambda^5}[e^{hc/\lambda kT}-1]^{-1}$$ Hint: $\lambda=c/\nu$

Problem 2 Assuming the following objects radiate as black bodies, what is the $\lambda_m$ for:
a) the sun at $T_s=5700K$
b) a mammal with a surface temperature $T=37 ^\circ C$
c) a lizard with a surface temperature $T=10 ^\circ C$
d) the clear night sky, radiation temperature $T=-33 ^\circ C$

Problem 3 a) The horns of a reindeer are at approximately internal body temperature, though the rest of the deer’s body is well insulated. If these horns are at a temperature ($T_0$) of 32°C, and they have a surface area of $1000 cm^2$, what is the heat loss rate if the environment is at a (radiation) temperature ($T_a$) of -10°C? Convert your answer to calories/minute. [See Gates 1968 for some actual organismal temperature measurements.] b) A bald mountaineer is outside on a clear night. Although he is heavily bundled in his down parka, he has forgotten his hat. If his bald pate has a surface area of $75 cm^2$ and is at a temperature of 32°C, what is his heat loss rate to the night sky which is at a (radiation) temperature of -45°C? [Is this why they say, “If your feet are cold, put on your hat.”?] Convert to calories/minute.

Problem 4 Using the Rayleigh criterion and the solution to Prob. 2b, determine the resolution of the infrared sensing system of a pit viper. Using reasonable physical parameters (e.g. body size of prey, how close the snake must be to strike, how accurate the strike has to be), determine the minimum separation of the pits. Does this agree with your intuition and/or experience?