The Dual Magic of Mirrors and Glass Slabs
Imagine a perfectly symmetric optical setup: two identical concave mirrors facing each other, with a glass slab suspended exactly in the middle. Right at the heart of this glass slab lies a tiny, glowing point source of light, S. The challenge? We need the final image of this source to form exactly where it started—back at S.
At first glance, this might seem like a straightforward reflection problem. But the presence of the glass slab introduces a beautiful optical illusion that we must carefully decode.
The Illusion of the Glass Slab
When light travels from a denser medium (like glass) into a rarer medium (like air), it bends away from the normal. Because of this refraction, an observer—or in our case, the concave mirror—does not see the source S at its actual physical location. Instead, it sees an apparent source S′ shifted slightly closer to the surface of the slab.
Let the total distance between the two mirrors be L. Since the setup is symmetric, the actual distance from the source S to either mirror is 2L.
However, the light must travel through half the thickness of the glass slab, which is 2t. The apparent shift Δx caused by this half-slab is given by the standard formula:
Δx=2t−n0t/2=2t(1−n01)
Because of this shift, the mirror perceives the light as coming from an apparent distance dapp, which is slightly less than the actual distance:
dapp=2L−Δx=2L−2t(1−n01)
This dapp is our master equation. Everything that happens next depends on where this apparent source S′ is located relative to the mirror's focal points.
Scenario 1
The Boomerang Effect
What is the most direct way for light to return to its origin? It must hit the mirror at exactly a 90-degree angle (normal incidence) so that it retraces its path perfectly.
For a spherical mirror, any light ray passing through its Center of Curvature (C) strikes the surface normally. Therefore, if the apparent source S′ lies exactly at the center of curvature, the light will boomerang right back to S.
The distance to the center of curvature is the radius R, which is twice the focal length (2f). Let's set our apparent distance equal to 2f:
Multiplying the entire equation by 2 to clear the fractions, we get:
Rearranging for L, we find our first valid distance:
This perfectly matches option (A).
Scenario 2
The Infinite Ping-Pong
Is there another way? Yes! Physics often provides multiple elegant paths to the same destination.
Imagine if the apparent source S′ was located exactly at the Focus (F) of the left mirror. We know that light originating from the focus becomes perfectly parallel to the principal axis after reflection.
These parallel rays will travel across the gap and strike the right mirror. And what does a concave mirror do to parallel rays? It converges them precisely at its own focus! Because the setup is perfectly symmetric, the focus of the right mirror coincides with the apparent position S′ on the right side, meaning the light is perfectly reconstructed back at the physical source S.
For this to happen, the apparent distance must equal the focal length f:
Again, multiplying by 2, we get:
Rearranging for L, we find our second valid distance:
This perfectly matches option (B).
The Grand Conclusion
By analyzing the optical illusion created by the glass slab and combining it with the fundamental rules of spherical mirrors, we discovered two entirely different physical mechanisms—the retracing path and the parallel path—that both satisfy the problem's condition.
This is why both (A) and (B) are correct. It is a brilliant reminder that in advanced physics, you must always ask yourself: "Is there another way this could happen?"
