Homework B
PHY4523
Due: February 10, 2012
1. Statistical Definition of Temperature
(a) A thermodynamic system is in equilibrium at 300 K. Find the percentage increase in
the number of microstates of the system when its energy is increased by 0.1 meV.
(b) (4.5) Show that the number of microstates   exp(E / k BT ) where E is change in
energy. Consider a 100 MHz photon coming out of a FM transmitter.
2. An atom carries magnetic moment, m. Consider a thermodynamic system consisted
of N non-interacting such atoms. This system is in thermal equilibrium at T and in the
presence of a magnetic field Bo. Each atom can be aligned either in the parallel (m) with
energy –mBo or anti-parallel (-m) direction with energy +mBo.
(a) Show that the total magnetic moment of this system (M) is
M (T , Bo )  M  Nm tanh(mBo  ) .
(b) What is the leading temperature dependence of M when mBo << 1 (high temperature
limit).
(c) Show that
1  M 
 M2  k BT   

  Bo T
where σM = (<M2> - <M>2)1/2 is the variance of the total magnetic moment. A similar
result was obtained for heat capacity.
Use the fact  M2  N m2 .
3. Let’s work on the same magnetic system in Prob. #2 in the microcanonical ensemble.
You will see how awkward it is to solve this problem.
4. (19.3)

5. (19.6) Check your answer for  = 1 and 2. You will need to use (n)  x n1e  x dx .

0
See C.1 of the textbook.
6. (19.7)
7. (20.6) There is an error in equation (20.48). The numerator should be 2n( g B ) 2  o .
Follow the recipe of (20.5).
8. (20.7)
9. (20.8)