Question

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness \( h \) in terms of the angle of incidence \( i \) and angle of refraction \( r \), if the glass slab is placed in air medium?

• A. \( \frac{h \, \tan(i - r)}{\tan r} \)
• B. \( \frac{h \, \cos(i - r)}{\sin r} \)
• C. \( h \)
• D. \( \frac{h \, \sin(i - r)}{\cos r} \)

Hint:

The lateral shift is affected by the thickness of the glass slab and the angles at which the light enters and exits the slab. Understanding the geometry of refraction in a parallel-sided slab is essential for deriving the shift formula.

Answer 1

The Correct Option is D

The lateral displacement of a light ray refracted through a parallel-sided glass slab is determined by the ray's geometric trajectory and the principles of refraction.

Concept: Refraction occurs at both the air-glass and glass-air interfaces as a light ray traverses a parallel-sided glass slab. The exiting ray is parallel to its incident path but exhibits a sideways displacement, termed "lateral shift."

The lateral shift \( S \) is calculated using the following formula, which depends on the geometry of the situation:

\(S = \frac{h \cdot \sin(i - r)}{\cos r}\)

where:

• \(h\) represents the slab's thickness,
• \(i\) is the angle of incidence,
• \(r\) is the angle of refraction.

Derivation:

1. Upon entering the slab, the ray bends towards the normal due to the increased refractive index of glass relative to air.
2. The horizontal component of the ray's path within the slab results in a different traversed distance compared to propagation in air, leading to lateral displacement.
3. The shift is determined by considering the path difference at the slab's exit face, employing trigonometric calculations.
4. The lateral shift, \( S \), is obtained by projecting the slab's thickness onto the refracted angle:
5. \(\(S = h \cdot \sin(i - r) \cdot \csc r = \frac{h \cdot \sin(i - r)}{\cos r}\)\)

This derivation aligns with the correct option and accurately accounts for both the angle of incidence and the angle of refraction. The emergent ray is parallel to the incident ray but laterally displaced.

Conclusion: The formula for the lateral shift of a light ray passing through a parallel-sided glass slab in air is \(\frac{h \cdot \sin(i - r)}{\cos r}\). Therefore, the fourth provided option is correct:

\(\frac{h \cdot \sin(i - r)}{\cos r}\)