Animated Solution for Physics - Optics: In a Young's double slit experiment, 15 fringes are observed on a small portion of the screen when light of wavelength 500 nm is used. Ten fringes are observed on the same section of the screen when another light source of wavelength $\lambda$ is used. Then, the value of $\lambda$ is (in nm) ...... .

Imagine you are standing in a dark room, looking at the beautiful, alternating bright and dark bands of a Young's Double Slit Experiment (YDSE) projected onto a screen. You take a marker and draw a small, fixed window on the screen. Let's call the length of this window $y$.

When you shine a light of wavelength ${\lambda }_1=500$ nm through the slits, you count exactly $15$ fringes fitting perfectly inside your window. But when you switch to a mysterious second light source with an unknown wavelength $\lambda$, you only count $10$ fringes in that exact same window. Where did the other fringes go? Let's unravel this mystery using the physics of wave optics.

To solve this, we need to understand what determines the size of a fringe. The width of a single fringe, denoted by $\beta$, is governed by the formula:

Here, $D$ is the distance to the screen, and $d$ is the separation between the slits. If one fringe has a width of $\beta$, then the total length occupied by $n$ fringes is simply $n$ times $\beta$:

The crucial piece of information in the problem is that both sets of fringes are observed on the same section of the screen. This means the physical length $y$ is identical for both light sources. Therefore, we can equate the lengths:

Since we haven't moved the screen or changed the slits, $D$ and $d$ are constants. They elegantly cancel out from both sides, leaving us with a powerful and simple inverse relationship:

This equation tells us a profound physical truth: for a fixed distance on the screen, the number of fringes is inversely proportional to the wavelength. A longer wavelength creates wider fringes, so fewer of them can fit into our window.

Now, it is just a matter of plugging in the numbers. We know $n_1=15$, ${\lambda }_1=500$ nm, and $n_2=10$. Let's substitute these into our master equation:

Solving for $\lambda$:

The mysterious second light source has a wavelength of $750$ nm. Because $750$ nm is longer than $500$ nm, its fringes are wider, which perfectly explains why only $10$ of them could fit into the space where $15$ of the narrower fringes previously fit. Physics always makes sense when you look closely!