I have question about Maxwell's 2nd equation

[yang hg]

2nd maxwell's equation is ∇⋅B = 0. Then Can ∇⋅H be non-zero? I know that there is anisotropic media regarding permeability. If ∇⋅H can be non-zero in anisotropic media, I think it show that there is magnetic monopole because I think magnetic polarization and current is same essentially. Is this idea correct? If not, teach me what is wrong. thanks in advance.^^

 

[Orodruin]

Can ∇⋅H be non-zero?

Yes. If the material is not isotropic or homogeneous.

If ∇⋅H can be non-zero in anisotropic media, I think it show that there is magnetic monopole because I think magnetic polarization and current is same essentially. Is this idea correct?

No. A magnetic monopole would be a source for $\vec B$.

 

[yang hg]

No. A magnetic monopole would be a source for →BB→\vec B.

maxwell's second equation imply there is no magnetic monopole. But you tell me monopole is source for B. It is contradiction.​

And I cannot understand that ∇⋅H can be non-zero. I'm confused B with H. What makes B and H have this different property? Plz help me.

 

[Orodruin]

maxwell's second equation imply there is no magnetic monopole. But you tell me monopole is source for B. It is contradiction.

No, it is not. It is precisely the fact that there is no source on the right-hand side of $\nabla \cdot \vec B = 0$ that tells you there is no monopole. A monopole by definition would mean that the right-hand side would be non-zero.

 

[yang hg]

No, it is not. It is precisely the fact that there is no source on the right-hand side of $\nabla \cdot \vec B = 0$ that tells you there is no monopole. A monopole by definition would mean that the right-hand side would be non-zero.

I get it. Then what is meaning of ∇⋅H ≠ 0? ie. if ∇⋅H = 1, what is physics meaning of 1?

 

[Orodruin]

That would be dimensionally inconsistent.

Since $\vec H = \vec B/\mu_0 - \vec M$, $\nabla \cdot \vec H = - \nabla \cdot \vec M$. The right-hand side would therefore be a source of magnetisation.

 

[essenmein]

From an intuitive understanding perspective I prefer the integral form.

The way I understand gauss's law is this: (quite possible that it is wrong lol)

Integral of B over a closed surface S = 0. Ie Magnetic field lines are loops, if field leaves the surface it must come back in somewhere, then, B cannot exist without H. If considering something that is pre magnetized (as in above), then the little unpaired electrons in that magnetized material are producing the H to create B given the reluctance path of the magnetic loop (that's where load lines come in for designing with permanent magnets). Ie the M in the equation above is actually Br/u (where Br = remnant flux density).

So far as I know we have not ever identified a magnetic mono pole, so therefore integral of B or H over a closed surface = 0.

 

[snowflakesarepowder]

2nd maxwell's equation is ∇⋅B = 0. Then Can ∇⋅H be non-zero? I know that there is anisotropic media regarding permeability. If ∇⋅H can be non-zero in anisotropic media, I think it show that there is magnetic monopole because I think magnetic polarization and current is same essentially. Is this idea correct? If not, teach me what is wrong. thanks in advance.^^

You have everything correct but need to have delta right side up ∆. I am sorry that Maxwell made an error as is should be it Laplace operator right side up. So by cubing the H you can make correct calculations. I am not trying to make this up you can try it if you like. I can only tell you what I know.

 

[Orodruin]

You have everything correct but need to have delta right side up ∆

No, this is incorrect. There is a difference between $\nabla$ and the Laplace operator $\Delta$.

 

[Dale]

You have everything correct but need to have delta right side up ∆

As @Orodruin mentioned the $\nabla$ is correct, not $\Delta$. You can easily verify this at: https://en.wikipedia.org/wiki/Maxwell's_equations#Formulation_in_SI_units_convention