Statistical physics at constant temp

[oddiseas]

Homework Statement

A system of 32 spin-½ dipoles (each of moment μ) is held at a constant temperature in an external magnetic field, where on average, 20 dipoles are aligned with the magnetic field of strength B.
a) What is the temperature, in terms of given parameters?
b) What is the standard deviation in the mean energy, in terms of given parameters?

Homework Equations

The Attempt at a Solution

a) the net magnetic moment is 20mu-12m=8mu
p(spin up)=20/32
p(spin down)=12/32

Now i am a bit cofused in using this information in the boltzman distribution.

(p₂)/(p₁)=e^-(E₂-E₁)/(kT)

The energy for the dipoles is given by E=-muB
The net magnetic moment is 8mu and thus the total energy of the system in this state is E=-8muB.

So is the energy difference in the case simply the total energy of the system which is E=-8muB, or is it E(12)-E(20)
ie:
E(12)=12muB
E(20)=-20muB
Anyway i did the following:

(3/5)=e^-(8muB)/(kT)

Also is there are restriction on what (p₂)/(p₁) is interpreted as?
can i take it to be 12/20 or 20/12 without changing the result?

 

[_coyote_]

Solving this equation for T yields:T=(8muBln(3/5))/kb) Now to calculate the standard deviation i used the following formula:σ=sqrt((E²)-(E)²)/N where E is the mean energy, E² is the mean energy squared and N is the number of dipoles.Since E=-8muB we get:σ=sqrt((64μ²B²)-(-64μ²B²))/32σ=sqrt(128μ²B²/32)σ=(4μBsqrt(2))/sqrt(2)