Obtain an expression for the magnetic susceptibility asso- ciated with the orbital motion of free electrons in the presence of a uniform magnetic field H,

The grand partition function for this system was written  as

Since βµH/mc << 1, we can use Euler´s formula to rewrite this expression

In the first integral, it is convenient to introduce the new variable

so that it becomes easy to integrate over k_z. In the second in- tegral, we just call ε = k_z²h²/2m. It is now straightforward to obtain an expression for the zero-field susceptibility [check the excellent article by R. B. Dingle, "Some magnetic properties of metals. I. General introduction and properties of large systems of electrons", Proc. Roy. Soc. A211, 500 (1952)].