Mathematical modeling of the free surface of a magnetic liquid surrounding a magnetized plate

Abstract

We propose a mathematical model that allows one to describe the shape of the free surface of a magnetic liquid surrounding a thin flat magnetized plate with infinite magnetic susceptibility. The Maxwell equations and the hydrostatics equations for magnetic liquid with the corresponding boundary conditions on the free surface of the magnetic liquid are solved. By methods of complex analysis, we find the distribution of the magnetic field and the pressure in the magnetic liquid surrounding the plate. An equation for the shape of the free surface of a magnetic liquid near the plate is written, and graphs of equilibrium shapes of the free surface of the magnetic liquid in a neighborhood of the plate are given for various volumes of the liquid and the angle between the plate and the vector of the external magnetic field.