Question

In a single-slit diffraction pattern, there is a bright central maximum surrounded by successively dimmer higher-order maxima. Farther out from the central maximum, eventually no more maxima are observed. Is this because the remaining maxima are too dim? Or is there an upper limit to the number of maxima that can be observed, no matter how good the observer's eyes, for a given slit and light source?

Short answer

Answer: There is an upper limit to the number of maxima that can be observed for a given slit and light source in a single-slit diffraction pattern, regardless of the observer's vision. The maxima are not observed beyond a certain point due to the condition for constructive interference, not because of being too dim.

Step-by-step solution

Understand Single-Slit Diffraction Pattern

A single-slit diffraction pattern occurs when light passes through a small slit and interferes in such a way that certain angles create constructive interference (bright fringes or maxima) and other angles create destructive interference (dark fringes). The pattern is characterized by a central maximum and other alternating bright and dark fringes on both sides of this central maximum.

Derive the condition for single-slit diffraction maxima

To find the angles at which the maxima occur, we use the condition for constructive interference, which is given by the formula: d sin(θ) = mλ Where d is the width of the slit, θ is the angle formed by the light path with respect to the central maximum, m is the order of the maxima (0, 1, 2, ...), and λ is the wavelength of the light.

Determine the range of possible angles and m values

The constructive interference angles, θ, can vary from 0 (which corresponds to the central maximum) up to 90 degrees (which corresponds to the side edge of the screen). Also, the sin(θ) values range between 0 and 1. Additionally, m values must be integers (m=0, 1, 2,...). We can rewrite the condition for maxima as: sin(θ) = mλ / d Now, using the fact that sin(θ) can only vary between 0 and 1: 0 ≤ mλ / d ≤ 1

Investigate the limit of m values

From the inequality in step 3, we can find the upper limit for m: mλ / d ≤ 1 ⇒ m ≤ d / λ This inequality indicates that there is indeed an upper limit to the number of maxima that can be observed for a given slit and light source. As m approaches d/λ, the angle θ will approach 90°, meaning that the bright fringes will be close to the edge of the screen, and no further maxima will be observed beyond this point.

Conclusion

In a single-slit diffraction pattern, there is an upper limit to the number of maxima that can be observed for a given slit and light source, regardless of the observer's vision. This limit depends on the width of the slit and the wavelength of the light, and the maxima are not observed beyond a certain point due to the condition for constructive interference rather than because of being too dim.

Key concepts

Constructive Interference in Single-Slit Diffraction

When light passes through a single slit, it creates an intriguing pattern of alternating bright and dark fringes. This phenomenon is known as single-slit diffraction. The bright fringes, or maxima, appear due to "constructive interference". This occurs when the waves of light add up in such a way that they enhance each other, leading to increased brightness at specific angles.

• Constructive interference arises when the path difference between light rays is an integer multiple of the light's wavelength.
• This is expressed mathematically as:
  \[ d \sin(\theta) = m\lambda \] where:
  • \(d\) is the width of the slit
  • \(\theta\) is the angle of the light path
  • \(m\) is the order of the maxima
  • \(\lambda\) is the wavelength

Understanding this model helps us predict where the bright fringes will occur around a central bright spot, known as the central maximum.

Diffraction Maxima and Their Limits

As you explore single-slit diffraction, you'll notice that beyond the glorious central maximum, there exists a series of weaker successive maxima. These "diffraction maxima" appear at angles determined by the same condition used for constructive interference.
However, there's a practical limit to what can be observed. This ties back to the condition for constructive interference, expressed as:

• The sine of the angle \(\theta\) can only take values between 0 and 1.
• Thus, \(\sin(\theta) = \frac{m\lambda}{d}\) can only be valid for \( m \leq \frac{d}{\lambda} \).

Once you hit the limit of \( m \), no additional maxima can be seen. This isn't because the fringes get dimmer, but rather, because further maxima simply don't exist beyond this geometrical constraint.

Understanding Wavelength of Light

The "wavelength of light" \(\lambda\) is a fundamental factor in all interference patterns, including single-slit diffraction. It represents the distance between consecutive peaks of a light wave.
A crucial insight is that the wavelength determines the spacing and specific location of maxima in a diffraction pattern:

• Light with a longer wavelength, such as red light, will result in diffraction maxima being more spread out.
• Conversely, shorter wavelengths, like blue light, lead to closer spaced fringes.

This dependency highlights why asking, "Is wavelength important in diffraction?" leads to a resounding yes—it directly influences both the angle of occurrence for each maximum and the number of maxima within observable limits.