A prevalent theory circulating among the non-scientific community is that the intensive deployment of base stations over the territory significantly increases the level of electromagnetic field (EMF) exposure and affects population health. To alleviate this concern, in this work, we propose a network architecture that introduces tethered unmanned aerial vehicles (TUAVs) carrying green antennas to minimize the EMF exposure while guaranteeing a high data rate for users. In particular, each TUAV can attach itself to one of the possible ground stations at the top of some buildings. The location of the TUAVs, transmit power of user equipment and association policy are optimized to minimize the EMF exposure. Unfortunately, the problem turns out to be a mixed integer non-linear programming (MINLP), which is non-deterministic polynomial-time (NP) hard. We propose an efficient low-complexity algorithm composed of three submodules. Firstly, we propose an algorithm based on the greedy principle to determine the optimal association matrix between the users and base stations. Then, we offer two approaches, modified K-mean and shrink and realign (SR) process, to associate each TUAV with a ground station. Finally, we put forward two algorithms based on the golden search and SR process to adjust the TUAV's position within the hovering area over the building. After that, we consider the dual problem that maximizes the sum rate while keeping the exposure below a predefined value, such as the level enforced by the regulation. Next, we perform extensive simulations to show the effectiveness of the proposed TUAVs to reduce the exposure compared to various architectures. Eventually, we show that TUAVs with green antennas can effectively mitigate the EMF exposure by more than 20% compared to fixed green small cells while achieving a higher data rate.