A light ray enters through a right angled prism at point P with the angle of incidence 30° as shown in figure. It travels through the prism parallel to its base BC and emerges along the face AC. The refractive index of the prism is :

The correct option is √5/2.

1. Analyzing the Prism and Geometry:
From the given figure, we see a right-angled prism with refracting angle $A=90°$ at vertex A.
The light ray enters face AB at point P with an angle of incidence $i=30°$.
Inside the prism, the refracted ray travels parallel to the base BC.

Let $B$ be the angle at vertex B of the right-angled triangle ABC. Since the ray inside the prism is parallel to the base BC, the angle of refraction $r_1$ at face AB satisfies:
$r_1+B=90°$  ⇒  $r_1=90°-B$

Also, the ray emerges along the face AC, which means total internal reflection occurs at the critical angle $i_c$ at face AC, or the angle of emergence is $e=90°$.
The angle of incidence at face AC inside the prism is $r_2=i_c$.
Since the ray inside is parallel to BC, by interior alternate angles with face AC:
$r_2=C$
Because $A=90°$, we have $B+C=90°$, which gives $B=90°-C=90°-r_2$.
Therefore, $r_1=90°-(90°-r_2)=r_2$.
Also, since $r_1+r_2=A=90°$, we get:
$r_1=r_2=45°$

Wait, let's re-verify using Snell's Law at face AC:
At face AC, the ray emerges along the surface, so the critical angle condition gives:
$sin\left(r_2\right)=\frac{1}{\mu }$

2. Applying Snell's Law at Face AB:
At the first refracting surface AB:
$1·sin\left(i\right)=\mu ·sin\left(r_1\right)$
Given $i=30°$:
$sin(30°)=\frac{1}{2}=\mu ·sin\left(r_1\right)$  ⇒  $sin\left(r_1\right)=\frac{1}{2\mu }$

3. Relating Angles of Refraction:
Since $r_1+r_2=A=90°$, we have:
$r_2=90°-r_1$
$cos\left(r_1\right)=sin\left(r_2\right)=\frac{1}{\mu }$

Using the fundamental trigonometric identity ${sin}^2\left(r_1\right)+{cos}^2\left(r_1\right)=1$:
${\left(\frac{1}{2\mu }\right)}^2+{\left(\frac{1}{\mu }\right)}^2=1$

4. Solving for the Refractive Index ($\mu$):
$\frac{1}{4{\mu }^2}+\frac{1}{{\mu }^2}=1$
$\frac{1+4}{4{\mu }^2}=1$
$\frac{5}{4{\mu }^2}=1$
${\mu }^2=\frac{5}{4}$
$\mu =\frac{\sqrt{5}}{2}$

Thus, the refractive index of the prism is √5/2.