The electromagnetic intensity thermally radiated by a body of temperature is given by where
This is known as the Stefan-Boltzmann law. Show that this law follows from equation (9-46). (Note: Intensity, or power per unit area, is the product of the energy per unit volume and distance per unit time. But because intensity is a flow in a given direction away from the blackbody, the correct speed is not . For radiation moving uniformly in all directions, the average component of velocity in a given direction is .)
The intensity is
Expression for energy of photons in container (U) is given by-
Speed of light in vacuum
Volume of container
Alternate expression for intensity -
To start, equation (2) can be rewritten slightly:
Use that the average component of velocity in any particular direction is , set v equal to , and use equation (1) for the energy in equation (3):
Substitute for .s and for in the above equation and obtain the equation as given below.
The diagram shows two systems that may exchange both thermal and mechanical energy via a movable, heat-conducting partition. Because both E and V may change. We consider the entropy of each system to be a function of both: . Considering the exchange of thermal energy only, we argued in Section 9.2 that was reasonable to define as . In the more general case, is also defined as something.
a) Why should pressure come into play, and to what might be equated.
b) Given this relationship, show that (Remember the first law of thermodynamics.)
To obtain equation (9-42), we calculated a total number of fermions as a function of assuming . starting with equation . But note that is the denominator of our model for calculating average particle energy, equation (9.26). its numerator is the total (as opposed (o average particle) energy'. which we’ll call here. In other wonts. the total system energy is the average particle energy times the total number of particles (n). Calculate as a function of
And use this to show that the minimum energy of a gas of spin fermions may be written as
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