A ray of light passes through an equilateral glass prism such that the angle of incidence (i) = angle of emergence (e). If the angle of emergence is 0.75 times the angle of prism, the refractive index of the prism will be

$\sqrt{2}$

The correct answer is Option (3) → $\sqrt{2}$

Given:

Angle of prism, $A$

Angle of incidence = Angle of emergence = $i$

$e = i$ and $e = 0.75A$

For minimum deviation in a prism:

$\delta_m = 2i - A$

and $\sin i = \mu \sin \frac{A}{2}$

Given that $i = e = 0.75A$

Using Snell’s law relation at minimum deviation:

$\mu = \frac{\sin i}{\sin (A/2)}$

Substitute $i = 0.75A$:

$\mu = \frac{\sin (0.75A)}{\sin (A/2)}$

For an equilateral prism, $A = 60^\circ$:

$\mu = \frac{\sin (0.75 \times 60)}{\sin (30)}$

$\mu = \frac{\sin (45)}{0.5}$

$\mu = \frac{0.7071}{0.5} = 1.414$

Final Answer: $\mu = 1.414$