Question

 In an experiment with a rectangular glass slab, for an angle of incidence of ${{60}^{\circ }}$in air, angle of refraction is measured to be ${{r}_{1}}$ . When the glass slab is replaced by a hollow slab filled with water, angle of refraction is measured to be ${{r}_{2}}$. Then:
A. \[{{r}_{2}}~\]= \[{{r}_{1}}~\]
B. \[{{r}_{2}}~\]> \[{{r}_{1}}~\]
C. \[{{r}_{2}}~\] < \[{{r}_{1}}~\]
D. None of the above

Answer 1

Hint: We can use Snell’s law to find a relation between angle of refraction and refractive index of medium. After getting a relation between them, we can compare the refractive indexes of glass and water, and then use our relation to compare the angle of refraction in different mediums.

Complete step by step answer:
 Refraction is the phenomenon of bending of light rays when they travel from a medium to another. If a light ray travels from medium of refractive index\[{{\mu }_{1}}\], making an angle $\theta $ (angle between the normal at boundary of mediums and incident ray) to a medium of refractive index\[{{\mu }_{2}}\], making an angle $r$ (angle between the normal of boundary of mediums and refracted ray), then by Snell’s law we can say that
${{\mu }_{1}}\sin \theta ={{\mu }_{2}}\sin r$
In the given problem, the first medium is air (refractive index =1) and$\theta ={{60}^{\circ }}$. For the first case, the second medium is glass slab. So, we can write
$1\sin {{60}^{\circ }}={{\mu }_{g}}\sin {{r}_{1}}$,
where ${{\mu }_{g}}$ is the refractive index of glass and ${{r}_{1}}$ is angle of refraction.
$\sin {{r}_{1}}=\dfrac{1}{2{{\mu }_{g}}}$ .. (1)
For the second case, the second medium is water. So, we can write
$1\sin {{60}^{\circ }}={{\mu }_{w}}\sin {{r}_{2}}$,
where ${{\mu }_{w}}$ is the refractive index of glass and ${{r}_{2}}$ is the angle of refraction.
$\sin {{r}_{2}}=\dfrac{1}{2{{\mu }_{w}}}$ … (2)
Dividing equation 1 and 2
$\dfrac{\sin {{r}_{1}}}{\sin {{r}_{2}}}=\dfrac{{{\mu }_{w}}}{{{\mu }_{g}}}$ … (3)
We know that refractive index of water (1.33) is less than refractive index of glass (1.5)
Therefore mathematically we can write,
${{\mu }_{g}}>{{\mu }_{w}}$
So,
$\dfrac{{{\mu }_{w}}}{{{\mu }_{g}}}<1$
Putting value of $\dfrac{{{\mu }_{w}}}{{{\mu }_{g}}}$ from equation 3, we get
$\dfrac{\sin {{r}_{1}}}{\sin {{r}_{2}}}<1$
$\sin {{r}_{1}}<\sin {{r}_{2}}$
Since, sine is an increasing function, if sine of some value is greater than sine of other value then the value (with greater sine) will be greater than the value (with smaller sine). So, we can say
${{r}_{2}}>{{r}_{1}}$
Hence, the correct option is B.
Note: Refractive index of a medium depends on velocity of light in that medium. If velocity of light in one medium is greater than velocity of light in another medium then the refractive index of the first medium will be smaller than the refractive index of the second medium. This is the reason why the refractive medium of glass is greater than the refractive index of water.