Question

In two separate Young's double-slit experimental set-ups and two monochromatic light sources of different wavelengths are used to get fringes of equal width. The ratios of the slits separations and that of the wavelengths of light used are 2:1 and 1:2 respectively. The corresponding ratio of the distances between the slits and the respective screens (D₁/D₂) is_______.}

Hint:

If you decrease the wavelength (making the fringes narrower), you must increase the screen distance or decrease the slit separation to maintain the same fringe width.

Answer 1

Correct Answer: 4

Step 1: Understanding the Concept:
Fringe width (\(\beta\)) in YDSE is the distance between two consecutive bright or dark fringes. If the fringe widths are equal for two different setups, we equate the fringe width formulas for both cases.

Step 2: Key Formula or Approach:
1. Fringe width \(\beta = \frac{\lambda D}{d}\).
2. Given \(\beta_1 = \beta_2 \implies \frac{\lambda_1 D_1}{d_1} = \frac{\lambda_2 D_2}{d_2}\).
Step 3: Detailed Explanation:
Given: Ratios of slit separations: \(d_1/d_2 = 2/1\). Ratios of wavelengths: \(\lambda_1/\lambda_2 = 1/2\). Condition \(\beta_1 = \beta_2\): \[ \frac{\lambda_1 D_1}{d_1} = \frac{\lambda_2 D_2}{d_2} \] Rearranging for the ratio of screen distances: \[ \frac{D_1}{D_2} = \left( \frac{\lambda_2}{\lambda_1} \right) \times \left( \frac{d_1}{d_2} \right) \] Substitute the ratios: \[ \frac{\lambda_2}{\lambda_1} = \frac{2}{1} \] \[ \frac{d_1}{d_2} = \frac{2}{1} \] \[ \frac{D_1}{D_2} = 2 \times 2 = 4 \]
Step 4: Final Answer:
The ratio (D₁/D₂) is 4.