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We propose a rigorous model of the electromagnetic field in a conductive circular cylinder illuminated by an arbitrarily shaped and oriented current-carrying coil placed nearby for nondestructive evaluation at eddy-current frequencies. The model is based on a dyadic field formulation, and its modularity enables handy generalizations to a multilayered cylindrical structure that might be affected by various defects. We give a variety of numerical results about the primary electric field within the cylinder in both canonical and practical cases in order to illustrate pros and cons of the solution method. The results agree quite well with known analytical results (which are available only in highly symmetrical cases), and fairly well with results yielded at high computational cost by an industrial finite-element code in more general cases. We discuss in detail the conditions for reaching a good numerical accuracy (involving some suitable regularization) while preserving both generality and small computational burden of the resulting computer code.