$$Z_N=Z_1^N$$
This reminds us of the Gibbs paradox.
Next step:
$$Z_N=\frac{Z_1^N}{N!},$$
but since $$m=-\frac{dF}{dB}$$
it still doesn't change magnetization as the relation to \(Z_1\) still hasn't changed. So the ultimate solution would be
$$Z_N=\frac{(N+Z_1-1)!}{N!(Z_1-1)!},$$
and
$$Z_N=\frac{Z_1!}{N!(Z_1-N)!}.$$
$$Z=e^{0\beta\mu_B B}+e^{\beta\mu_B B}+e^{-\beta\mu_B B}=1+2\cosh(\beta\mu_B B)$$
The photoelectric effect suggests that the energy of electromagnetic field for monochromatic light should be a function of the frequency \(\omega\). Since waves are not generally monochromatic, to generalize this, we look for an expression of \(\omega\) in terms of the electromagnetic field (electric/magnetic) itself: since $$\textbf{E}(z,t)=E_0\sin(kz-\omega t)\hat{\textbf{x}}, \ \ \textbf{B}(z,t)=\frac{1}{c}E_0\cos(kz-\omega t)\hat{\textbf{y}},$$ we find $$\omega=\frac{1}{E_0}\frac{\Arrowvert\frac{\partial\textbf{E}}{\partial t}\Arrowvert}{\sqrt{1-(\frac{E}{E_0})^2}},$$
