The conventional integral equation governing the electric field inside dielectric particles is reformed to bridge and to provide mathematical foundations for analytic techniques widely used to estimate such a field. The solution of the reformed equation inside a dielectric slab explained how inner-field formulations based on the Rayleigh, the Rayleigh-Gans, the quasi-static, and the Shifrin approximations can be supported by the particles. It also confirmed the approach employed to reform the integral equation. The analysis performed uncovered the differences between the depolarization tensor characterizing electrostatic fields inside the particles and the source dyadic resulting from the extraction of the singularity of the integral equation kernel.
