The field resulting from the illumination by a localized time-harmonic low-frequency source (typically a magnetic dipole) of a voluminous lossy dielectric body placed in a lossy dielectric embedding is determined within the framework of the localized nonlinear approximation by means of a low-frequency Rayleigh analysis. It is sketched (1) how one derives a low-frequency series expansion in positive integral powers of (jk), where k is the embedding complex wavenumber, of the depolarization dyad that relates the background electric field to the total electric field inside the body; (2) how this expansion is used to determine the magnetic field resulting outside the body and how the corresponding series expansion of this field, up to the power 5 in (jk), follows once the series expansion of the incident electric field in the body volume is known up to the same power; and (3) how the needed nonzero coefficients of the depolarization dyad (up to the power 3 in (jk)) are obtained, for a general triaxial ellipsoid and after careful reduction for the geometrically degenerate geometries, with the help of the elliptical harmonic theory. Numerical results obtained by this hybrid low-frequency approach illustrate its capability to provide accurate magnetic fields at low computational cost, in particular, in comparison with a general purpose method-of-moments code. ¿ 2000 American Geophysical Union