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?
Show Hint
- \( \frac{h \, \tan(i - r)}{\tan r} \)
- \( \frac{h \, \cos(i - r)}{\sin r} \)
- \( h \)
- \( \frac{h \, \sin(i - r)}{\cos r} \)
The Correct Option is D
Approach Solution - 1
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:
- As the ray enters the slab, it bends towards the normal due to the higher refractive index of the glass compared to air.
- 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.
- To find the shift, we consider the path difference at the exit face of the slab, using trigonometry:
- 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:
- \(\(S = h \cdot \sin(i - r) \cdot \csc r = \frac{h \cdot \sin(i - r)}{\cos r}\)\)
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:
\(\frac{h \cdot \sin(i - r)}{\cos r}\)
Approach Solution -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:
- 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.
- The lateral shift \(d\) is the horizontal component of this path difference, which can be expressed as \(L \sin(i - r)\).
- 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 \(\frac{h \, \sin(i - r)}{\cos r}\).
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