Question

The refractive index of diamond is $2.47$ and that of glass is $1.51$. How much faster does light travel in glass than in diamond?

Answer 1

The refractive index of any medium is the ratio of the speed of light in vacuum to the speed of light in that medium. So using that formula we can find the ratio of the speed of light in glass to the speed of light in diamond. And hence find the number of times by which light travels faster in glass than in diamond.

Formula used: To solve this problem, we use the following formula
$\Rightarrow n = \dfrac{c}{v}$
where $n$ is the refractive index of the medium,
$v$ is the speed of light in that medium
and $c$ is the speed of light in vacuum.

Complete step by step answer:
The refractive index of a medium is the amount by which a ray bends when it is travelling from one medium to another. The light ray bends while travelling from one medium to another because the speed of light is different in different media.
Now, the refractive index of any medium is given by the ratio of the speed of light in vacuum to the speed of light in that medium. So,
$\Rightarrow n = \dfrac{c}{v}$
Now for the glass, the refractive index is given by,
$\Rightarrow {n_g} = \dfrac{c}{{{v_g}}}$
So the speed of light in glass is given by,
$\Rightarrow {v_g} = \dfrac{c}{{{n_g}}}$
Similarly, for diamond, the speed of light in diamond is given by,
$\Rightarrow {v_d} = \dfrac{c}{{{n_d}}}$
Now taking the ratio of the velocity of the speed of light in glass to that of diamond we get,
$\Rightarrow \dfrac{{{v_g}}}{{{v_d}}} = \dfrac{{\dfrac{c}{{{n_g}}}}}{{\dfrac{c}{{{n_d}}}}}$
$\Rightarrow \dfrac{{{v_g}}}{{{v_d}}} = \dfrac{c}{{{n_g}}} \times \dfrac{{{n_d}}}{c}$
Now cancelling out the ‘$c$’ we get,
$\Rightarrow \dfrac{{{v_g}}}{{{v_d}}} = \dfrac{{{n_d}}}{{{n_g}}}$
But from the question, we have ${n_d} = 2.47$ and ${n_g} = 1.51$. Therefore,
$\Rightarrow \dfrac{{{v_g}}}{{{v_d}}} = \dfrac{{2.47}}{{1.51}}$
Doing the division we get,
$\Rightarrow \dfrac{{{v_g}}}{{{v_d}}} = 1.63$
By taking ${v_d}$ from the L.H.S to the R.H.S we get,
$\Rightarrow {v_g} = 1.63{v_d}$
Therefore the speed of light in glass is $1.63$ times greater than the speed of light in diamond.

Note:
The refractive index of any medium is always greater than one. This is because, from the definition of the refractive index, the denominator of the ratio can never be more than the numerator as the speed of light in any medium can never exceed the speed of light in vacuum.