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The $T$-matrix, often obtained with Waterman's extended boundary conditionmethod (EBCM), is a widely-used tool for fast calculations of electromagneticscattering by particles. Here we investigate the quasistatic or long-wavelengthlimit of this approach, where it reduces to an electrostatics problem. We firstpresent a fully electrostatic version of the EBCM/T-matrix method (dubbedES-EBCM). Explicit expressions are then given to quantitatively express thelong-wavelength limit of the EBCM matrix elements in terms of those of theES-EBCM formalism. From this connection we deduce a number of symmetryproperties of the ES-EBCM matrices. We then investigate the matrix elements ofthe ES-EBCM formalism in the special case of prolate spheroids. Using thegeneral electrostatic solution in spheroidal coordinates, we derive fullyanalytic expressions (in the form of finite sums) for all matrix elements.Those can be used for example for studies of the convergence of the $T$-matrixformalism. We illustrate this by discussing the validity of the Rayleighhypothesis, where analytical expressions highlight clearly the link withanalytical continuation of series.

Electrostatic limit of the $T$-matrix for electromagnetic scattering: Exact results for spheroidal particles