$(a)$ For the same angle of incidence $45^{\circ}$,the angle of refraction in two transparent media $I$ and $II$ is $20^{\circ}$ and $30^{\circ}$ respectively. Out of $I$ and $II$,which medium is optically denser and why?
$(b)$ Light enters from air to diamond,which has a refractive index of $2.42$. Calculate the speed of light in diamond,if the speed of light in air is $3.00 \times 10^{8} \text{ m s}^{-1}$.

(N/A) According to Snell's Law,$n_1 \sin(i) = n_2 \sin(r)$. For a constant angle of incidence $i$,the refractive index $n$ is inversely proportional to $\sin(r)$. Since the angle of refraction $r$ is smaller in medium $I$ $(20^{\circ})$ compared to medium $II$ $(30^{\circ})$,medium $I$ has a higher refractive index and is therefore optically denser.
$(b)$ The refractive index $n$ is given by the formula $n = \frac{c}{v}$,where $c$ is the speed of light in air (or vacuum) and $v$ is the speed of light in the medium.
Given $n = 2.42$ and $c = 3.00 \times 10^{8} \text{ m s}^{-1}$.
$v = \frac{c}{n} = \frac{3.00 \times 10^{8}}{2.42} \approx 1.24 \times 10^{8} \text{ m s}^{-1}$.