Cyber-physical security of power systems under power injection attacks that alter generation and loads is studied. The system operator employs Phasor Measurement Units (PMUs) for detecting such attacks, while attackers devise attacks that are unobservable by such PMU networks. For the NP-hard problem of finding the sparsest unobservable attacks, it is shown that the solution has a simple form with probability one, namely, min (κ(GM),M) + 1, where κ(GM) is the vertex connectivity of an augmented graph, and M is the number of PMUs. The constructive proof allows one to find the entire set of the sparsest unobservable attacks in polynomial time. Furthermore, the geometric interpretation of unobservable attacks leads to a natural characterization of their potential impacts. With optimized PMU deployment, the sparsest unobservable attacks and their potential impact as functions of the number of PMUs are evaluated numerically for IEEE 30, 57, 118, 300-bus systems and Polish 2383, 2737, 3012-bus systems. It is observed that, as more PMUs are added, the maximum potential impact among all the sparsest unobservable attacks drops quickly until it reaches the minimum sparsity.