Question

The minimum distance from the central maxima at which the two wavelengths coincide for the bright fringes produced.

Hint:

For two different wavelengths, their bright fringes coincide at a point where their fringe orders coincide.

Answer 1

Step 1: Condition for Coincidence of Bright Fringes.
For the two wavelengths to coincide at a point, the path difference for both wavelengths should be the same. That is, the distances for the bright fringes for both wavelengths must be equal. For the first wavelength \( \lambda_1 = 6000 \, \text{Å} \) and the second wavelength \( \lambda_2 = 5000 \, \text{Å} \), let the order of the fringe for both wavelengths be the same, i.e., \( m_1 = m_2 \).
Step 2: Solving for the Minimum Distance.
Using the formula for the fringe positions and setting them equal: \[ y_1 = \frac{m \lambda_1 L}{d} \quad \text{and} \quad y_2 = \frac{m \lambda_2 L}{d} \] To make the fringes coincide, we set \( y_1 = y_2 \): \[ \frac{m \lambda_1 L}{d} = \frac{m \lambda_2 L}{d} \] Thus, the two wavelengths coincide at the minimum distance where \( m_1 = m_2 \). The minimum distance is found when the first coincidence occurs at the least common multiple of the positions of the fringes for both wavelengths.