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In this paper, we look at the characterization of the far-field regions of antennas located over a ground plane. For antennas radiating in free space, the far field starts at a distance 2L2/lambda, where L is' the effective size of the antenna, and lambda is the wavelength. The question now is if this same radiating antenna is placed at a height H over a ground plane, then where does the far field of that antenna start? The goal of this paper is to demonstrate that for antennas radiating with either polarizations located over a ground plane, the far field starts at a distance 2H2/lambda (here, H ≫ L). We illustrate the validity of some rules of thumb through numerical simulations, and by using the definition of the far field as the region where the radial component of the field is negligible compared to the other components, and for regions where the ratio of the electric field to the magnetic field is characterized by eta, the characteristic impedance of free space. We also look at the validity of this rule of thumb when antennas are located over an imperfect ground plane. Finally, we examine the phenomenon of height-gain in wireless cellular communications. We illustrate that under the current operating scenarios, where the base-station antennas are deployed over a tall tower, the field strength actually decreases with the height of the antenna over a realistic ground, and there is no height gain. Therefore, to obtain a scientifically meaningful operational environment, the vertically polarized base-station antennas should be deployed closer to the ground. When deploying antennas over tall towers, it may be more advantageous to use horizontally polarized antennas than vertically polarized antennas for communication in cellular environments. Numerical examples are presented to illustrate these cases.