Okay so I want to solve for refractive index. I'm given the dielectric constant and the permeability M. From what I understand the refractive index is frequency dependent. I'm just unsure what equation to use. n=(Er)^1/2 does not work because the dielectric constant for water is 80.1 and the known index is 1.333. what do I use??

To determine the refractive index when given the dielectric constant and permeability, you can use a modified equation known as the complex refractive index. This modified equation takes into account the frequency dependence of the refractive index.

The equation relating the complex refractive index (n) with the dielectric constant (ε_r) and permeability (μ) is given by:

n = √(ε_r * μ)

However, it appears that you mentioned the dielectric constant for water (ε_r) is 80.1, and the known refractive index is 1.333. This discrepancy suggests that there might be additional factors or corrections needed to accurately calculate the refractive index for your particular system.

It's important to note that the refractive index of a material can vary with the wavelength of light or frequency. In this case, you have provided the dielectric constant and permeability, but it seems that these values are not sufficient to directly calculate the refractive index for water.

If you are working with the refractive index of water, you may need additional data, such as the wavelength or frequency of the light you are studying. There are empirically determined equations or dispersion relationships specific to different materials that can be used to accurately determine refractive index at a specific wavelength or frequency.