This paper shows the applicability of the integral equation formulation of the measured equation of invariance (IE-MEI) to two-dimensional dielectric scatterers. That is, a relationship between the scattered electric and magnetic fields, which is derived from the new formulation of the IE-MEI, is applicable to lossless dielectric materials as well as perfect electric conductors (PEC). In addition, we show that the IE-MEI does not suffer from internal resonance problems. These two facts are validated by numerical examples for a circular cylinder and a square cylinder illuminated by Transverse Magnetic (TM) plane wave or a TM line source very close to the scatterers. The numerical results calculated by the IE-MEI agree well with the ones by moment methods that employ combined field formulations with exact boundary conditions.