Asymptotic analysis of the natural system modes of coupled bodies in the large-separation low-frequency regime

We examine the natural system modes (characteristic frequencies and currents) of two coupled bodies in the limit of large separation. It is known that when objects are oriented such that they may interact electromagnetically, natural modes of the coupled system occur. These modes differ from, but may be related to, the natural modes of the isolated bodies. For example, the first antisymmetric and symmetric system frequencies of two identical bodies separated by some intermediate distance spiral around the dominant natural frequency of the isolated body as separation is varied. As separation further increases, these system resonances tend toward the origin in the complex frequency plane, rather than approaching the isolated body-dominant natural frequency. Here we treat an N-body scattering problem in the limit of large separation by replacing the bodies with equivalent dipole moments. The natural frequencies are obtained as singular points in the scattering solution. For the special case of two coupled objects, a simple equation for the natural system frequencies is obtained that shows that the real radian-system frequency approaches the origin as 1/r, independent of the relative orientation and type of the two bodies. The damping coefficient approaches the origin approximately logarithmically as a function of the body orientation and type. Using this formulation, the natural system modes of two coupled wires are investigated for large separation between the wires and compared to an integral equation solution.