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The effect of surrounding an electrically small dipole antenna with a shell of double negative (DNG) material (εr<0 and μr<0) has been investigated both analytically and numerically. The problem of an infinitesimal electric dipole embedded in a homogeneous DNG medium is treated; its analytical solution shows that this electrically small antenna acts inductively rather than capacitively as it would in free space. It is then shown that a properly designed dipole-DNG shell combination increases the real power radiated by more than an order of magnitude over the corresponding free space case. The reactance of the antenna is shown to have a corresponding decrease. Analysis of the reactive power within this dipole-DNG shell system indicates that the DNG shell acts as a natural matching network for the dipole. An equivalent circuit model is introduced that confirms this explanation. Several cases are presented to illustrate these results. The difficult problem of interpreting the energy stored in this dipole-DNG shell system when the DNG medium is frequency independent and, hence, of calculating the radiation Q is discussed from several points of view.