Consider an Ising model with an infinite-range interaction such that each spin interacts equally strongly with all other spins:

$$\hat{H}=-c\sum _{i<j}{\sigma }_i{\sigma }_j-\mu {\mu }_{0}H\sum _i{\sigma }_i$$

Express this Hamiltonian in terms of the parameter $L(=N^{-1}{\Sigma }_i{\sigma }_i)$ and show that, in the limit $N\to \infty$ and $c\to 0$, the mean field theory, with $J=Nc/q$, is exact for this model.