Cracks in conductors are detected through changes in the impedance of a coil that induces current in the material. In order to gain insight into the physics of the inspection, we have developed a theoretical and computational model that predicts the signals due to cracks in circular cylindrical holes using a boundary element calculation. In formulating the problem, the electromagnetic field is decomposed into transverse electric and transverse magnetic scalar modes. The effect of a planar crack in an electromagnetic field is represented by an electric current dipole layer orientated normal to the crack surface. The dipole density is determined by the integral equation whose dyadic kernel ensures that the tangential electric and magnetic fields are continuous at the surface of the hole. Instead of solving this equation, a numerical approximation is found in the form of a discrete system of linear algebraic equations formed using either boundary and volume elements depending respectively, on whether the crack opening is negligible or not. Because the kernel embodies the interface conditions at the surface of the hole, a discrete approximation of the field is only necessary in the flaw domain which means that relatively few unknowns are needed. The probe impedance variation has been computed for both ideal cracks, defined as having negligible opening but impenetrable to current, and open cracks/slots. Open crack model predictions of coil impedance variations with position relative to a semi-elliptical axial crack are in good agreement with measurements.