Biot Savart Law with Different Magnetic Permeabilities

[BrandonBerchtold]

I would like to make a program that produces a 2D heat map showing the magnitude of the magnetic field produced by a finite length solenoid. The heat map would show the field strength along the radial and axial directions of the solenoid.

I plan to divide the conductor into "infinitessimally" small lengths (on the order of 1-10 microns long), and sum up the total field contributions from each of these lengths for every axial and radial location along the coil to generate the 2D heat map, using Biot Savart Law.

My question is as follows: if a core is added to the solenoid, the field strength increases significantly, even outside of the core. Does Biot Savart Law account for this when the field is being calculated outside of the core? If I am summing the infinitessimally small field contributions of each current element, how would the locations outside of the core know that the field strength has increased?

Is there a way to deal with this problem similarly to how magnetic circuits are solved in transformers with air gaps, as they are still able to solve for the magnetic flux as it passes through both an air gap and the ferrite core.

 

[Charles Link]

In the surface current model of magnetism, there are surface currents on the cylindrical iron core with the exact same geometry as the current in the solenoid. Biot-Savart can be used to compute the magnetic field from these surface currents. In general for a solenoid with current per unit length $K=nI$ where $n$ is the number of turns per unit length, the surface current per unit length is $\mu_r-1$ times that of the surface current per unit length $K$ of the solenoid, where $\mu_r$ is the relative magnetic permeability of the core material.
Alternatively, the pole method for computing the magnetic field gets the exact same answer, and is much simpler to compute than a Biot-Savart calculation on the surface currents.