EMFIE and MEFIE Formulations for Numerical Solution of Single or Composities Bodies of Revolution Úrsula C. Resende, Fernando J. S. Moreira and Odilon M. C. Pereira-Filho Dept. Electronics Engineering, Federal University of Minas Gerais Av. Pres. Antonio Carlos 6627, Belo Horizonte, MG, CEP 31270-901, Brazil Abstract  In this paper, we present two new accurate formulations, EMFIE (electric-magnetic field integral equation) and MEFIE (magnetic-electric field integral equation), for the solution of the electromagnetic scattering by dielectric and composites bodies of revolution. The new formulations are genereted by the stantard electric and magnetic field surfaces integral equations. We also report numerical results are comparated with analitical solution to demostrate the accuracy and capability of the proposed formulation. Key-Words  Electric and magnetic field integral equations, eletromgnetic sacattering by bodies of revolution, Method of moments. I. INTRODUCTION Electromagnetic scattering from conducting, dielectric and composite bodies is an important and challenging problem in the field of computacionnal electromagnetics. Analitical solutions are available for only very limited geometries such as spheres, discs, cylinders, etc. For bodies having an arbitrary shape, one has to resort to some approximate numerical technique. A variety of approaches have been developed to study this propblem, which include the method of momentos (MoM), the finite element method (FEM), and the finite-difference time-domain (FDTD) method. When the bodies are homogeneous, MoM is perefed because the problem can be formulated in terms of surface integrals over the conducting and dielectric surfaces [1]-[3]. In special for bodies of revolution (BOR´s) the problem is formulated in terms of integrals over generatrixes [4]-[6]. For perfectly conducting (PEC) BOR´s the problem has been exhaustively studied and the most acurrate formulations are EFIE (electric field integral equation) and CFIE (combined field integral equation that is a linear combination of EFIE and MFIE - magnetic field integral equation) for open and closed conductiong bodies, respectively [5]. For dielectric BOR´s, many combinations of EFIE and MFIE have been investigated [6]-[8]. Some of these are the EFIE, MFIE, CFIE, PMCHWT, and Müller integral equations [6]-[8]. Such solutions (or proper combinations of them) have also been used in the analysis of scattering by Úrsula C. Resende, resendeursula@gmail.com, Fernando J. S. Moreira, fernandomoreira@ufmg.br, Odilon M. C. Pereira-Filho, odilon@cpdee.ufmg .br, Tel +55-31-34994848. This work was partially supported by CNPq, Brazil, under Covenant 471750/2004-2. Fig. 1. Generic axially bodies of revolution. a) Dielectric body. b) Coated PEC body. c) Coated dielectric body. d) Body with dielectric and conducting regions e) Body with two dielectric regions composite BOR’s [8]-[12]. For bodies with regions of different materials, as them show in the Fig. 1 d) and 1 e), the most used. formulation is PMCHWT [12]. For layered bodies, as them show in the Fig. 1 b) and 1 c), many formulations have been evalueted [8],[9]. The reported investigations are generally conducted for electrically small BOR’s (i.e., with dimensions of the order of the wavelength) [6]-[12]. In the present work we present two new accurate formulations for solution of the electromagnetic scattering by dielectric and composites bodies, as those shown in Fig. 1. For that, the plane-wave scattering from dielectric and composite spheres of different electrical sizes and different relative permittivity (ε r ) values are adopted as study case and the results are compared with analytical data, Mie-series [13]. The study shows that the new formulations are accurate for scattering analysis by different type of BOR´s. II. PROBLEM FORMULATION For a homogeneous body, as that shown in Fig. 1, with permittivity ε 1 and permeability µ 1 , immersed in an infinite e) d) µ 0 ,ε 0 µ 1 ,ε 1 a) µ 1 ,ε 1 µ 1 ,ε 1 µ 0 ,ε 0 µ 0 ,ε 0 b) c) µ 2 ,ε 2 σ→∞ µ 0 ,ε 0 µ 0 ,ε 0 µ 1 ,ε 1 σ→∞ µ 2 ,ε 2 µ 1 ,ε 1