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Chapter 9: Q77E (page 409)
At what wavelength does the human body emit the maximum electromagnetic radiation? Use Wien's law from Exercise 79 and assume a skin temperature of.
The wavelength at which human body emits maximum electromagnetic radiation .
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The electrons’ contribution to the heat capacity of a metal is small and goes to as . We might try to calculate it via the total internal energy, localid="1660131882505" , but it is one of those integrals impossible to do in closed form, and localid="1660131274621" is the culprit. Still, we can explain why the heat capacity should go to zero and obtain a rough value.
(a) Starting withexpressed as in equation (34), show that the slope atis.
(b) Based on part (a), the accompanying figure is a good approximation to when is small. In a normal gas, such as air, whenis raised a little, all molecules, on average, gain a little energy, proportional to . Thus, the internal energy increases linearly with , and the heat capacity, , is roughly constant. Argue on the basis of the figure that in this fermion gas, as the temperature increases from to a small value , while some particles gain energy of roughly , not all do, and the number doing so is also roughly proportional to localid="1660131824460" . What effect does this have on the heat capacity?
(c)Viewing the total energy increase as simply = (number of particles whose energy increases) (energy change per particle) and assuming the density of states is simply a constant over the entire range of particle energies, show that the heat capacity under these lowest-temperature conditions should be proportional to . (Trying to be more precise is not really worthwhile, for the proportionality constant is subject to several corrections from effects we ignore).
Discusses the energy balance in a white dwarf. The tendency to contract due to gravitational attraction is balanced by a kind of incompressibility of the electrons due to the exclusion principle.
(a) Matter contains protons and neutrons, which are also fanions. Why do the electrons become a hindrance to compression before the protons and neutrons do?
(b) Stars several times our Sun's mass has sufficient gravitational potential energy to collapse further than a white dwarf; they can force essentially all their matter to become neutrons (formed when electrons and protons combine). When they cool off, an energy balance is reached like that in the white dwarf but with the neutrons filling the role of the incompressible fermions. The result is a neutron star. Repeat the process of Exercise 89. but assume a body consisting solely of neutrons. Show that the equilibrium radius is given by
(c) Show that the radius of a neutron star whose mass is twice that of our Sun is only about .
The Stirling approximation., is very handy when dealing with numbers larger than about . Consider the following ratio: the number of ways Nparticles can be evenly divided between two halves of a room to the number of ways they can be divided withon the right andon the left.
(a) Show, using the Stirling approximation, that the ratio is approximatelyfor large.
(b) Explain how this fits with the claim that average behaviours become more predictable in large systems.
Show that in the Iimit of large numbers, the exact probability of equation (9-9) becomes the Boltzmann probability of (9-17). Use the fact that , which holds when .
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