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)/(dn) = (1)/(2)`
As `r_1 = r_2 :. i_1 = theta = 60^@`
`:. (d theta)/(dn) = (1)/(2 cos 60^@) = 1` or `m = 1`.
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