Question

Find the least distance from the central maximum where the bright fringes due to both the wavelengths coincide.

Hint:

For fringe coincidence in double slit interference, the condition is that the fringe widths for both wavelengths must be the same.

Answer 1

Minimum Distance from Central Maximum for Coincident Bright Fringes

Parameters Provided:

• Wavelength 1: \( \lambda_1 \)
• Wavelength 2: \( \lambda_2 \)
• Slit-to-screen distance: \( D \)
• Slit separation: \( d \)

Bright Fringe Location Formula:

The position of the \( m \)-th bright fringe for a given wavelength \( \lambda \) is calculated as: \[ y_m = \frac{m \lambda D}{d} \]

Condition for Fringe Coincidence:

For bright fringes from both wavelengths to align, their positions must be equal: \[ \frac{m \lambda_1 D}{d} = \frac{n \lambda_2 D}{d} \] Simplified to: \[ m \lambda_1 = n \lambda_2 \] Which yields the ratio: \[ \frac{m}{n} = \frac{\lambda_2}{\lambda_1} \]

Illustrative Example:

• Wavelength 1 (\( \lambda_1 \)): \( 600 \, \text{nm} \)
• Wavelength 2 (\( \lambda_2 \)): \( 450 \, \text{nm} \)
• Screen distance (\( D \)): \( 1 \, \text{m} \)
• Slit separation (\( d \)): \( 0.5 \, \text{mm} \) or \( 0.5 \times 10^{-3} \, \text{m} \)

Step 1: Determine Fringe Order Ratio:

Calculate the ratio of fringe orders: \[ \frac{m}{n} = \frac{\lambda_2}{\lambda_1} = \frac{450 \times 10^{-9}}{600 \times 10^{-9}} = \frac{3}{4} \]

The smallest integer values satisfying this ratio are \( m = 3 \) and \( n = 4 \).

Step 2: Compute the Coincidence Distance:

The minimum distance \( y \) where fringes overlap is: \[ y = \frac{m \lambda_1 D}{d} = \frac{3 \times 600 \times 10^{-9} \times 1}{0.5 \times 10^{-3}} = 3.6 \, \text{mm} \]

Final Result:

The bright fringes from both wavelengths coincide at the minimum distance of 3.6 mm from the central maximum.