MTH 455 Homework #5

1. What is the temperature of the tungsten filament of a regular 100W bulb (when it's on!) if the length of the filament is 5 cm, the diameter is 0.1mm, and the emissivity is 0.5? (This is just arithmetic using Stefan's Law.)

2. Show that if T(x,t) is a (smooth enough) solution of the heat equation then so is T_x(x,t).

3. For a material diffusion problem, let C be the concentration of the diffusing substance (in kg/m³, say). Thus C is analogous to the temperature T. The 1D heat equation becomes the diffusion equation C_t = &kappa C_xx, and in 3D, C_t = &kappa times the 3D Laplacian of C.

Using the steps below, solve the following problem. Suppose 1 microgram (1 microgram = 10^-6g) of liquid perfume is sprayed into the air in an auditorium where the air is still. If the human threshold for detecting the vapor is 10^-11g/cm³, how close would you need to be to the point of release in order to smell the perfume?

Use the 3D version of the fundamental solution (the point-source solution):

Here M denotes the mass of perfume originally aerosolized. Proceed as follows:
(a) Verify that the total amount of perfume is M for all t. (Check your Calc III book if necessary to recall how to do integrals in spherical coordinates.)
(b) Find the time of maximum perfume density at a distance r from the point of release.
(c) Find what the maximum concentration itself is (at a distance r from the point of release). Observe that it is independent of the diffusivity of the perfume!
(d) Determine at what distance the maximum concentration equals the detection threshold, and give your answer to the original question.
(e) Can you tell when you will get the momentary whiff of perfume if you are sitting right at this distance?

4. (a) What is the phase relationship between (the real parts of) e^{i&omega t} and ie^{i&omega t}?
(b) What is the phase relationship between (the real parts of) e^{i&omega t} and (i-1)e^{i&omega t}?

1. Show that there is only one physically meaningful (i.e. non-negative) solution of the equilibrium condition given at the bottom of p3 of Text 3A for the reversible reaction A + B <-> C.