A monochromatic beam of light is incident at `60^@` on one face of an equilateral prism of refractive inder `n` and emerges from the opposite face making an angle `theta` with the normal. For `n = sqrt(3)`, the value of `theta` is `60^@ and (d theta)/(dn) = m`. The value of `m` is.

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

A

Explanation

Here, `i_(1) = 60^(@), n = sqrt(3)`. `A = 60^(@)`
`n = (sin i_(1))/(sin r_(1))` or `sqrt(3) = (sin 60^(@))/(sin r_(1))`
`sin r_(1) = (sqrt(3)//2)/(sqrt(3)) = (1)/(2) :. r_(1) = 30^(@)`

As `/_O + /_A = 180^(@), /_O = 120^(@)`
`/_O + r_(1) + r_(2) = 180^(@), :. r_(1) = r_(2) = 30^(@)`
`n = (sin theta)/(sin r_(2))`
`sin theta = n sin r_(2) = n sin 30^(@) = (n)/(2)`
Differentiating w.r.t. `n`, we get
`cos theta(d theta)/(d n) = (1)/(2)`
As `r_(1) = r_(2) :. i_(1) = theta = 60^(@)`
`:. (d theta)/(d n) = (1)/(2 cos 60^(@)) = 1 or m = 1`