Consider an atomic spin which can have exactly two configurations with respect to an applied magnetic field, parallel or antiparallel. Starting from the probability of finding a particular spin in that configuration, at a given temperature, derive an expression for the paramagnetic susceptibility of a solid consisting of N atomic spins. :

b$.$ Within the quantum theory, derive an expression for the paramagnetic susceptibility of a material.

c$.$ Sketch the temperature dependence of the results in $($a$)$ and $($b$).$ Is the temperature dependence the same? Why or why not?