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In this paper, we are concerned with the gradient estimate of the electricfield due to two nearly touching dielectric inclusions, which is a centraltopic in the theory of composite materials. We derive accurate quantitativecharacterisations of the gradient fields in the transverse electromagnetic casewithin the quasi-static regime, which clearly indicate the optimal blowup rateor non-blowup of the gradient fields in different scenarios. There are mainlytwo novelties of our study. First, the sizes of the two material inclusions maybe of different scales. Second, we consider our study in the quasi-staticregime, whereas most of the existing studies are concerned with the staticcase.

Gradient estimates for electric fields with multi-scale inclusions in the quasi-static regime