The figure shows a constant deviation prism ABCD. The incident ray is PQ and the emergent ray is ST. Although it is made up of one piece of glass, but it is equivalent to two `30^(@)-60^(@)-90^(@)` prism and one `45^(@)-45^(@)90^(@)` prism.The angle `theta_(1)` is the angle of incidence on face AB. The path of the ray inside the prism is indicated in the figure. for this prism `mu=2sintheta_(1)`.
Q. The ratio `(theta_(1))/(theta_(2))` is

Options

A. `(1)/(2)`
B. `1`
C. `(2)/(3)`
D. `(1)/(3)`

Answer

A

Explanation

let `r` be angle of refraction at `theta_(1)`
`mu=(sintheta_(1))/(sinr),sintheta_(1)=(mu)/(2)impliessinr=(1)/(2),r=30^(@)`
considering ABC as a triangular prism `r+r^(')=A`
`30^(@)+r^(')=75^(@)impliesr^(')=45^(@)`
Thus ray is incident on AB at `45^(@)` it is reflected at the same angle on to CD.
From the figure `angleRSD=360^(@)-(45^(@)+135^(@)+60^(@))=120^(@)`
(considering the quadrilateral)
`because` angle of incidence at `CD=120^(@)-90^(@)=30^(@)`
`impliestheta_(1)=theta_(2)`
`because` total deviation of ray `=(theta_(1)-30^(@))+(180^(@)-90^(@))-(theta_(2)-30^(@))=90^(@)`