Animated Solution for Physics - Optics: In Young's double slit experiment, light of $500\text{ nm}$ is used to produce an interference pattern. When the distance between the slits is $0.05\text{ mm}$, the angular width (in degree) of the fringes formed on the distance screen is close to

Finding the Angular Width

Imagine you are standing in a dark room, looking at the classic Young's Double Slit setup. Light passes through two incredibly narrow slits, creating a beautiful, rhythmic pattern of bright and dark fringes on a distant screen. While we often talk about the linear width of these fringes—how many millimeters wide they are on the screen—there is another, arguably more fundamental way to measure them: the angular width.

The angular width, denoted by $\theta$, is simply the angle that one complete fringe subtends when viewed from the slits. Geometrically, if the linear fringe width is $\beta$ and the screen is at a distance $D$, the angle in radians is $\theta =\frac{\beta }{D}$.

Since we know that the linear fringe width is given by $\beta =\frac{\lambda D}{d}$, substituting this into our angle equation yields a beautifully simple result:

Notice something fascinating here? The distance to the screen, $D$, has completely vanished from the equation! This means that no matter how far away you move the screen, the angular width of the fringes remains absolutely constant. It is an intrinsic property of the light's wavelength $\lambda$ and the slit separation $d$.

In physics, formulas naturally output angles in radians. However, our problem specifically demands the answer in degrees. This is a classic trap where many students lose easy marks. To convert radians to degrees, we must multiply our result by $\frac{180}{\pi }$.

Now, let's carefully substitute the given values. The wavelength $\lambda$ is $500\text{ nm}$, which is $500\times 10^{-9}\text{ m}$. The slit separation $d$ is $0.05\text{ mm}$, which is $0.05\times 10^{-3}\text{ m}$.

Let's simplify the powers of 10 to avoid any silly mistakes:

The $5$s cancel out perfectly, leaving us with:

Since $\pi \approx 3.14$, we can estimate the final value:

And there we have it! The angular width of the fringes is approximately $0.57^{\circ }$.