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

Express this Hamiltonian in terms of the parameter \(L\left(=N^{-1} \Sigma_{i} \sigma_{i}\right)\) and show that, in the limit \(N \rightarrow \infty\) and \(c \rightarrow 0\), the mean field theory, with \(J=N c / q\), is exact for this model.