Step 1: Understanding the Concept:
For total internal reflection (TIR) to occur at the hypotenuse surface of the prism, the angle of incidence \(i\) must be greater than or equal to the critical angle \(\theta_{c}\). Step 2: Key Formula or Approach:
The condition for TIR is \(\sin i \geq \sin \theta_{c}\), where \(\sin \theta_{c} = \frac{1}{\mu}\). Step 3: Detailed Explanation:
The prism is given as \(AB = BC\) with \(\angle B = 90^{\circ}\). This implies \(\angle A = \angle C = 45^{\circ}\).
The ray enters the prism normally through face \(AB\), so it strikes the hypotenuse \(AC\) without any deviation.
The angle of incidence \(i\) at the surface \(AC\) can be found from the geometry:
The normal to \(AC\) makes an angle of \(45^{\circ}\) with face \(AB\). Thus, the angle of incidence at \(AC\) is \(i = 45^{\circ}\).
For the ray to undergo TIR and travel parallel to the base as shown:
\[ \sin i \geq \frac{1}{\mu} \]
\[ \sin 45^{\circ} \geq \frac{1}{\mu} \]
\[ \frac{1}{\sqrt{2}} \geq \frac{1}{\mu} \implies \mu \geq \sqrt{2} \]
Therefore, the minimum value of the refractive index is \(\sqrt{2}\). Step 4: Final Answer:
The minimum refractive index is \(\sqrt{2}\).
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