Q) The absolute refractive indices of glass and water are [tex]$\frac{8}{2}$[/tex] and [tex]$\frac{4}{3}$[/tex] respectively. If the speed of light in glass is [tex]$2 \times 10^8 \, m/s$[/tex], what is the speed of light in water?

A. [tex]$\frac{9}{4} \times 10^8 \, m/s$[/tex]
B. [tex]$\frac{5}{2} \times 10^8 \, m/s$[/tex]
C. [tex]$\frac{7}{3} \times 10^8 \, m/s$[/tex]
D. [tex]$\frac{16}{9} \times 10^8 \, m/s$[/tex]



Answer :

Let's solve the problem step-by-step:

1. Calculate the absolute refractive index of glass and water:

The absolute refractive index of glass is given by [tex]\(\frac{8}{2}\)[/tex].
[tex]\[ n_{\text{glass}} = \frac{8}{2} = 4.0 \][/tex]

The absolute refractive index of water is given by [tex]\(\frac{4}{3}\)[/tex].
[tex]\[ n_{\text{water}} = \frac{4}{3} \approx 1.3333 \][/tex]

2. Determine the speed of light in glass:

The speed of light in glass is given as [tex]\(2 \times 10^8 \, \text{m/s}\)[/tex].

3. Find the speed of light in a vacuum (denoted as [tex]\(c\)[/tex]):

The refractive index is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium.

For glass, we have:
[tex]\[ n_{\text{glass}} = \frac{c}{v_{\text{glass}}} \][/tex]
Rearranging for [tex]\(c\)[/tex]:
[tex]\[ c = n_{\text{glass}} \times v_{\text{glass}} \][/tex]
Substituting in the given values:
[tex]\[ c = 4.0 \times 2 \times 10^8 \, \text{m/s} = 8 \times 10^8 \, \text{m/s} \][/tex]

4. Calculate the speed of light in water:

Using the refractive index formula for water:
[tex]\[ n_{\text{water}} = \frac{c}{v_{\text{water}}} \][/tex]
Rearranging for [tex]\(v_{\text{water}}\)[/tex]:
[tex]\[ v_{\text{water}} = \frac{c}{n_{\text{water}}} \][/tex]
Substituting in the values:
[tex]\[ v_{\text{water}} = \frac{8 \times 10^8 \, \text{m/s}}{\frac{4}{3}} \][/tex]
Simplifying this, we get:
[tex]\[ v_{\text{water}} = \frac{8 \times 10^8 \, \text{m/s}}{1.3333} \approx 6 \times 10^8 \, \text{m/s} \][/tex]

Therefore, the speed of light in water is approximately [tex]\(6 \times 10^8 \, \text{m/s}\)[/tex].

None of the given options exactly matches [tex]\(6 \times 10^8 \, \text{m/s}\)[/tex], so it appears there may be an issue with the provided choices or the exact representation of values. However, based on the detailed calculations, we have determined that:

[tex]\[ v_{\text{water}} = 6 \times 10^8 \, \text{m/s} \][/tex]