Question

Assertion (A): A ray of light is incident normally on the face of a prism. The emergent ray will graze along the opposite face of the prism when the critical angle at glass-air interface is equal to the angle of the prism.
Reason (R): The refractive index of a prism depends on angle of the prism.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Both Assertion (A) and Reason (R) are false.

Hint:

For a ray incident normally on a prism, the angle of incidence at the second face is the prism angle \( A \). Grazing occurs when \( \sin A = \frac{1}{n} \), which relates to the critical angle, but the refractive index does not depend on the prism angle.

Answer 1

The Correct Option is C

Step 1: Analyze Assertion (A).
The ray is incident normally on the first face of the prism (\( i_1 = 0 \)), so the angle of refraction \( r_1 = 0 \). At the second face, the angle of incidence is \( r_2 = A \) (where \( A \) is the prism angle). For the emergent ray to graze the second face, the angle of emergence \( e = 90^\circ \). By Snell’s law at the second face: \[ n \sin r_2 = \sin e \quad \Rightarrow \quad n \sin A = 1 \quad \Rightarrow \quad \sin A = \frac{1}{n} \] The critical angle \( \theta_c \) at the glass-air interface is: \[ \sin \theta_c = \frac{1}{n} \] The assertion states \( \theta_c = A \), so \( \sin \theta_c = \sin A \), which holds when \( \sin A = \frac{1}{n} \). Thus, the emergent ray grazes the second face when \( A = \theta_c \), making Assertion (A) true. Step 2: Analyze Reason (R).
The refractive index \( n \) of the prism material is a property of the material and depends on the wavelength of light, not on the prism angle \( A \). The angle \( A \) is a geometrical property of the prism, and \( n \) is independent of it. Thus, Reason (R) is false. Step 3: Conclusion.
Assertion (A) is true, but Reason (R) is false, so the correct option is (C).