In this paper, we develop reduced models to approximate the solution of the electromagnetic scattering problem in an unbounded domain which contains a small perfectly conducting sphere. Our approach is based on the method of matched asymptotic expansions. This method consists in defining an approximate solution using multi-scale expansions over outer and inner fields related in a matching area. We make explicit the asymptotics up to the second order of approximation for the inner expansion and up to the fifth order for the outer expansion. We validate the results with numerical experiments which illustrate theoretical orders of convergence for the asymptotic models requiring negligible computational cost.