The polarizability and corresponding dielectric relaxation of the Debye-Huckel (DH) atmosphere surrounding a charged rod-like polyelectrolyte immersed in an ionic solution of a symmetrical electrolyte is determined following the method developed by J. A. Fornes [Phys. Rev. E 57, 2110(1998)], Several formulas are given to estimate the DH atmosphere parameters, namely the polarizability at zero frequency, alpha(0), the relaxation time, tau, the cloud capacitance, C, the average displacement of the ionic cloud, delta, the square root dipole moment quadratic fluctuation, [p(2)](1/2), and the thermal fluctuating field, [E-2](1/2). The Poisson-Boltzmann equation is solved numerically to apply the theory to a highly charged polyelectrolyte such as DNA in solution, although formulas valid for the DH approximation are also given. A dispersion in the polarizability and correspondingly in the dielectric constant of these solutions in the microwave region is predicted. For instance, considering a DNA length of 1000 Angstrom, with its reduced linear charge density xi(0) = 4.25 and ionization factor gamma = 0.5, immersed in a NaCl solution (40 mM), we predict a polarizability of the DH atmosphere at zero frequency alpha(0) of 1x10(-33) Fm-2 (similar or equal to 6.1 x 10(6)) times greater than the mean value of the polarizability of water) and the corresponding fluctuating dipole moment p of 2.1 x 10(-27) Cm (similar or equal to 600 times greater than the permanent dipole moment of water molecule). The relaxation time and the average displacement of the ionic cloud are tau = 1.6 ns and delta=14. Angstrom, respectively. This displacement is produced by the thermal fluctuating field, which, in this case, at room temperature is [E-2](1/2) = 2 x 10(6) V/m,