# Relation between the magnetization and the magnetic field in permeable media

I am trying to re-write the constitutive equation of linear permeable media in a way that is more convenient for its numerical implementation. From most books I have seen that the constitutive relation is expressed as

$$\mathbf{B}=\mu\mathbf{H}$$

being $$\mu$$ the media magnetic permeability.

However, I would like to express $$\mathbf{B}$$ as a funcion of $$\mathbf{M}$$ using the relation $$\mathbf{B}=\mu_{0}\left(\mathbf{H}+\mathbf{M}\right)$$. If it is right the fact that for linear media the field induces a magnetization, I could write

$$\mathbf{B}=\mu_{0}\left(\frac{1}{\mu}\mathbf{B}+\mathbf{M}\right)$$

which gives

$$\mathbf{M}=\frac{\mu-\mu_{0}}{\mu\mu_{0}}\mathbf{B}=\bar{\chi}\mathbf{B}$$

now being $$\bar{\chi}$$ a susceptibility measure.

With this form, $$\mathbf{M}$$ plays the role of an induced magnetization. So, if permanent materials are considered, a new $$\mathbf{M}_{0}$$ should be added as

$$\mathbf{B}=\mu_{0}\left(\mathbf{H}+\mathbf{M}+\mathbf{M}_{0}\right)$$

Is this an accepted way to express the constitutive relation? Can you see any inconsistency?

What you have written down does not look right. The relation $${\bf B}=\mu_{0}({\bf H}+{\bf M})$$ always holds exactly; indeed, it gives the definition of the auxiliary field $${\bf H}$$. So your notation already has a problem.
However, it seems that what you really want to do is to split the magnetization into two pieces: a linear piece and another constant piece supplied by a ferromagnetic background. Unfortunately, this is not generally a good approximation for ferromagnets. In a ferromagnet, the constitutive relation $${\bf M}={\bf M}({\bf B})$$ is not typically an affine function (constant piece plus linear piece). The relation is actually extremely complicated and nonlinear, with saturation effects and dependence on the history of the sample (hysteresis). So your method is probably not going to be very useful.