In a double-slit experiment, the slit separation is 2.0 mm, and two wavelengths, 750 nm and 900 nm, illuminate the slits. A screen is placed 2.0 m from the slits. At what distance from the central maximum on the screen will a bright fringe from one pattern first coincide with a bright fringe from the other?

Question

BlackTom AI · Accepted Answer

Bright fringes occur at positions y = m λ L / d (m integer), so matching bright fringes from two wavelengths requires integer m1 and m2 with m1 λ1 = m2 λ2.

1) Relevant concepts
- Interference maxima for a double-slit: y_m = (m λ L) / d.
- Two patterns coincide when their mλ values are equal: m1 λ1 = m2 λ2.
- The central maximum (m = 0) always coincides; the problem asks for the first nonzero coincidence.

2) Step-by-step reasoning
- Given: λ1 = 750 nm, λ2 = 900 nm, d = 2.0 mm, L = 2.0 m.
- Form ratio: λ2 / λ1 = 900 / 750 = 6 / 5 = 1.2, so m1 / m2 = λ2 / λ1 = 6/5.
- Smallest integer pair with this ratio is m1 = 6 and m2 = 5 (smallest positive integers that satisfy 6/5).

3) Calculations (all intermediate steps)
- Use y = m1 λ1 L / d with m1 = 6:
  λ1 = 750 nm = 750 × 10^(-9) m
  d = 2.0 mm = 2.0 × 10^(-3) m
  L = 2.0 m
  m1 λ1 = 6 × 750 × 10^(-9) = 4500 × 10^(-9) = 4.5 × 10^(-6) m
  L / d = 2.0 / (2.0 × 10^(-3)) = 1000
  y = (4.5 × 10^(-6) m) × 1000 = 4.5 × 10^(-3) m = 4.5 mm

4) Why this is correct
- The chosen m1 = 6 and m2 = 5 are the smallest integers that satisfy m1 λ1 = m2 λ2, so this is the first nonzero fringe where both patterns have a bright fringe at the same screen position.

5) Why other options are incorrect
- 1.5 mm: yields m1 = 2 for 750 nm but m2 = 1.667 for 900 nm (not integer).
- 3.0 mm: yields m1 = 4 but m2 = 3.333 (not integer).
- 6.0 mm: yields m1 = 8 but m2 = 6.667 (not integer).
None of these give integer m for both wavelengths simultaneously.

6) Conclusion
- The first nonzero coincident bright fringe is at y = 4.5 mm from the central maximum, which matches the provided answer 4.5 mm.

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