A comparative study of modeling the magnetostatic field in a current-carrying plate containing an elliptic hole

The presence of crack-like defects can cause an uneven distribution of the electric current density in a cracked conductor. To investigate the perturbation of the magnetic field resulting from the disturbed electric current, computational modeling of the magnetostatics is attempted on an infinite conductive plate, which contains an elliptic hole and is subjected to uniform current flow at infinity. Both 2-D and 3-D analyses are considered in this study. The 2-D analysis requires certain crucial assumptions and the governing Maxwell's equations are solved analytically in elliptic coordinates. The 3-D numerical computation is based on superposition of the elementary solution, whose derivation utilizes the Biot-Savart law. To improve the efficiency of the 3-D calculation, an adaptive mesh refinement algorithm is implemented in the numerical discretization. Finally, through a comparative study, the validity of the introduced simplifications in the 2-D analysis is benchmarked with the 3-D computational results. The present study shows that the 2-D solution predicts the upper bound for the out-of-plane component of the magnetic field perturbed by the elliptical hole, whose semi-major axis does not exceed ten times the thickness of the plate.