Modelling deformations of solenoid generated magnetic field

Abstract

The concentric sphere problem admitted for the possibility of finding an effective permeability. We proposed using numerical methods to find the effective permeability of a two sphere system. We found that the permeability of the sphere with the larger radius dominated in the numerical value of the effective permeability. We showed the effect of magnetic shielding can be achieved by selecting the material of the shield to have a very high magnetic permeability. The cavity inside will be shielded from the effects of the external magnetic field. __The final component of the project consisted in extending the analysis to elliptical shapes. We developed the full theory behind deriving the tangent space to an elliptical manifold in R2 in order to produce the interface conditions for the ellipse. We proposed an approximation scheme based on perturbation theory to solve for the coefficients in a second order Taylor series in the ellipse parameter e. We found the field polarizes around the ellipse, concentrating strongly on one side and weakly on the other. We also observed consistency with our solution for a spherical object when the ellipse parameter was zero. __This work focused on the deformations of magnetic fields caused by magnetizable spherical objects in the presence of a solenoidal coil. We derived magnetostatic interface conditions and investigated the behavior of the magnetic field in different scenarios, including a sphere in a uniform B-field and concentric spheres with varying magnetic permeabilities. We also extended our analysis to elliptical shapes, developing a theory for deriving interface conditions and proposing an approximation scheme for solving the problem. Overall, our findings contribute to the understanding of magnetic field interactions with magnetizable materials and their deformation effects to the uniform magnetic fields produced by the solenoidal coils in an MRI. (from Conclusion)