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Physical bounds on the directivity Q-factor quotient and optimal current distributions are determined for antennas of arbitrary shape and size using an optimization formulation. A variational approach offers closed form solutions for small antennas expressed in the polarizability of the antenna structure. Finite sized antennas are solved using Lagrangian parameters in a method of moments formulation. It is also shown that the optimal charge density for a small antenna can be generated by several current densities. Numerical examples for small and large antennas are used to illustrate the results.