Math Problem Statement

give me scratch of third diffraction minimum in 2 slits

Solution

To locate the third diffraction minimum in a two-slit interference pattern, we consider the physics of diffraction and interference together. Here’s the outline or "scratch" for deriving it:


1. Diffraction Condition for a Single Slit:

The condition for diffraction minima in a single slit of width aa is: asinθ=mλ,m=±1,±2,±3,a \sin \theta = m \lambda, \quad m = \pm 1, \pm 2, \pm 3, \ldots For the third diffraction minimum, we set m=3m = 3: asinθ=3λa \sin \theta = 3 \lambda


2. Two-Slit Interference Pattern:

For two slits separated by a distance dd, the interference condition for constructive interference is: dsinθ=nλ,n=0,±1,±2,d \sin \theta = n \lambda, \quad n = 0, \pm 1, \pm 2, \ldots Similarly, the condition for destructive interference is: dsinθ=(n+12)λd \sin \theta = \left(n + \frac{1}{2}\right) \lambda


3. Combining Diffraction and Interference:

The two-slit diffraction pattern is modulated by the single-slit envelope. The interference fringes exist within this envelope. The third diffraction minimum occurs at the angle: sinθ=3λa\sin \theta = \frac{3 \lambda}{a} At this angle, no interference fringes are observed because the intensity is zero due to diffraction.


4. Conclusion:

The third diffraction minimum for a two-slit system occurs at an angle θ\theta satisfying: sinθ=3λa\sin \theta = \frac{3 \lambda}{a} where aa is the slit width, and λ\lambda is the wavelength of light.

Let me know if you'd like detailed calculations or a visual representation of this pattern.


Questions to Explore Further:

  1. How does slit separation dd influence the interference pattern's fringe spacing?
  2. What happens to the diffraction minima if the wavelength is doubled?
  3. Can we observe the third diffraction minimum if a<3λa < 3\lambda?
  4. How is intensity distributed across the interference fringes within the diffraction envelope?
  5. How does the two-slit pattern change in the limit dad \gg a?

Tip:

When analyzing interference patterns, always start with the single-slit diffraction envelope

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Math Problem Analysis

Mathematical Concepts

Diffraction
Interference
Trigonometry

Formulas

a sin θ = mλ (Diffraction minima for a single slit)
d sin θ = nλ (Constructive interference for two slits)
d sin θ = (n + 1/2)λ (Destructive interference for two slits)

Theorems

Huygens–Fresnel principle
Superposition principle

Suitable Grade Level

Undergraduate Physics