How Does the Divergence of Magnetic Field Relate to Magnetizing Fields?

[hotel]

Hi
I am trying to derive the relation between magnetic field strength in materials and magnetizing field from the $$\mathbf B$$ field. More exactly, my question is:

how do we get this expression

$$\nabla \centerdot \mathbf H = - \nabla \centerdot \mathbf M \\$$

knowing

$$\nabla \centerdot \mathbf B = 0 \\$$

and

$$\mathbf B=\mu_0(\mathbf H + \mathbf M )$$
?

Anyone can guid me how the first equation is related to the 2nd and 3rd equations?

thanku

 

[inha]

$$\nabla \cdot B=\nabla \cdot {\mu}_0(H+M)=0$$

Do you get the idea now?

 

[hotel]

I 'm not sure !?

but how do we get from this:

$$\nabla \cdot B=\nabla \cdot {\mu}_0(H+M)=0$$

to this:

$$\nabla \cdot H= -\nabla \cdot M$$

I can only see you have replaced B !

Or maybe you want me to think like this :
$$\nabla \cdot H=\nabla \cdot {\mu}_0H+{\mu}_0M$$
$$\nabla \cdot H=\nabla \cdot {\mu}_0M$$
assuming for a very small volume of material ?

or am I totally misunderstanding !?

 

[inha]

$$\nabla \cdot {\mu}_0(H+M)=0$$

divide $$\mu_o$$ out. you can do that since it's just a constant. then you have

$$\nabla \cdot H + \nabla \cdot M=0$$

which is what you're looking for once you move the M-term to the other side of the eq.