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Abstract Romashets and Vandas (2024) derived a method for the determination of Euler potentials at a spherical surface and applied it to the geomagnetic field. Here, we apply it to find Euler potentials at the source surface. A regular mesh defined by Euler potentials divides the source surface to surface elements with the same magnetic flux. By tracing magnetic-field lines away from the source surface, Euler potentials can be extended into the heliosphere. 

Abstract Finding the magnetic flux mapping in the ionosphere is very important. It would not only divide the surface into the elements with the same flux, but also indicate locations of conjugated points. It is important for studies of field aligned currents and bouncing of energetic charged particles and their precipitation. The existing methods involve numerical magnetic field lines tracing in the entire volume of the magnetosphere or numerical integration along assumed contour lines of the Euler potentials on the surface of the ionosphere. It is possible to determine the mapping with these methods near the magnetic equator, but not on middle latitudes and near and inside the polar caps. Our approach is to search for the Euler potentials as a sum of basic functions with their coefficients. Each basic function is a product of a sine or cosine of longitude multiplied bymand the Legendre polynomial of the colatitude angle cosine and of the ordern. Maxima ofmandnin this calculation were set to 13. The difference between the radial component from the cross product of the Euler potentials gradients and from International Geomagnetic Reference Field is less than 0.01 percent. We discuss the possibility of using orthogonal coordinates defined on the sphere's surface, which remain finite functions ofθandφeverywhere except for the vicinities of the North and South poles. The issues with numerical errors accumulated on long tracing are avoided when using this approach. 

The charge distribution within a hollow conducting cylinder with zero-thickness walls is calculated from the minimum potential energy (𝑈) consideration. The surface charge density consists of a diverging term (Jackson, 1975) and a sum of Legendre polynomials with the coefficients determined from the minimum 𝑈 approach. The sum converges. This allows to express the capacitance in closed form. It is in agreement with Butler (1980). We present electric field lines inside and outside of the cylinder. An electric field pattern can be studied in detail. Most of the numerical analysis is done for the conducting cylinder of the length equal to ten radii. The surface charge density near the edges diverges; and in the middle, it is twenty five percent less than that of a uniformly distributed charge. The self-energy of the conducting cylinder is about 5 percent lower than that of uniformly distributed surface charge.