Animated Solution for Physics - Optics: In the ideal double-slit experiment, when a glass-plate (refractive index 1.5) of thickness $t$ is introduced in the path of one of the interfering beams (wavelength $\lambda$), the intensity at the position where the central maximum occurred previously remains unchanged. The minimum thickness of the glass-plate is

Imagine you are standing in a dark room, looking at a screen illuminated by two tiny, coherent sources of light. What you see is not just a blob of light, but a beautiful, rhythmic pattern of bright and dark bands. This is the classic Young's Double Slit Experiment (YDSE), a profound demonstration of the wave nature of light.

At the exact center of the screen, point $O$, the light waves from both slits travel the exact same distance. Because they travel the same distance, they arrive perfectly in phase. Crest meets crest, trough meets trough, and they undergo constructive interference. This results in a bright spot, which we call the central maximum. Mathematically, the path difference $\Delta x$ at this point is zero.

Introducing the Glass Plate

Now, let's shake things up. We introduce a thin glass plate of thickness $t$ and refractive index $\mu =1.5$ directly in front of one of the slits.

What does this glass plate do? Light travels slower in a denser medium like glass compared to air. Even though the physical distance the light travels hasn't changed much, the optical path—the equivalent distance the light would have traveled in a vacuum in the same amount of time—has increased.

The glass plate introduces an additional optical path difference between the two beams. The formula for this extra path difference is beautifully simple:

Because of this new path difference, the entire fringe pattern shifts on the screen. The central maximum is no longer at point $O$; it has moved!

Unchanged Intensity

The problem gives us a fascinating constraint: despite the glass plate being introduced, the intensity at the original central point $O$ remains unchanged.

What does this mean physically? It means that point $O$ must still be a bright fringe! For constructive interference to occur at any point, the path difference must be an exact integer multiple of the wavelength $\lambda$.

So, we can set up our master equation by equating the path difference introduced by the glass plate to the condition for a maximum:

where $n=1,2,3,\dots$

Finding the Minimum Thickness

We want to find the minimum thickness $t$ of the glass plate. Let's rearrange our master equation to solve for $t$:

To make $t$ as small as possible, we must choose the smallest possible positive integer for $n$. That integer is $n=1$.

Now, we simply substitute our known values into the equation. We know $n=1$ and the refractive index $\mu =1.5$:

Since dividing by $0.5$ is the same as multiplying by $2$, we arrive at our final, elegant result:

And there we have it! The minimum thickness of the glass plate required to keep the intensity at the central point unchanged is exactly twice the wavelength of the light used. The correct option is (a).