Question

Velocity of light in water, glass and vacuum have the values ${{\text{V}}_{\text{w}}}\text{,}{{\text{V}}_{\text{g}}}$ and${{\text{V}}_{\text{c}}}$respectively. Which of the following relations is true?
A. ${{V}_{c}}>{{V}_{w}}>{{V}_{g}}$
B. ${{V}_{w}}>{{V}_{f}}$ but ${{V}_{w}}<{{V}_{c}}$
C. ${{V}_{w}}={{V}_{g}}$ but ${{V}_{w}}<{{V}_{c}}$
D. ${{V}_{c}}>{{V}_{w}}$ but ${{V}_{w}}>{{V}_{g}}$

Answer 1

Hint: We need to find the relation between velocity of light in water, glass and vacuum. We know the value of velocity of light in vacuum and the refractive index of water and glass. By using this we can find the velocity of light in water and glass. Then by comparing all three velocities we can find the relation between them.

Formula Used:
Refractive index of glass,
${{\mu }_{g}}=\dfrac{{{V}_{c}}}{{{V}_{g}}}$
Refractive index of water,
${{\mu }_{w}}=\dfrac{{{V}_{c}}}{{{V}_{w}}}$

Complete step by step answer:
Velocity of light in water, glass and vacuum is given in the question.
Velocity of light in water is given as ${{V}_{w}}$
Velocity of light in glass is given as ${{V}_{g}}$
Velocity of light in vacuum is given as ${{V}_{c}}$
We know that velocity of light in vacuum is $3\times {{10}^{8}}m/s$
${{V}_{c}}=3\times {{10}^{8}}m/s$
Let us find the velocity of light in glass.
We know that refractive index of glass is 1.5, i.e.
${{\mu }_{g}}=1.5$, were ‘${{\mu }_{g}}$’ is the refractive index of glass.
We know that refractive index of glass is the ratio of speed of light in vacuum and speed of light in glass, i.e.
${{\mu }_{g}}=\dfrac{{{V}_{c}}}{{{V}_{g}}}$
Therefore, velocity of light in glass,
$\begin{align}
  & {{V}_{g}}=\dfrac{{{V}_{c}}}{{{\mu }_{g}}} \\
 & {{V}_{g}}=\dfrac{3\times {{10}^{8}}}{1.5} \\
 & {{V}_{g}}=2\times {{10}^{8}}m/s \\
\end{align}$
Similarly we can calculate the velocity of light in water.
We know that refractive index of water is 1.33, i.e.
${{\mu }_{w}}=1.33$
Same as in the case of glass, we can write
${{\mu }_{w}}=\dfrac{{{V}_{c}}}{{{V}_{w}}}$
We know the value of the refractive index of water and velocity of light in vacuum.
Therefore,
\[\begin{align}
  & {{V}_{w}}=\dfrac{{{V}_{c}}}{{{\mu }_{w}}} \\
 & {{V}_{w}}=\dfrac{3\times {{10}^{8}}}{1.33} \\
 & {{V}_{w}}=2.25\times {{10}^{8}}m/s \\
\end{align}\]
Now we have velocity of light in glass, water and vacuum.
$\begin{align}
  & {{V}_{c}}=3\times {{10}^{8}}m/s \\
 & {{V}_{w}}=2.25\times {{10}^{8}}m/s \\
 & {{V}_{g}}=2\times {{10}^{8}}m/s \\
\end{align}$
From this we can see that ${{V}_{c}}>{{V}_{w}}>{{V}_{g}}$.
Hence the correct answer is option A.

Note:
We know that velocity of light decreases from rarer medium to denser medium.
When comparing water, glass and vacuum, vacuum is an optically rarer medium, therefore velocity of light in vacuum will be more than velocity of light in glass and water.
Now let us compare glass and water. Refractive index of glass is 1.5 whereas the refractive index of water is 1.33.
Therefore water is a rarer medium compared with glass. Hence the velocity of light in water will be more than the velocity of light in glass.
Therefore, velocity of light in vacuum is greater than velocity of light in water greater than velocity of light in glass.
${{V}_{c}}>{{V}_{w}}>{{V}_{g}}$