Calculation of the vector potential of the magnetic field in a non-conducting medium during crucible-free zone melting

The vector potential of the magnetic field is determined in the region between the inductor, the upper part of the floating zone, the liquid film, the feed rod, and the protective screen in an axisymmetric problem of crucible-free zone melting of a silicon sample with a radius of 5 $\div$ 10 cm. The boundary condition at infinity is transferred to a semicircular arc connecting the feed rod to the protective screen and located at a sufficiently large distance from the inductor, which allows the problem to be considered in a finite domain. This domain is conformally mapped onto a rectangle, within which the problem of determining the magnetic field vector potential is solved. The problem is reduced to solving Laplace's equation for the only non-zero component of the vector potential, A$_{\varphi}$, where $\varphi$ is the polar angle, with first- or second-type boundary conditions on the edges of the rectangle. The method can be applied to calculate the variable thickness and shape of the liquid film adjacent to the lower part of the feed rod, as well as the hydrodynamic flow within it.