A method is presented by which series solutions for the impedance change in an eddy‐current test probe due to closed cracks in a non‐magnetic, conducting half‐space can be derived at low frequency. The series solution is applicable for flaws whose dimensions are much smaller than the electromagnetic skin‐depth. The problem is formulated using an approach in which the flaw is represented by an equivalent distribution of current dipoles. The electric field scattered by the flaw is then written as an integral, over the flaw, of the product of the dipole density distribution and an appropriate Green’s function. Terms in the series expansion for the dipole density are calculated by solving the integral equation at each order in the chosen small parameter, using perturbation theory and a dual integral equation method. The impedance change due to the crack is then calculated from the dipole distribution using the reciprocity theorem. Example solutions are given for semi‐circular surface‐breaking cracks and for long, uniformly deep surface‐breaking cracks. Results are compared with other analytical solutions and the predictions of an independent numerical scheme, and very good agreement is observed.