We know that the speed of light in vaccuum can be expressed as $c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$ and thus the speed of light in vaccuum is thus $$v=\frac{1}{\sqrt{\mu\epsilon}}=\frac{c}{\sqrt{\kappa_e\kappa_b}}$$
And by using Snell's law, we also know that $n=\frac{c}{v}$ which turn out to be $v=\frac{c}{n}$. Thus, from these 2 equation we can actually seen that $$n=\sqrt{\kappa_e\kappa_b}.$$
When the frequency of light increases, $\kappa_e$ tends to decrease, while $\kappa_b$ remain unchanged, leading to $n$ become smaller. However theoretically, the $n$ should be increase with frequency. Where did I go wrong?
