A light source, which emits two wavelengths \( \lambda_{1} = 400\ \mathrm{nm} \) and \( \lambda_{2} = 600\ \mathrm{nm} \), is used in a Young’s double slit experiment. If the recorded fringe widths for \( \lambda_{1} \) and \( \lambda_{2} \) are \( W_{1} \) and \( W_{2} \) and the number of fringes for them within a distance ‘\( z \)’ on one side of the central maximum are \( n_{1} \) and \( n_{2} \), respectively, then which of the following statement is INCORRECT?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → From the central maximum, \( 3^{\mathrm{rd}} \) maximum of \( \lambda_{2} \) overlaps with \( 4^{\mathrm{th}} \) minimum of \( \lambda_{1} \).
In Young’s experiment the fringe width is \( W = \lambda D/d \), bright fringes (maxima) lie at \( y = n\lambda D/d \) and dark fringes (minima) at \( y = (n - 1/2)\lambda D/d \), with \( n = 1, 2, 3, \ldots \) counted from the central maximum. Work in units of \( D/d \).
Option (1) \( W_{2} \gt W_{1} \) — Correct. \( W \propto \lambda \), and \( \lambda_{2} \gt \lambda_{1} \), so \( W_{2} \gt W_{1} \).
Option (2) \( n_{1} \gt n_{2} \) — Correct. The number of fringes in a distance \( z \) is \( z/W \propto 1/\lambda \), so \( n_{1} \gt n_{2} \).
Option (3) From the central maximum, \( 3^{\mathrm{rd}} \) maximum of \( \lambda_{2} \) overlaps with \( 5^{\mathrm{th}} \) minimum of \( \lambda_{1} \). — Correct. Positions, in units of \( D/d \):
Option (4) From the central maximum, \( 3^{\mathrm{rd}} \) maximum of \( \lambda_{2} \) overlaps with \( 4^{\mathrm{th}} \) minimum of \( \lambda_{1} \). — Incorrect. The \( 4^{\mathrm{th}} \) minimum of \( \lambda_{1} \) is at \( 3.5 \times 400 = 1400 \), not at \( 1800 \), where the \( 3^{\mathrm{rd}} \) maximum of \( \lambda_{2} \) lies. This is the incorrect statement asked for.
The check for Option (3):
\( 3^{\mathrm{rd}}\ \text{maximum of}\ \lambda_{2}:\ 3 \times 600 = 1800 \)
\( 5^{\mathrm{th}}\ \text{minimum of}\ \lambda_{1}:\ (5 - 1/2) \times 400 = 4.5 \times 400 = 1800 \)