  For designing high-field electromagnets, the Lorentz force on coils must be
computed to ensure a support structure is feasible, and the inductance should
be computed to evaluate the stored energy. Also, the magnetic field and its
variation inside the conductor is of interest for computing stress and strain,
and due to superconducting quench limits. For these force, inductance, energy,
and internal field calculations, the coils cannot be naively approximated as
infinitesimally thin filaments due to divergences when the source and
evaluation points coincide, so more computationally demanding calculations are
usually required, resolving the finite cross-section of the conductors. Here,
we present a new alternative method that enables the internal magnetic field
vector, self-force, and self-inductance to be computed rapidly and accurately
within a 1D filament model. The method is applicable to coils for which the
curve center-line can have general noncircular shape, as long as the conductor
width is small compared to the radius of curvature. This paper extends a
previous calculation for circular-cross-section conductors [Hurwitz et al,
arXiv:2310.09313 (2023)] to consider the case of rectangular cross-section. The
reduced model is derived by rigorous analysis of the singularity, regularizing
the filament integrals such that they match the true high-dimensional integrals
at high coil aspect ratio. The new filament model exactly recovers analytic
results for a circular coil, and is shown to accurately reproduce full
finite-cross-section calculations for a non-planar coil of a stellarator
magnetic fusion device. Due to the efficiency of the model here, it is well
suited for use inside design optimization.
