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Capture effect has been shown as a physical layer (PHY) phenomenon of modern wireless devices that improves the performance of wireless local area networks (WLANs) in terms of throughput. In this paper, however, we explore the effect of PHY capture in the domain of energy efficiency. Analysis model that takes into account the effect of PHY capture is backed up by ns-2 simulations show that capture effect improves energy efficiency of WLAN by 20%. This improvement, however, results in unfairness, i.e, a group of nodes located far away from the Access Point (AP) is three times less energy efficient than the group of nodes located closer to the AP. To resolve the unfairness caused by the capture effect, furthermore, this paper proposes a joint strategy of adaptive transmission power control (ATXPR) and contention window adjustment (CWADJ). Namely, a node that suffers transmission failure due to another node capturing the channel steps up its transmission power according to the transmission power control algorithm and refrains from increasing its contention window according to contention window adjustment mechanism, respectively. Our proposed joint strategy is 99% fair while maintaining overall energy efficiency of the network.