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Does the magnitude of the refractive index affect reflection? Will a ray be more reflected the higher the refractive index of the other medium is?

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Absolutely. See Fresnel's equations.

These show that reflection depends on angle of incidence, polarization, and refractive index.

The simplest case (for reflection at normal incidence, where polarization doesn't matter) tells us that the reflectance is

$$R = \left|\frac{n_1-n_2}{n_1+n_2}\right|^2$$

You can see that when $n_1=n_2$, there is no reflection; and that the intensity of reflection is the same whether you go from $n_1$ to $n_2$ or the other way around. Finally, when $n_1\gg n_2$, the reflectance gets very close to 1.

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