1. Given situation

• The prism is a right-angled prism with angle $90^\circ$.
• A ray is incident perpendicularly on one face, so it enters the prism without refraction.
• It then strikes the glass-air interface and is totally internally reflected.
• The angle of reflection is $45^\circ$.

2. Interpret the angle inside the prism

For reflection, the angle of reflection equals the angle of incidence.

So at the glass-air interface, $i = r = 45^\circ$ where $i$ is the angle of incidence on the glass-air surface.

3. Condition for total internal reflection

For total internal reflection at a glass-air boundary, $i > c$ where $c$ is the critical angle.

Also, $\sin c = \frac{1}{n}$ for glass of refractive index $n$ with air outside.

Since total internal reflection occurs at $i=45^\circ$, $c < 45^\circ$

Therefore, $\sin c < \sin 45^\circ = \frac{1}{\sqrt{2}}$

But $\sin c = \frac{1}{n}$ so $\frac{1}{n} < \frac{1}{\sqrt{2}}$

Hence, $n > \sqrt{2}$

4. Check options

• A: $n > \frac{1}{\sqrt{2}}$ — true but not the required conclusion.
• B: $n > \sqrt{2}$ — correct.
• C: $n < \frac{1}{\sqrt{2}}$ — false.
• D: $n < \sqrt{2}$ — false.

Therefore the correct option is: $\boxed{B}$