This paper deals with an interaction problem between a solid and an electromagnetic field in the frequency domain. More precisely, we aim to compute both the magnetic component of the scattered wave and the elastic vibrations that take place in the solid elastic body. To this end, we solve a transmission problem holding between the bounded domain $\Omega_S \subset R^3$ representing the obstacle and a sufficiently large annular region surrounding it. We point out here that (following Voigt's model, cf. [12]) we only allow the electromagnetic field to interact with the elastic body through the boundary of $\Omega_S$. We apply the abstract framework developed in the work [3] by A. Buffa to prove that our coupled variational formulation is well posed. We define the corresponding discrete scheme by using the edge element in the electromagnetic domain and standard Lagrange finite elements in the solid domain. Then we show that the resulting Galerkin scheme is uniquely solvable, convergent and we derive optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments.