The refractive index of the material of a prism is `sqrt(2)` and the angle of the prism is `30^(@)`. One of the two refracting surfaces of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is

Options

A. `30^(@)`
B. `45^(@)`C
C. `60^(@)`C
D. zero

Answer

B

Explanation

According to the question, the figure of mentioned prism is given as

(since, there Is no refraction at the face AC) Given, Refractive index of the material of prism, `mu=sqrt(2)`
Angle of prism, A=`30^(@)`
If the ray OR has to retrace its path after reflection (as per the given condition), then the ray has to fall normally on the surface AC.
This means, `angleARO = angleORC = 90^(@)`
In `DeltaAOR, angleAOR + angleOAR=180^(@)`
`rArr angleAOR = 180^(@) -120^(@)=60^(@)`..........(i)
As we know, `angleAOR + angler_(1)=90^(@)`
`rArr angler_(1)=90^(@)-60^(@)=30^(@)` from Eq. (i)
Applying Snell's law at the face AB, we get
`mu = sini/sinr_(1)`
Substituting the given values, we get
`sqrt(2)=(sini)/(sin30^(@))`
rArr sini =sin30^(@) xx sqrt(2)`
=`1/2 xx sqrt(2)` `(therefore sin 30^(@) =1/2)`
`=1/sqrt(2)`
or `i=sin^(-1)(1/sqrt(2))=45^(@)` (`therefore sin45^(@)=1/sqrt(2))`
The angle of incidence of the ray on the prism is `45^(@)`.