The article shows that in a magnetic system with a thin-walled balanced winding close to a force-free one, a significant increase in the parameter theta=W_Mγ/Msigma_M, is possible, which, according to the virial theorem, characterizes the ratio of the energy of the magnetic system W_M to the weight of equipment with a material density γ, where under the action of electromagnetic forces there appears a mechanical stress sigma_M. In a quasi-force-free magnetic system, the main part of the winding is in a state of local equilibrium, and only a relatively small part of the equipment is subject to stress. This part determines the weight of the entire system, and this weight can be minimized. The configurations of balanced thin-walled windings are developed, at the boundaries two boundary conditions are fulfilled simultaneously - the absence of the induction component normal to the boundary and the constancy of the product of induction and radius. The authors consider an example of a system consisting of a main part - a sequence of balanced "transverse" modules in the form of flat discs and end parts, consisting of a combination of "transverse" modules and "longitudinal" ones, having the form of rings elongated along the axis with balanced end parts. It is shown that in the system under consideration, the characteristic dimensionless parameter theta with an unlimited increase in the number of elements of the main part can reach a value of about 24, and when the number of these elements changes within 20 -40, it changes from 6 to 9. Keywords: Quasi-force-free magnetic field, field energy, mass of a magnetic system, virial theorem, minimization of the ratio of mass to energy.

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