The width of the central maximum in a single-slit diffraction pattern is primarily determined by the wavelength of light, the width of the slit, and the distance to the screen. It is often expressed as either an angular width or a linear width. A key principle is that the slit width is inversely proportional to the width of the central maximum, as stated in the reference.
Understanding the Central Maximum in Diffraction
When light passes through a narrow slit, it diffracts, spreading out and creating an interference pattern on a screen placed behind the slit. The brightest and widest part of this pattern is the central maximum, located directly opposite the slit. It is flanked by alternating dark and bright fringes (minima and secondary maxima).
Angular Width of the Central Maximum
The angular width provides a measure of how much the central maximum spreads out, regardless of the distance to the screen. As per the reference, the angular width of the central maximum is defined as the angular distance between the two first-order minima on either side of the pattern's center. This angular width is denoted by 2θ.
The first minimum occurs at an angle θ where destructive interference happens. For a single slit of width 'a', the condition for the first minimum is given by the formula:
a sin θ = λ
where:
ais the width of the slitθis the angle to the first minimum from the centerλis the wavelength of the light
For small angles (which is often the case in diffraction experiments), sin θ is approximately equal to θ (measured in radians). Thus, we can simplify the condition for the first minimum:
a θ ≈ λ
θ ≈ λ / a
Since the central maximum extends from -θ to +θ, the total angular width (2θ) of the central maximum is approximately:
Angular Width ≈ 2λ / a
This formula clearly shows that the angular width is directly proportional to the wavelength and inversely proportional to the slit width 'a'.
Linear Width of the Central Maximum
The linear width is the actual physical width of the central maximum measured on a screen. This width depends on the angular width and the distance 'D' from the slit to the screen.
If 'D' is the distance to the screen, the linear width 'W' can be found using trigonometry. For small angles, tan θ ≈ θ. Since the first minimum is at an angle θ, its distance 'y' from the center on the screen is approximately y = D tan θ ≈ D θ.
The central maximum spans from -y to +y, so its total linear width 'W' is 2y. Using the small angle approximation for θ (θ ≈ λ/a):
W = 2y ≈ 2 * D * θ
W ≈ 2 * D * (λ / a)
Thus, the linear width (W) of the central maximum on a screen is approximately:
Linear Width ≈ 2λD / a
This formula shows the linear width is directly proportional to the wavelength λ and the screen distance D, and inversely proportional to the slit width a.
Key Relationship: Slit Width and Central Maximum Width
As highlighted in the reference, a crucial aspect of the diffraction formula is the inverse relationship between the slit width and the width of the central maximum:
In the diffraction formula, the slit width is inversely proportional to the width of the central maximum.
This means:
- If the slit width (
a) is reduced (made narrower), the angular and linear widths of the central maximum increase. - If the slit width (
a) is increased (made wider), the angular and linear widths of the central maximum decrease.
This relationship is fundamental to understanding diffraction patterns. A very narrow slit causes significant spreading (diffraction), resulting in a wide central maximum. A wide slit causes less noticeable spreading, resulting in a narrower central maximum that more closely resembles the geometric shadow of the slit.
Summary Formulas
| Measure | Formula (approx. for small angles) | Dependent On | Relationship to Slit Width ('a') |
|---|---|---|---|
| Angular Width (2θ) | ≈ 2λ / a |
Wavelength (λ), Slit Width (a) | Inversely Proportional |
| Linear Width (W) | ≈ 2λD / a |
Wavelength (λ), Slit Width (a), Screen Distance (D) | Inversely Proportional |
Practical Insights
Understanding the width of the central maximum is important in various applications, such as:
- Optical instruments: Diffraction limits the resolution of telescopes and microscopes. The size of the central maximum of the diffraction pattern of a point source (like a star or a tiny object) determines how close two sources can be and still be distinguished.
- Spectroscopy: Diffraction gratings (essentially multiple slits) are used to separate light into its different wavelengths, forming spectra. The principles governing the width of diffraction features apply here as well.
- Holography: Diffraction is the key principle behind creating and viewing holograms.
The formulas for the width of the central maximum provide a quantitative way to predict and understand how the properties of light and the diffracting object affect the resulting pattern.
