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Proximity effect

In the lab the current density is defined by measuring the voltages on the conductor segments, the phase of current is measured separately by the digital phasemeter.

Problem type
Plane-parallel problem of time-harmonic electromagnetic field.

Geometry
Copper rods Eddy currents in the copper rods Copper Air b O a D30 128

Steel tubes eddy currents in the steel tubes Steel Air b O a e f D84 d70 250

Given
Relative magnetic permeability of copper and air μ= 1.
Relative magnetic permeability of steel μ= 100.
Total current I=300 A. Frequency f=50 Hz.
Electric conductivity of copper σ=57000000 S/m.
Electric conductivity of steel σ=10000000 S/m.

Task
Find the current density distribution along the cross section of long parallel conductors. Two types of conductors are analyzed: two copper rods and two steel tubes.

Solution
Due to problem symmetry only upper-right quarter aOb is defined, and at the axes of symmetry the boundary conditions are set.
At the horizontal axis of symmetry (line Oa) Ht=0. At the vertical axis of symmetry Ob Bn=0. Equation B=rot A in the cylindrical coordinate system leads to A=const at the axis Ob. Field fades at the infinity so at the other boundaries A=0.

Results:
Current density distribution along the line Oa for copper rods.
Current density distribution for copper rods

Current density distribution along the steel tube perimeter (from the point e to point f clockwise).
Current density distribution along the steel tube perimeter

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