By adopting the usual cold plasma dielectric tensor, it is demonstrated that a rapidly oscillating electric field gives rise to slow time scale drifts, which cause density modifications near antennas. In the presence of a strong magnetic field, the poloidal gradients of the field are at the origin of radial displacements of the plasma while radial field gradients have the potential to trigger density inhomogeneity along the antenna. The radio frequency-induced plasma drifts are more prominent at higher power and for more evanescent modes. It is discussed that the usual cold plasma dielectric tensor is derived neglecting nonlinear effects and zero-order drifts, and therefore does not uniformly allow the capture of the wave–particle interaction near the antenna self-consistently, necessitating a more detailed description to capture both wave and particle effects on the one hand, and global wave propagation and local sheath effects, on the other. A strategy is proposed to complement the model with other needed ingredients enabling one to capture the dynamics on the fast and slow time scales.