Thermal/Statistical Physics Paramagnet

[NIQ]

Problem Set:

http://www.physics.utoronto.ca/%7Epoppitz/hw2.pdf

I'm having a problem with I.3

I got I.1 (answer is given on sheet)
For I.2 I found T(N,U,B) the following way
$$T = \frac{\tau}{k_B}$$
$$\frac{1}{\tau} = (\frac{\partial \sigma}{\partial U})_{N,V}$$
$$\therefore T(N,U,B) = \frac{2mB}{k_B} [ln(\frac{N}{2} - \frac{U}{2mB}) - ln(\frac{N}{2} + \frac{U}{2mB})]^{-1}$$

The problem I was having with I.3 is that I don't know how to go about finding the maximum magnetization. I remember hearing in class that the magnetization is maximized when the temperature is 0 but... it wouldn't make sense if B=0 and m can't be 0 so there's no other way of making that equation equal to 0.

I know the following is true:
$$M = -\frac{U}{B} = 2ms$$

But I'm not quite sure what I can do with this...

Any help would be greatly appreciated, thanks!

 

[Gokul43201]

Can you not simply rewrite T(N,U,B) as U(T,N,B) ?

PS : Is this course being taught by Yong Baek Kim ?

 

[NIQ]

Ok I'll try to do that and let you know.

And my course is being taught by Erich Poppitz.

 

[beee]

I guess the only thing to do is take a limit of M(B, T, N) as T goes to zero. That seems to work perfectly and agrees with common sense in the end.