Question

A ray of light is incident at an angle \( i \) on a parallel sided glass slab of thickness \( d \) and gets refracted into the slab at angle \( r \). Draw a ray diagram to show its path as it emerges out of the slab. Hence, obtain an expression for the lateral shift of the ray. Under what condition will the shift be minimum?

Hint:

The lateral shift of the ray inside a parallel sided slab depends on the thickness of the slab, the angle of incidence, and the angle of refraction. The shift will be minimum when the incident ray is normal to the slab surface (i.e., \( i = r \)).

Answer 1

Let’s consider the ray diagram for a parallel sided glass slab: 1. A ray of light is incident on the glass slab at an angle \( i \). 2. The ray is refracted inside the glass at an angle \( r \). 3. The ray then emerges out of the slab at the same angle \( r \) (because the glass slab is parallel-sided).

From the diagram, we see that after the light emerges from the slab, it undergoes a lateral shift due to the thickness \( d \) of the slab. Derivation of Lateral Shift: Let the lateral shift be \( x \), which is the horizontal displacement of the emergent ray from the original path. The light undergoes refraction twice: 1. First, when it enters the slab at angle \( i \), it refracts to angle \( r \). 2. Second, when it exits the slab, it refracts back to the same angle \( r \). Let the thickness of the slab be \( d \), and the distance traveled inside the slab be \( d \sec r \) (since the ray is traveling at angle \( r \)). The lateral shift \( x \) is given by the following formula: \[ x = d \left( \frac{\sin(i - r)}{\cos r} \right) \] where: - \( i \) is the angle of incidence, - \( r \) is the angle of refraction, - \( d \) is the thickness of the slab. Condition for Minimum Shift: The lateral shift will be minimum when the angle of incidence \( i \) is equal to the angle of refraction \( r \), i.e., when \( i = r \). This would make the ray pass straight through the slab without any lateral shift. Thus, the condition for the minimum lateral shift is when the incident angle is equal to the refracted angle (\( i = r \)).