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Exercise N 3. Proximity effect

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.

Experiment
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
Linear plane-parallel problem of time-harmonic electromagnetic field.

Geometry
Due to problem symmetry only upper-right quarter aOb is defined, and at the axes of symmetry the boundary conditions are set.

Pic.1. Copper rods

Pic.2. Steel tubes

Given data
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 Sm/m.
Electric conductivity of steel σ=10000000 Sm/m.

Boundary conditions
At the horizontal axis of symmetry (line Oa) H_t=0. At the vertical axis of symmetry Ob B_n=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 along the steel tube perimeter (from the point e to point f clockwise).
  

Please, see problems lab3_Cu.zip and lab3_Fe.zip.