Help me understand Biot-Savart law

Hi,

I’m posting this question on physics forum, since I didn’t get the answer on electromagnetism forum… Sorry for doubling…

As a student I’m having difficulties understanding the Biot-Savart law…

The problem originates in basic definition of magnetic induction B and its relation with magnetization M and magnetic field H. The induction B in ferromagnetic materials can be written as:

B=n0*(H+M)

where H is the filed that is not affected by the presence of iron material and M is the material contribution to the field… The above equation can be rewritten in well known expression

M=x*H; (1+x)=nr  B=n0*nr*H

So now the question – In above equations, H is the filed that would be produced, for instance, by the current in the vacuum, right? So, H is not affected by the presence of the iron? Thus, this filed can be calculated using Biot-Savart…. We can then calculate induction B, by simply multiplying this value with the material constants?

This is of course not correct, since this way all the geometry (flux path) is neglected. For example, if we would like to calculate the B in a tooth of the electric motor, then we would simple calculate H using Biot-Savart expression and multiply this value with n0*nr…. If above presumption would be correct, the FEM would not be needed at all….

Can somebody please point me where in above thinking I made a mistake?

Thank you in advance,

Vincenc

Re: Biot-Savart Law

The Calculation with the law of Biot-Savart is only
valid for homogen space. For example in air (vacuum)
or in some magnetic material. And in 'classical' electromagnetic
we observe just macroscopic phenomenon.
If you have a halfspace, you can introduce image sources, in order
to take the effects of the inhomogeneity into account.
If your problem is so inhomogen, I think FEM is your tool.

regards

Re: Biot-Savart Law

Thank you for your reply!

I do understand that in such problems FEM is the most suitable tool, but what puzzles me in the moment is the basic understanding of the above equations …things I really should understand well.

Basically – if we look at the following eq. B=n0*(H+M)…Is magnetic filed H in this eq. affected by the presence of the iron or not? I was taught that H is material independent and M is the material contribution to the field ... Presuming this, H can be estimated using Biot-Savart… Because also M can be expressed with filed H, it seems, that all the problems in magnetism could be simply solved using Biot-Savart …I know this is terrible wrong, but at this moment I can’t see where I made a mistake in above thinking…

I would be very grateful if someone would show me where I made a wrong presumption…

Thanks in advance

Added after 28 minutes:

Hi, me again….

Just wanted to say, that I’m familiar with the reduced scalar potential formulation, on the basis of which the B can indeed be calculated by Biot-Savart combined with FEM code:

H=Hm+Hs; Hs is calculated using Biot Savart and Hm (material contribution to the filed) can be expressed using scalar potential --> Hm=-gradF. This way we get:
B=n0*nr*(Hs-gradF)

This seems very logical to me: Hs is material independent. On the other hand, calculation of gradF takes all the geometries’ and materials’ properties of the given problem into account.

How can then be simply said that in eq. B=n0*(H+M) magnetization M can be easily replaced with (1+x)*H, so we get B=n0*nr*H ?

Thanks