## 1. Introduction

[2] The conventional discretization in method of moments (MOM) of the magnetic field integral equation (MFIE) shows in general huge inaccuracies in the computed radar cross section (RCS) at very low frequencies [*Zhang et al.*, 2003]. As shown by *Zhang et al.* [2003], these huge RCS inaccuracies come from the nonnull radiating nonsolenoidal static component of the current arising in the MFIE. The irruption of these huge inaccuracies in the computed RCS depends ultimately on the degree of cancellation of the contributions from the nonnull nonsolenoidal static current in the far-field computation. For a curved object like a sphere, the error appears for extremely low electrical dimensions, below 10^{−15}*λ* [*Zhang et al.*, 2003]. However, this inaccuracy in the computed RCS for a sharp-edged object is first observed at electrical dimensions around 10^{−3}*λ*.

[3] *Zhang et al.* [2003] propose an implementation of the MFIE free from the observed inaccuracy at very low frequencies by discarding the static nonnull nonsolenoidal contribution of the current in the computed RCS after the iterative computation of the solenoidal and nonsolenoidal components of the current through the *jk*-Taylor's expansion around *k* = 0, the static limit. The whole procedure becomes rather time consuming because it is required the inversion of the resulting impedance matrix at each iterative step, although it is true that in the very low frequency regime few terms are required.

[4] In section 2, we discuss with examples how the appearance of the error in the computed RCS with a conventional discretization of the MFIE for sharp-edged objects can be shifted to much smaller electrical dimensions with the adoption of uniform meshes. We justify that in this case the overall cancellation in the computed far-field from the nonsolenoidal static contributions is much better reached than with a nonuniform discretization. This represents a simple strategy to extend the validity of the conventional Rao-Wilton-Glisson (RWG) discretization of the MFIE deeper inside of the Rayleigh frequency region, down to the extremely low frequency regime, with no need to implement other more complicated and time-consuming schemes. This strategy, though, can only be applied to objects that are amenable to uniform discretization.

[5] We then provide two other approaches relying on the MOM discretization of integral equations of the second kind that become free from this observed inaccuracy and lead to the right *λ*^{−4}-asymptotic RCS scaling deep inside the Rayleigh frequency region for any object under analysis:

[6] In section 3, we present a modification of the MOM-MFIE implementation at very low frequencies provided by *Zhang et al.* [2003]. We propose to simplify notably the process described by *Zhang et al.* [2003] in two steps. We first compute the static component of the current, and second, the remaining contributions. This strategy requires two matrix inversions for a solution valid at any frequency.

[7] In section 4, we present a new integral equation, the electric magnetic field integral equation (EMFIE), which we implement under a MOM discretization with the loop-star basis functions [*Wu et al.*, 1995]. The EMFIE, just like the MFIE, represents a second-kind integral equation and inherits the advantageous properties of the matrices resulting from MOM-MFIE implementations regarding stable and good condition numbers. The loop-star discretization of the EMFIE provides a zero static nonsolenoidal component of the current which allows the *λ*^{−2}-asymptotic scaling of the real part of the nonsolenoidal current observed in the electric field integral equation (EFIE) at very low frequencies.