Solution of quasi-static axisymmetric problem for a single-layer quasi-force-free magnet

A basic model of a cylindrical quasi-force-free solenoid for the generation of strong and superstrong magnetic fields was considered in this paper for the resulting stress-strain state under the influence of current and magnetic induction. For a single-layer model, the parameterization problem satisfying the local Maxwell equations for the magnetic quantities in poloidal and toroidal directions has been solved. Analytical minimization problem of the integral radial component of the Lorentz mechanical force through conductor thickness was performed by taking into account the introduced parameterization of the system. In the formulation of linear elasticity theory, it was analytically considered the quasi-static axisymmetric problem of mechanics of a deformable solid under the action of a volumetric radial force. Taking into account the introduced parameterization, it was found the optimal induction ratio in the conductor from the point of view of minimizing the maximum value of the von Mises stress normalized to magnetic pressure mechanical stress.