Why is polarization in dielectric material linearly proportional to the E-field. T.y.
Do you mean why is proportional or why there is an epsilon there?
That's an approximation in the case of small values of E. Even in the linear case, the polarization does not only depend on the electric field at the same time and co-ordinate but in principle on all previous times and other locations. One speaks of temporal and spatial dispersion as these effects are responsible for the refractive index to be a function of frequency.
The regime of strong fields where non-linear terms in E become important is called non-linear optics.
Ok, but say we have some idealized isotropic medium can you explain why dipole moments induced will be proportional to the electric field if it small, please?
Leaving aside some special materials like ferroelectrics, the induced dipole moment will be a smooth function of E and you can use a Taylor expansion in E and keep only the first term.
In quantum mechanics this corresponds to using only 1st order pertrubation theory for the calculation of the dipole moment.
I don't know of any theory that says it is proportional.....what we observe is what DrDu has posted....its an approximation....and in fact the exact relationship is non linear.
If you are looking for a conceptual explanation, here you go. Approximate a dielectric material as a large collection of tiny spheres packed together. Assume each sphere is small enough that the applied electric field across its interior is constant. Then each sphere responds independently to the distinct applied electric field its feels at its center. It is a fairly straightforward calculation to show in general that a dielectric sphere placed in a previously uniform electric field will respond as if it were a perfect electric dipole at its center pointing in the direction of the applied field, i.e. the material response field is the same as that due to a aligned dipole. Finding the total material response field of an extended object then just involves adding up (integrating) all the induced dipole fields of all the little spheres. The material's response field therefore ends up being essentially the polarization of the material, which by definition is the average dipole density.
Another way to learn when the linear approximation is sufficient would be to compare the typical field strength in an atom due to the nucleus with the external field strength of the applied field.
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