Animated Solution for Physics - Optics: Consider the diffraction pattern obtained from the sunlight incident on a pinhole of diameter $0.1\mu \text{m}$. If the diameter of the pinhole is slightly increased, it will affect the diffraction pattern such that

The phenomenon of diffraction often defies our everyday intuition. If you shine a flashlight through a large window, you get a large patch of light on the wall. But in the quantum and wave realm, the rules flip: a bigger hole creates a smaller spot!

Let's dive into the fascinating physics of pinhole diffraction and understand exactly why this happens.

Imagine sunlight passing through a microscopic pinhole. Because light is a wave, it doesn't just travel in a straight line; it bends and spreads out as it squeezes through the tiny opening. This bending creates a diffraction pattern on the screen, characterized by a bright central spot (called the Airy disk) surrounded by fainter, alternating dark and bright rings.

The size of this central bright spot is determined by its angular spread, $\theta$. For a circular aperture of diameter $a$, the position of the first dark ring is given by the formula:

Here, $\lambda$ is the wavelength of the light. Notice the position of $a$ in this equation—it's in the denominator! This inverse relationship is the heart of the problem.

If we increase the diameter $a$ of the pinhole, the value of the fraction $\frac{\lambda }{a}$ decreases. Consequently, the angular spread $\theta$ decreases. The wave doesn't need to bend as much, so the diffraction pattern shrinks. The size of the central maximum decreases.

Now, what happens to the brightness or intensity of the spot? Intensity is a measure of how much light energy reaches the screen per unit area.

When we increase the diameter of the pinhole, we are physically making the hole larger. A larger hole has a greater area ($A=\frac{\pi a^2}{4}$), which means it allows more photons of sunlight to pass through every second.

Because more light energy is entering the system, and it is being concentrated into an even smaller central spot, the peak intensity of the diffraction pattern will dramatically increase. The intensity increases.

Combining our two physical insights: 1. A larger diameter reduces the wave spreading, so the size decreases. 2. A larger diameter allows more light to enter, so the intensity increases.

Therefore, the correct answer is that its size decreases, but intensity increases.

Elite Educator Insight: Did you notice the specific numerical values given in the problem? The pinhole diameter is $0.1\mu \text{m}$, but visible sunlight has a wavelength $\lambda$ of around $0.4$ to $0.7\mu \text{m}$. Because $\lambda >a$, the equation $\sin \theta =1.22\frac{\lambda }{a}$ yields a value greater than 1! Physically, this means the light would spread out spherically in all forward directions without forming distinct dark rings. However, the conceptual intent of the JEE examiners was to test the inverse relationship $\theta \propto 1/a$, which remains mathematically valid as we scale the diameter up.