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This is not the definition of the refractive index. Its definition is $n^2 = \epsilon_r$. The relative dielectric "constant" is in turn defined as $\epsilon_r = 1+ \chi_e$ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

In the above I have assumed an isotropic medium, at rest, with linear response and relative magnetic permeability $\mu=1$. This includes metals. Within these limitations the treatment is generally valid.

To answer your question, there is no fundamental argument against setting $n=1/\sqrt{\epsilon_r}$. It is the conventional definition, again with the limitations given.

This is not the definition of the refractive index. Its definition is $n^2 = \epsilon_r$. The relative dielectric "constant" is in turn defined as $\epsilon_r = 1+ \chi_e$ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

In the above I have assumed an isotropic medium, at rest, with linear response and relative magnetic permeability $\mu=1$. This includes metals. Within these limitations the treatment is generally valid. In the most general case the dielectric "constant" is a rank 2 lLorentz tensor where the elements depend on the frequency and on the direction of $\vec k$.

To answer your question, there is no fundamental argument against setting $n=1/\sqrt{\epsilon_r}$. It is the conventional definition, again with the limitations given.

This is not the definition of the refractive index. Its definition is $n^2 = \epsilon_r$. The relative dielectric "constant" is in turn defined as $\epsilon_r = 1+ \chi_e$ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

In the above I have assumed an isotropic medium, at rest, with linear response and relative magnetic permeability $\mu=1$. Within these limitations the treatment is generally valid.

To answer your question, there is no fundamental argument against setting $n=1/\sqrt{\epsilon_r}$. It is the conventional definition, again with the limitations given.

This is not the definition of the refractive index. Its definition is $n^2 = \epsilon_r$. The relative dielectric "constant" is in turn defined as $\epsilon_r = 1+ \chi_e$ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

In the above I have assumed an isotropic medium, at rest, with linear response and relative magnetic permeability $\mu=1$. This includes metals. Within these limitations the treatment is generally valid.

To answer your question, there is no fundamental argument against setting $n=1/\sqrt{\epsilon_r}$. It is the conventional definition, again with the limitations given.

This is not the definition of the refractive index. Its definition is $n^2 = \epsilon_r$. The relative dielectric "constant" is in turn defined as $\epsilon_r = 1+ \chi_e$ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

This is not the definition of the refractive index. Its definition is $n^2 = \epsilon_r$. The relative dielectric "constant" is in turn defined as $\epsilon_r = 1+ \chi_e$ where the polarisation $P = \epsilon_0 \chi_e E$ is related to the applied field by the electric susceptibility. See https://en.wikipedia.org/wiki/Electric_susceptibility.

In the above I have assumed an isotropic medium, at rest, with linear response and relative magnetic permeability $\mu=1$. Within these limitations the treatment is generally valid.

To answer your question, there is no fundamental argument against setting $n=1/\sqrt{\epsilon_r}$. It is the conventional definition, again with the limitations given.