We have computed the impedance variations of coils due to cracks in bore holes using a boundary element method. The problem is formulated using a scalar analysis of the electromagnetic field into transverse electric and transverse magnetic modes. A formal solution of the scalar equations gives the potentials in integral form with scalar kernels. The kernels take account of cross-coupling between the modes that is a consequence of ensuring the continuity of the tangential electric and magnetic fields at the surface of the hole. A dyadic kernel, formed using the scalar Green's kernels, incorporates the field continuity conditions at the surface of the hole. As a results, the flaw field is simply expressed as an integral over the flaw region. Following an earlier calculation for planar structures, the effect of a crack is represented by a current dipole layer. An electric integral equation determines the dipole density at the crack but instead of solving the equation directly, a numerical approximation is found from a system of linear simultaneous equations derived using a boundary elements approximation. The probe impedance variation can be computed for infinitesimally thin cracks and for slots with a small opening. Numerical predictions of the coil impedance due to the crack have been compared with results from finite element calculations showing close agreement.