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As the voltage is applied at both sides of the dielectric material(ex. front surface +5V and rear surface 0V), the energy is stored in the material, right?

I heard it's because the dielectric molecules slightly rotate due to the electric field. If so, does the dielectric constant of the material also differ from the one without the electric field(DC or AC)? Would the answer be valid for every dielectric material?

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We do not know what the permittivity is when it is not in an electric field, after all, it is an electric phenomenon. In practice, the permittivity of most material is nearly constant over a very large range of bias but if the bias voltage is large enough then it will surely change; after all, it can break down, burn, etc. Electric circuits are designed to operate to take this into account and there are limits beyond which the permittivity can change. In fact, there are nonlinear circuits that are based on the intentional variability of permittivity in certain materials but commercial components called capacitors are supposed be essentially independent of the bias voltage as long as one observes those limits.

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