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Cylindrical coaxial coils interaction force

magnetic force between coaxial coils, electromagnetic FEA finite element analysis of cylindrical coils

In the lab the forces are measured by electrotic balance on which one of the coils is installed. Tests are performed with currents varying, and different conductors and ferromagnetics are placed between coils.

Problem type
Plane-parallel problem of DC magnetics, AC magnetics.

Geometry
Due to symmetry of the formulation only upper half of the problem (above ab line) is defined, and at the axis of symmetry (line ab) the boundary conditions are set.

Given
Current density in the coil j = 0.1 A/mm².
Relative magnetic permeability of the steel disc μ= 1000.
Electrical conductivity of steel disc σ= 10 MS/m.
Electrical conductivity of aluminum disc σ=37 MS/m.
Coils are wound by insulated wire, so cross-section conductivity in coils σ= 0 S/m.

Task
Find the force between the coils carring current with and without the shield between them.

Solution
Along the horizontal symmetry axis (line ab) B_n =0.
Equation B=rot A in cylindrical coordinate system leads to A=const at the axis ab. The field fades at the infinity, so due to the condition of continuity of the potential A=0 at the line ab.

Results:
Field distribution around the coils: