Physical attacks, which aim to render one or more sensor nodes non-operational by physically capturing and destroying them, are among the most serious security threats in Wireless Sensor Networks (WSNs). In case of dense deployment (a desirable property in the design of WSNs) multiple sensor nodes acquire redundant (highly correlated) data. As a result, even if some nodes are dead, the remaining nodes can successfully complete the sensing task. However, as the number of nodes in the network decreases, the remaining nodes are burdened with the extra load (energy dissipation). In this study, we investigate the energy cost of survivability in the presence of physical attacks through a novel Linear Programming framework. We explore the energy dissipation characteristics of the network for different physical attack scenarios.