# Magnetic field behaviour around non-magnetic and magnetic materials

The purpose of this report is to present a mathematical model to describe the induced magnetic field B in free space.

Induced magnetic field is easily modelled with a set of virtual wires moving with a certain speed. The position, peed and the number of these wires depends on some material constants such as the permeability of the medium. This model is very similar to the well known method of images used in electrostatics. This method will be discussed in a more accurate way in chapter 4 and 5.

In order to build and test the accuracy of our model we need some data on how magnetic fields behave in reality. We have then used a computer program called Opera 3d to simulate these magnetic fields. The results given by this program are analysed in chapter 3. Observe that our mathematical model discussed in chapter 4 and 5 have nothing to do with Opera 3d. Opera 3d uses a different algorithm to simulates magnetic fields.

Opera 3d uses finite element techniques to analyse time varying electromagnetic fields. The program uses the total and reduced magnetic scalar potentials in non-conducting media thereby correcting the cancellation errors associated with the simple reduced potential approach. In conducting media Opera 3d uses the magnetic vector and electric scalar potentials which are directly coupled with the potentials in the exterior. The permeability is calculated using the magnitude of the field at any point giving a representation of saturation effect in ferromagnetic materials.

In chapter 4 we will take in consideration the theory describing the induced magnetic field in the upper half plane for both non-magnetic and high magnetic mediums. Our model consisting in the virtual wires is in fact an interpretation of the solution of the set of PDEs describing the magnetic field in this region. In chapter 4 we will only discuss the magnetostatic case of our model.

In chapter 5 we will discuss our model for time varying magnetic fields and test its accuracy in the non-magnetic case. We will also discuss the application of our model on high-magnetic materials and some of the difficulties encountered in that area.

In chapter 6 we will give some indications on why our model for the magnetic case should work.

Author: Jomni, Yassin

Source: LuleĆ„ University of Technology