Magnetic Field of Current loop around the Core

[FelixTheWhale]

Hello friends, I am trying to find how calculate the magnetic field created by current loop considering that there is a some geometry of material that can enhance the magnetic field. I thought it can be solved by multipling by permeability μ but realized that the iron core changes the picture of magnetic qualitatively.

To illustrate this i attached image for: only coil (1) and coil around the iron bar (2).

For the first picture, it is easy to integrate by Bio-Savart-Laplace law, but..
How to find he field for any point at the second picture?

Spoiler: Illustrations

Or, more complicated case:

 

[Hesch]

The latter case is easy if the width of the airgap is small ( the magnetic field in the airgap is almost homogenous ).

From Ampere's law you know that
∫ H⋅ds = N*I ( complete circular path )
Also you know that H_iron = μ_r,iron * H_air

In case of the rod/solenoid, you don't quite know the path of the magnetic field. Some numeric calculation will be needed. ( See my avatar ).

 

[FelixTheWhale]

Nice image! What can you advise to make a numeric calculation? Is it possible to apply a vector potential (B=curl (A)) to have the solution for complex geometry?

 

[Hesch]

What can you advise to make a numeric calculation?

I have used high precision Biot-Savart calculation. Just use a 3D compass and follow the field direction from some point, until the field bites its own tail.