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We consider an asymptotic representation formula for scattered electromagnetic waves caused by penetrable scatterers of low volume contained in some homogeneous three-dimen- sional bounded domain. The only regularity assumption on the support of the scatterers is that it is measurable and well-separated from the boundary of the domain. Our analysis extends re- sults originally obtained in (Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction, Math. Model. Numer. Anal. 37 (2003), 159-173) for steady state voltage potentials to time-harmonic Maxwell's equations. 1. Introduction. In this work we study time-harmonic electromagnetic waves in a smoothly bounded domain filled with a homogeneous medium, which we call the background domain and accordingly the background medium. We assume that this domain contains a penetrable object, the scatterer, that is a subdomain on which the electromagnetic properties of the medium dier from that of the background medium. The only regularity assumptions on this subdomain are that it is measurable and well separated from the boundary of the background domain. We are interested in the asymptotic behavior of scattered electromagnetic waves as the volume of the scatterer tends to zero. Observing that away from the scatterer the scattered field decreases at the same rate as the volume of the scatterer, we derive an analytic expression for the leading order term in the corresponding asymptotic expansion. Due to the generality of this formula higher order terms or even their precise order in terms of the volume of the scatterer are not obtained. For time- harmonic Maxwell's equations such an expansion has so far only been studied for the special case of diametrically small scatterers by Ammari, Vogelius, and Volkov (7) (see also (23) for a corresponding result for perfectly conducting scatterers in an unbounded layered background medium). Our analysis extends the work of Capdeboscq and Vogelius (14), where a similar representation formula has been established for perturbations in electrostatic poten- tials due to low volume conductivity inhomogeneities (cf. (15-17) for further studies in this direction). This formula generalizes and unifies earlier results obtained for the special case of diametrically small conductivity inhomogeneities (see, e.g., Friedman and Vogelius (21), Cedio-Fengya, Moskow, and Vogelius (18), or Ammari and Kang (3)) as well as for conductivity inhomogeneities of small thickness (see, e.g., Beretta et al. (11, 12)). Similarly, the general formula for time-harmonic Maxwell's equations discussed in this article allows to recover the formula for diametrically small scatter- ers from (7) but also yields new asymptotic formulas for practically important cases like thin penetrable cracks or thin tubes. Furthermore, our results can be extended

A general perturbation formula for electromagnetic fields in presence of low volume scatterers