Show that the angular width of the first diffraction fringe is half of that of the central fring ?

To understand the relationship between the angular widths of the diffraction fringes, we need to delve into the principles of diffraction patterns, particularly as they occur in a single-slit diffraction setup. This phenomenon can be beautifully illustrated using the principles of wave optics.

Defining Key Concepts

In a single-slit diffraction experiment, light passes through a narrow slit and spreads out, creating a pattern of bright and dark fringes on a screen. The central maximum is the brightest fringe, flanked by alternating dark and bright fringes. Understanding the angular positions of these fringes will help us derive the relationship between their widths.

Angular Position of Fringes

The angular position of the minima in a single-slit diffraction pattern is determined by the formula:

• a sin(θ) = mλ

Here, a is the width of the slit, θ is the angle of diffraction, m is the order of the minima (1, 2, 3, ...), and λ is the wavelength of the incident light.

Finding the First Minimum

For the first minimum (m = 1), the equation becomes:

• a sin(θ) = λ

This means that the angle θ for the first minimum can be approximated as:

• sin(θ) ≈ θ (in radians) when θ is small.

Substituting this approximation gives:

• θ₁ ≈ λ/a

Central Maximum and Its Width

The central maximum spans from the first minimum on one side to the first minimum on the other side, effectively doubling the angle we just calculated:

• Width of central maximum (Δθ) = θ₁ + θ₁ = 2θ₁

First Diffraction Fringe Width

The first diffraction fringe, which is the bright fringe immediately adjacent to the central maximum, is defined by the same first minimum on one side. Therefore, its angular width is simply:

• Width of first fringe = θ₁

Establishing the Relationship

To show that the angular width of the first diffraction fringe is half of that of the central maximum, we can summarize our findings:

• Angular width of the first fringe = θ₁ = λ/a
• Angular width of the central maximum = 2θ₁ = 2(λ/a)

From this, it is clear that:

• Angular width of the first fringe = 1/2 * Angular width of the central maximum

Conclusion

This shows that indeed, the angular width of the first diffraction fringe is half that of the central maximum. This relationship highlights the intricate nature of wave interference and diffraction, emphasizing how the spreading of light can create such distinct patterns. Understanding these concepts not only deepens your grasp of optics but also showcases the beauty of physics in action.