Question

A ray falls on a prism \(ABC\) (AB = BC) and travels as shown in the figure. The minimum refractive index of the prism material should be:

• A. \( \frac{4}{3} \)
• B. \( \sqrt{2} \)
• C. \(1.5\)
• D. \( \sqrt{3} \)

Hint:

For minimum refractive index, angle of incidence at the reflecting face equals the critical angle.

Answer 1

The Correct Option is B

Concept: For total internal reflection: \[ \sin C = \frac{1}{\mu} \]

Step 1: Geometry of prism Given \(AB = BC\) and angle at \(B = 90^\circ\), the prism is an isosceles right triangle. \[ \angle A = \angle C = 45^\circ \]

Step 2: Condition for minimum refractive index For minimum \(\mu\), the ray should undergo just TIR: \[ i = C \] From geometry, the angle of incidence at the reflecting face is: \[ i = 45^\circ \]

Step 3: Apply critical angle condition \[ \sin C = \frac{1}{\mu} \Rightarrow \sin 45^\circ = \frac{1}{\mu} \] \[ \frac{1}{\sqrt{2}} = \frac{1}{\mu} \Rightarrow \mu = \sqrt{2} \] Conclusion \[ \mu = \sqrt{2} \]