References: Griffiths, David J. (2007) Introduction to Electrodynamics, 3rd Edition; Prentice Hall – Problem 6.17.
Another example of using the auxiliary field to find the magnetic field. This time, we have a long wire of radius carrying a uniform current . The wire is made of a linear metal so that the magnetization is directly proportional to .
where is the magnetic susceptibility. Given this proportionality, the field is also directly proportional to :
where is the magnetic permeability.
From the symmetry of the setup, is circumferential, so we can take a circular integration path to get
From this we get the field for
Outside the wire, and the enclosed free current is just , so for
The magnetization is
so the free currents are
The total bound current is