Maxwell theory [tex]n=\sqrt{\epsilon_r \mu_r[/tex] Refractive index for water is [tex]n=1,33[/tex]. For water [tex]\epsilon_r=81, \mu_r=1[/tex] so it should be [tex]n=9[/tex] Why we have so big anomaly for water?
The index of refraction and the reduced electric permittivity are both frequency dependent quantities, they are not constants. The constant value of 81 you refer to is the reduced electric permittivity of room temperature water in the limit that the frequency goes to zero. A frequency of zero is clearly a bad approximation for the optical frequencies where index of refraction is typically measured. My guess is that the index of refraction of water for very low frequency radiation probably *is* around 9 ... that is probably part of the reason why radio waves cannot penetrate the water very effectively, whereas shorter wavelength visible light penetrates much further.