Animated Solution for Physics - Optics: Orange light of wavelength $6000\times 10^{-10}\text{ m}$ illuminates a single slit of width $0.6\times 10^{-4}\text{ m}$. The maximum possible number of diffraction minima produced on both sides of the central maximum is ......... .

Imagine a single slit illuminated by a vibrant orange light. As the light waves squeeze through this narrow opening, they don't just travel in a straight line. Instead, they spread out, interfering with one another to create a beautiful diffraction pattern on a distant screen. This pattern features a brilliant, wide central maximum, flanked by a series of alternating dark and bright fringes fading into the distance.

To pinpoint exactly where the shadows fall—the dark fringes or minima—we rely on a fundamental equation of wave optics:

Here, $a$ represents the width of the slit, $\lambda$ is the wavelength of the incident light, and $n$ is an integer representing the order of the minimum ($n=1,2,3,\dots$). Rearranging this to solve for the angle $\theta$, we get:

We are tasked with finding the maximum possible number of these minima. This requires us to think about the physical limits of our setup. For a fringe to actually form on the screen, the diffracted light ray must eventually intersect the screen.

If the angle $\theta$ reaches $90^{\circ }$, the light ray travels perfectly parallel to the screen and will never hit it. Therefore, the angle $\theta$ must be strictly less than $90^{\circ }$, which mathematically means:

Substituting our expression for $\sin \theta$, we establish the critical boundary condition:

Now, let's bring in the specific values provided in the problem. We are given: - Slit width, $a=0.6\times 10^{-4}\text{ m}$ - Wavelength, $\lambda =6000\times 10^{-10}\text{ m}$

To avoid any silly mistakes, it is always best to convert these into standard scientific notation: - $a=6\times 10^{-5}\text{ m}$ - $\lambda =6\times 10^{-7}\text{ m}$

Plugging these into our inequality:

The $6$ cancels out beautifully, leaving us with:

Since $n$ must be an integer strictly less than $100$, the highest possible order for a minimum on one side of the central maximum is $99$.

But remember, the diffraction pattern is perfectly symmetric. For every minimum above the central bright fringe, there is a corresponding minimum below it. Therefore, to find the total number of minima produced on both sides, we simply multiply by $2$:

And there we have it! A total of $198$ dark fringes will paint the screen.