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The problem of an infinitely long annular cylinder whose inner and outer surfaces are subjected to known surrounding temperatures and are traction-free is considered in the presence of an axial uniform magnetic field. The problem is in the context of generalized magneto-thermoelasticity theory with one relaxation time. The Laplace transform with respect to time is used. A numerical method based on a Fourier-series expansion is used for the inversion process.Numerical computations for the temperature, displacement and stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically.