Question

A single slit diffraction pattern is formed with white light. For what wavelength of light does the $3^{\text{rd}}$ secondary maximum in the diffraction pattern coincide with the $2^{\text{nd}}$ secondary maximum in the pattern of red light of wavelength $6000\ \text{\AA}$?

• A. $4500\ \text{\AA}$
• B. $3500\ \text{\AA}$
• C. $4000\ \text{\AA}$
• D. $5000\ \text{\AA}$

Hint:

When two positions coincide in wave optics, always set up a direct ratio equation. For secondary maxima, the path difference condition is an odd multiple of half wavelengths, leading to the relation $2.5\lambda_{\text{red}} = 3.5\lambda_{\text{unknown}}$. This directly isolates the parameter ratio without needing to compute screen parameters!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
Two spectral lines from different wavelengths coincide at a certain order in a diffraction pattern. Determine the wavelength ratio using path difference conditions.

Step 2: Key Formula or Approach:
For secondary maxima, the path difference is an odd multiple of half-wavelengths. At coincidence, m₁λ₁ = m₂λ₂, where the integer multipliers correspond to the respective fringe orders.

Step 3: Detailed Explanation:
Setting up the equality for the given orders: 2.5λ_red = 3.5λ_unknown. Solving for the ratio λ_unknown/λ_red = 2.5/3.5 = 5/7. This direct ratio method isolates the wavelength relationship without requiring slit separation, screen distance, or any other geometric parameters of the experimental setup.

Step 4: Final Answer:
The unknown wavelength is (5/7) times the red wavelength.