Calculation of the Vector Potential of the Magnetic Field in a Nonconductive Medium during Floating Zone Melting

The vector potential of the magnetic field in a region between the induction coil, upper part of the floating zone, liquid film, feed rod, and protective shield has been determined in an axisymmetric problem of floating zone melting of a silicon cylinder 5–10 cm in radius. The boundary condition at infinity has been brought to some semicircle connecting the feed rod and protective shield and located at a sufficiently large distance from the coil, which has made it possible the problem to be treated in a finite region. This region has been conformally mapped onto a rectangle, in which the problem of determining the magnetic field vector potential has been solved. The problem reduces to solving Laplace’s equation for the only nonzero component of the vector potential $${A_\varphi }$$ , where $$\varphi $$ is the azimuthal angle, with first- or second-order boundary conditions on the rectangle sides. The proposed method can be used to evaluate the variable thickness and shape of the liquid film adjacent to the lower part of the feed rod and the hydrodynamic flow in the film.