This paper addresses the variation of the attenuation characteristics of the Specific Absorption Rate (SAR) in a lossy medium as a function of the distance between an antenna and the medium with different EM-source sizes. Analysis and measurements were performed using a dipole antenna at 900 MHz and a COST244 cubic phantom. From this, an empirical equation has been derived, representing the attenuation characteristics of the SAR. The equation takes into consideration an energy loss due to the spatial spread of electromagnetic waves. In the case where an antenna is placed more than λ/2π away from the medium, the attenuation characteristics of the SAR are those obtained from plane waves in the lossy medium. In the case where a half-wavelength dipole antenna is located close to the medium, at a distance of less than λ/2π, the attenuation characteristics of the SAR are calculated from an equation that includes a loss caused by the spread of energy as a cylindrical wave. Moreover, when the length of antenna is short, it is found that a spatial attenuation factor appropriate to a spherical wave should be taken into account.