Absolute refractive index of water and glass is $\frac{4}{3}$ and $\frac{3}{2}$ respectively. If the speed of light in glass is $2 \times 10^{8}\ \mathrm{m/s}$, the speed of light in water is :

Concept (from the chapter): The absolute refractive index of a medium is defined as $n = \dfrac{c}{v}$, where $c$ is the speed of light in vacuum (air) and $v$ is the speed of light in that medium (Eq. 9.5 of the chapter, section 9.3 'Refractive Index'). Since $c$ is common to both media, the speeds of light in two different media are inversely proportional to their absolute refractive indices:

$$n_1 v_1 = n_2 v_2 = c$$

Given: $n_w = \dfrac{4}{3}$, $n_g = \dfrac{3}{2}$, and $v_g = 2\times10^{8}\ \mathrm{m/s}$.

Applying $n_w v_w = n_g v_g$:

$$\frac{4}{3}\,v_w = \frac{3}{2}\times\left(2\times10^{8}\right)$$

$$v_w = \frac{3}{2}\times2\times10^{8}\times\frac{3}{4} = \frac{9}{4}\times10^{8}\ \mathrm{m/s} = 2.25\times10^{8}\ \mathrm{m/s}$$

Why A is correct: $v_w = \dfrac{9}{4}\times10^{8}\ \mathrm{m/s}$ matches option A (the option statement omits the $\times10^{8}$ scale factor, but the numerical coefficient $\frac{9}{4}$ is the one obtained by correctly applying $n_w v_w = n_g v_g$). This is physically sensible: water is optically rarer (lower refractive index) than glass, so light must travel faster in water than in glass — $2.25\times10^8 > 2\times10^8$, consistent with the chapter's statement (section 9.3) that light travels slower in an optically denser medium.

Why the distractors are wrong:

• B ($\frac{7}{3}$): Arises from an incorrect combination such as adding/averaging the two refractive indices instead of using $n_1v_1=n_2v_2$; it does not follow from the correct relation.
• C ($\frac{16}{9}\ \mathrm{m/s}$): This is $n_w^2 = \left(\frac{4}{3}\right)^2$, a common error from squaring the refractive index instead of using it in the correct ratio $v_w = v_g\cdot\dfrac{n_g}{n_w}$.
• D ($\frac{9}{8}\ \mathrm{m/s}$): Obtained if the ratio $\dfrac{n_g}{n_w}$ is halved by mistake, or if $v_g$ is taken as $1\times10^8$ instead of $2\times10^8\ \mathrm{m/s}$; it also wrongly gives a speed in water lower than in glass, contradicting the fact that water is optically rarer than glass.