Question

When a glass slab is placed on a cross made on a sheet, the cross appears raised by $1cm$. The thickness of the glass is$3cm$. Then the refractive index of glass is:
a.$3$
b. $\dfrac{3}{2}$
c. $2$
d. $\dfrac{3}{{\sqrt 2 }}$

Answer 1

When attempting questions based on refractive index, keep in mind the various laws of refraction and the various formulas relating to the same. Keep in mind the concepts and formulas concerning refractive index, thickness of slab, critical angle and other factors.

Complete step-by-step answer:
We know the laws of refraction states that the incident ray, refracted ray and normal all lie in the same plane. And from Snell’s law of refraction we also know that the ratio of the sine of angles of incidence and transmission is equal to the ratio of the refractive index of the materials at the interface.
Refractive index measures the bending of a ray of light when passing from one medium to another, and is also equal to velocity of light of a given wavelength in empty space divided by its velocity in a substance. In this question the concerned formula will be $x=t(1-\dfrac{1}{u})$
Basically what we need to keep in mind is that the medium in which light travels slowest will have the highest refractive index. When we talk about refraction through a rectangular glass slab, we also talk about the principle of reversibility of light. We keep in mind that when a light ray enters the glass slab it bends towards normal because it is going from a rarer to denser medium.
We know that separation between the slabs , assuming it to be $x$ is $1cm$ as given in question.
The thickness, assuming it to be $t$, is given to us as $3cm$.
The formula to derive the refractive index is given as;
\Rightarrow $$$$x=t(1-\dfrac{1}{u})
Where $u$ is the refractive index we have to find.
Putting values given to us in the above mentioned formula we get;
$\Rightarrow 1=3(1-\dfrac{1}{u})$
$\Rightarrow \dfrac{1}{3}=1-\dfrac{1}{u}$
$\Rightarrow \dfrac{2}{3}=\dfrac{1}{u}$
So from above we get;
$\Rightarrow u=\dfrac{3}{2}$
So therefore the refractive index for the question will be $\dfrac{3}{2}$

So, the correct answer is “Option B”.

Note: When we talk about optics, we know that the angle of incidence to which the angle of refraction is ${{90}^{\circ }}$is called the critical angle. Relation between the critical angle and refractive index can be established as the critical angle being inversely proportional to the refractive index.