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

To determine the lateral shift of a light ray refracted through a parallel-sided glass slab, we need to consider the geometric path taken by the ray as well as the optical principles governing refraction. 

Concept: When a light ray passes through a parallel-sided glass slab, it undergoes refraction at both the air-slab interface and the slab-air interface. The ray exits parallel to its original path but displaced sideways. This displacement is what we refer to as the "lateral shift."

The lateral shift \( S \) is determined by the geometry of the setup, given by the formula:

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

where:

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

Derivation:

1. As the ray enters the slab, it bends towards the normal due to the higher refractive index of the glass compared to air.
2. Within the slab, its horizontal component causes the ray to travel a different path length than if it traveled in air, leading to lateral displacement.
3. To find the shift, we consider the path difference at the exit face of the slab, using trigonometry:
4. The length of the path through the slab in terms of horizontal distance (lateral shift, \( S \)) is given by projecting the thickness on the refracted angle:
5. \(\(S = h \cdot \sin(i - r) \cdot \csc r = \frac{h \cdot \sin(i - r)}{\cos r}\)\)

This matches the correct option, ensuring that it accounts for both the angle of incidence and refraction. The ray ultimately emerges parallel to the initial path but laterally shifted.

Conclusion: The correct formula for the lateral shift in a parallel-sided glass slab when placed in an air medium is \(\frac{h \cdot \sin(i - r)}{\cos r}\), and thus the correct answer is the fourth option given:

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

Answer 2

To find the lateral shift \(d\) of a ray refracted through a parallel-sided glass slab of thickness \(h\), we use the geometry of refraction. When a light ray enters a glass slab with an angle of incidence \(i\) and refracts at an angle \(r\), the path of the light ray inside the slab creates a lateral shift.

The lateral shift \(d\) is given by:

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

Here's how we derive the formula:

1. The refracted ray travels a horizontal distance \(L\) inside the slab, which can be given by \(L = h \sec r\), since \(h\) is the perpendicular distance (thickness of the slab) and \(r\) is the angle of refraction.
2. The lateral shift \(d\) is the horizontal component of this path difference, which can be expressed as \(L \sin(i - r)\).
3. Substitute \(L = h \sec r\) into the expression for the lateral shift:

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

Therefore, the correct answer is \(\frac{h \, \sin(i - r)}{\cos r}\).