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Consider the figures I and II

, they consist of similar circuits with identical batteries. There are two different slabs connected in the circuit, both are geometrically identical and have the same conductivity $\sigma_1$, but have different dielectric constants $k_1$ and $k_2$ respectively, will yhe electric field inside these slabs be the same or different as a current runs through them?

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My idea: We know that $$J = \sigma E$$ But i think that this is considering that there are know polarisation effects by the material in which J exists. Now in linear homogeneous material, electric field gets reduced by a factor of k , we call this dielectric constant, so in this case also it should become $$J/ \sigma k$$ Hence we conclude that the electric fields must be different. I dont know if this is correct or not, can someone please verify or correct my concept

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