In Differection the band width of central maxima is X = 2x How it proved?

To understand why the bandwidth of the central maxima in diffraction is expressed as "X = 2x," we need to delve into the principles of wave interference and diffraction patterns. This concept is often illustrated using a single-slit diffraction experiment, which provides a clear visual representation of how light behaves when it encounters an obstacle.

Understanding Diffraction Patterns

When light passes through a narrow slit, it spreads out instead of traveling in straight lines. This spreading creates a pattern of bright and dark regions on a screen, known as a diffraction pattern. The central bright region is called the central maxima, and it is flanked by alternating dark and bright fringes.

Key Variables in the Diffraction Equation

To analyze the diffraction pattern, we can use the following variables:

• w: Width of the slit
• L: Distance from the slit to the screen
• x: Distance from the center of the central maxima to the first minimum on either side
• X: Total width of the central maxima

Deriving the Bandwidth of Central Maxima

The position of the first minimum in the diffraction pattern can be derived from the single-slit diffraction formula:

a sin(θ) = mλ

Here, a is the slit width, θ is the angle of diffraction, m is the order of the minimum (with m = 1 for the first minimum), and λ is the wavelength of the light used.

For small angles, we can use the small angle approximation, where sin(θ) ≈ tan(θ) ≈ θ (in radians). Thus, we can rewrite the equation as:

aθ = λ

From this, we can find the angle θ for the first minimum:

θ = λ/a

Calculating the Width of the Central Maxima

The distance x from the center to the first minimum on one side can be expressed in terms of the angle and the distance to the screen:

x = L tan(θ) ≈ L sin(θ) = L(λ/a)

Since the central maxima extends from the first minimum on one side to the first minimum on the other side, the total width X of the central maxima is:

X = 2x

Substituting for x, we get:

X = 2(Lλ/a)

Visualizing the Concept

Imagine a water wave passing through a narrow opening. As the wave exits the slit, it spreads out, creating a series of ripples. The central ripple is the most pronounced, representing the central maxima. The width of this central ripple is determined by how far the wave spreads, which is influenced by the slit width and the wavelength of the wave.

Conclusion

In summary, the expression "X = 2x" for the bandwidth of the central maxima in diffraction arises from the relationship between the position of the minima and the geometry of the diffraction pattern. By understanding the underlying principles of wave behavior, we can derive this important relationship and appreciate the fascinating nature of light and its interactions with obstacles.