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Exercise N 2. Force of interaction of two cylindrical coaxial coils
Task
Find the force applied to the coils with current with and without the shield between them.
Experiment
Forces are measured by use of digital balance on which one of coils is installed (see the right picture below). Tests are performed with currents varying, and different conductors and ferromagnetics are placed between coils.
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 data
Current density in the coil j=100000 A/m^{2}.
Relative magnetic permeability of air, aluminum and copper coils μ= 1.
Relative magnetic permeability of the steel shield μ= 1000.
Electrical conductivity of steel σ= 10000000 Sm/m.
Electrical conductivity of aluminum σ=37000000 Sm/m.
Coils are wound by insulated wire, so crosssection conductivity in coils σ= 0 Sm/m.
Boundary conditions
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 coils:
 Timeharmonic magnetic field, f=50 Hz.
Shield type 
Interaction force, mN 
Problem 
No shield 
0.533 
lab2.pbm 
Steel 
0.098 
lab2_Fe.pbm 
Aluminum 
0.282 
lab2_Al.pbm 
 DC magnetic field
Shield type 
Interaction force, mN 
Problem 
No shield 
0.533 
lab2c.pbm 
Steel 
0.138 
lab2c_Fe.pbm 
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