Question

The refractive index of air with respect to glass is 2/3. The refractive index of diamond with respect to air is 12/5. Then the refractive index of glass with respect to diamond will be:
A. $\dfrac{5}{8}$
B. $\dfrac{8}{5}$
C. $\dfrac{5}{{18}}$
D. $\dfrac{{18}}{5}$

Answer 1

Hint: In this question we start with writing the refractive index of air with respect to glass that is ${\mu _{\dfrac{a}{g}}} = \dfrac{{{\mu _a}}}{{{\mu _g}}}$ and refractive index of diamond with respect to air that is ${\mu _{\dfrac{d}{a}}} = \dfrac{{{\mu _d}}}{{{\mu _a}}}$, then we solve for refractive index of glass with respect to diamond and get ${\mu _{\dfrac{g}{d}}} = \dfrac{{{\mu _g}}}{{{\mu _d}}}$.

Complete Step-by-Step solution:
The refractive index of a material is a dimensionless quantity that defines how fast light travels through the material. It is defined as the ratio of the speed of light in a vacuum to the phase velocity of light in the medium.
Let us take $c$be the velocity of light and $v$ be the phase velocity of light in the material. Then refractive index can be written as
$\mu = \dfrac{c}{v}$
Now let us assume the
Aabsolute refractive index of glass=${\mu _g}$
Absolute refractive index of diamond=${\mu _d}$
Absolute refractive index of air=\[{\mu _a}\]
The refractive index of air with respect to glass is ${\mu _{\dfrac{a}{g}}} = \dfrac{{{\mu _a}}}{{{\mu _g}}} = \dfrac{2}{3} = 0.66$
The refractive index of diamond with respect to air is ${\mu _{\dfrac{d}{a}}} = \dfrac{{{\mu _d}}}{{{\mu _a}}} = \dfrac{{12}}{5} = 2.4$
We are asked to find the refractive index of glass with respect to diamond that is
${\mu _{\dfrac{g}{d}}} = \dfrac{{{\mu _g}}}{{{\mu _d}}}$
Multiplying and dividing with\[{\mu _a}\], we get
$ \Rightarrow {\mu _{\dfrac{g}{d}}} = \dfrac{{{\mu _g}}}{{{\mu _d}}} = \dfrac{{{\mu _g} \times {\mu _a}}}{{{\mu _d} \times {\mu _a}}}$
\[ \Rightarrow {\mu _{\dfrac{g}{d}}} = \dfrac{{{\mu _g} \times {\mu _a}}}{{{\mu _d} \times {\mu _a}}} = \dfrac{{{\mu _g}}}{{{\mu _a}}} \times \dfrac{{{\mu _a}}}{{{\mu _d}}}\]
\[ \Rightarrow {\mu _{\dfrac{g}{d}}} = \dfrac{{{\mu _g}}}{{{\mu _a}}} \times \dfrac{{{\mu _a}}}{{{\mu _d}}} = \dfrac{2}{3} \times \dfrac{{12}}{5} = \dfrac{8}{5}\]
So we get the refractive index of glass with respect to diamond that is
${\mu _{\dfrac{g}{d}}} = \dfrac{8}{5}$, hence option B is correct.

Note: For these types of questions we need to know about the refractive index. We need to understand the difference between the absolute and relative refractive index. After that we are required to know the expression for the relative refractive index for example refractive index of glass with respect to diamond can be written as ${\mu _{\dfrac{g}{d}}} = \dfrac{{{\mu _g}}}{{{\mu _d}}}$.