A diverging beam of light from a point source S having divergence angle alpha falls symmetrically on

Q: Why doesn't the refractive index $n$ affect the final divergence angle?

The refractive index $n$ determines how much the rays bend *inside* the slab and thus affects the lateral displacement. However, because the slab has parallel sides, the bending at the first surface is exactly undone at the second surface, making the final angle independent of $n$.

Q: Does the thickness $t$ of the slab change the divergence angle?

No. The thickness $t$ only increases the lateral shift of the rays. The thicker the slab, the further the emergent ray is shifted from its original path, but it will always remain parallel to the original path.

Q: What if the glass slab was a prism instead?

If the surfaces were not parallel (like in a prism), the emergent rays would not be parallel to the incident rays. In that case, the divergence angle would change, and it would depend on both the refractive index and the angle of the prism.

Q: Are the emergent rays exactly the same as the incident rays?

They are parallel, but not collinear. The emergent rays are laterally displaced outwards compared to where they would have been if the glass slab wasn't there.

Q: Why is the divergence angle $2i$?

The problem states the beam falls symmetrically. If the normal is drawn down the center, the left extreme ray has an angle of incidence $i$, and the right extreme ray has an angle of incidence $i$. The total angle between them is $i + i = 2i = \alpha$.

Question

A diverging beam of light from a point source $S$ having divergence angle $\alpha$ falls symmetrically on a glass slab as shown. The angles of incidence of the two extreme rays are equal. If the thickness of the glass slab is $t$ and its refractive index is $n$, then the divergence angle of the emergent beam is

Options:

(A) zero

(B) $\alpha$

(C) $\sin^{-1}(1/n)$

(D) $2\sin^{-1}(1/n)$

Sigma Percentile · Accepted Answer

Have you ever looked through a thick glass window and noticed that while objects might seem slightly shifted, their overall shape and size remain perfectly intact? This everyday phenomenon holds the key to solving our problem about the diverging beam of light.

### Analyzing the Setup
Imagine a point source $S$ emitting a beam of light. The rays spread out, creating a divergence angle $\alpha$. This beam symmetrically strikes a glass slab of thickness $t$ and refractive index $n$. Our goal is to find out what happens to this divergence angle after the light passes completely through the slab.

### The Journey Through the Glass
When the light rays hit the top surface of the glass slab, they transition from a rarer medium (air) to a denser medium (glass). According to Snell's Law, they bend **towards** the normal.

Let the angle of incidence for the extreme rays be $i$. The refraction at the first surface is governed by:
$$ 1 \cdot \sin i = n \cdot \sin r $$

The rays travel through the slab and hit the bottom surface. Here, the geometry of the parallel slab dictates that the angle of incidence at this second surface is exactly $r$.

As the rays exit the glass and re-enter the air, they bend **away** from the normal. Applying Snell's Law again:
$$ n \cdot \sin r = 1 \cdot \sin e $$

### The Master Equation
If we combine our two equations, a beautiful mathematical cancellation occurs:
$$ \sin i = \sin e \implies i = e $$

What does this mean physically? It means that the angle at which the ray emerges ($e$) is exactly equal to the angle at which it originally entered ($i$). The emergent rays are perfectly parallel to the incident rays!

### Final Conclusion
Because the emergent rays are parallel to their corresponding incident rays, the angle between them cannot change. The entire beam has simply undergone a lateral shift—a slight displacement to the side—but its angular spread remains completely unaffected.

Therefore, the divergence angle of the emergent beam is exactly the same as the incident beam: $\alpha$.

This elegant result reminds us that a parallel-sided glass slab acts like a perfect optical translator. It shifts the path of light but preserves its direction, ensuring that the world we see through a window remains undistorted.

Select Answer:

Visualized Solution

Analyzing the Setup

Refraction at the First Surface

Refraction at the Second Surface

Parallel Emergence

Final Divergence Angle

The Sigma Insight: Refraction and Total Internal Reflection

Analyzing the Setup Imagine a point source $S$ emitting a beam of light

The Journey Through the Glass When the light rays hit the top surface of the glass slab, they transition from a rarer medium (air) to a denser medium (glass)

The Master Equation

Final Conclusion Because the emergent rays are parallel to their corresponding incident rays, the angle between them cannot change

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