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: Evaluate Assertion (A).
When the ray strikes the prism's first face normally (\( i_1 = 0 \)), the angle of refraction \( r_1 \) is 0. At the second face, the angle of incidence becomes \( r_2 = A \) (where \( A \) is the prism angle). For the emergent ray to just skim the second face, the angle of emergence \( e \) must be \( 90^\circ \). Applying Snell's law at the second face yields:\[n \sin r_2 = \sin e \quad \Rightarrow \quad n \sin A = 1 \quad \Rightarrow \quad \sin A = \frac{1}{n}\]The critical angle \( \theta_c \) at the glass-air boundary is defined by:\[\sin \theta_c = \frac{1}{n}\]Since the assertion states \( \theta_c = A \), it implies \( \sin \theta_c = \sin A \), which is true if \( \sin A = \frac{1}{n} \). Therefore, the emergent ray grazes the second face precisely when \( A = \theta_c \). Assertion (A) is thus valid.Step 2: Evaluate Reason (R).
The refractive index \( n \) is an intrinsic property of the prism material and is contingent upon the wavelength of light, not the prism angle \( A \). The angle \( A \) is a geometric characteristic of the prism, and \( n \) is independent of it. Consequently, Reason (R) is invalid.Step 3: Conclusion.
Assertion (A) is true, while Reason (R) is false. The correct option is (C).