A monochromatic beam of light is incident at 60° on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle θ(n) with the normal. For n=√3, the value of θ is 60° and dθ/dn = m. The value of m is?

From the geometry we have
r1 + r2 = 60°
Using Snell's law at the left side we have
sin60°/sinr1 = n or sinr1 = √3/(2n)
Using Snell's law at the right end
sinr2/sinθ = 1/n or sinθ = nsinr2 = nsin(60° - r1) = n(√3/2cosr1 - 1/2sinr1)
or = n(√3/2√(1 - sin²r1) - 1/2sinr1)
or = n(√3/2√(1 - 3/(4n²)) - √3/(4n))
Solving we get
sinθ = √3/4√(4n² - 3) - √3/(4n)
Differentiating
cosθdθ = √3/4 × 1/2√(4n² - 3) × 8n dn
or dθ/dn = √3n/√(4n² - 3) × 1/cosθ
= √3 × √3/√4 × 3 - 3 × 1/(1/2) = 2