A monochromatic light is incident on a Young's double slit setup that has a slit separation of 30.0 μm. The resultant bright fringe separation is 2.15 cm on a screen 1.20 m from the double slit. What is the separation between the third-order bright fringe and the zeroth-order bright fringe?

Question

BlackTom AI · Accepted Answer

For Young’s double-slit the fringe spacing β = λL/d, and the position of the m-th bright fringe relative to the central (zeroth) maximum is y_m = mβ. Thus the separation between the third-order bright fringe and the zeroth-order bright fringe is y_3 − y_0 = 3β.

1) Relevant concepts
- Fringe spacing (distance between adjacent bright fringes) β = λL/d.
- Position of m-th bright fringe y_m = mβ (measured from central maximum).
- Therefore separation between m-th and central bright fringe = mβ.

2) Step-by-step reasoning
- Given: slit separation d = 30.0 μm, screen distance L = 1.20 m, and adjacent bright fringe separation β = 2.15 cm.
- We want separation between third-order (m = 3) and zeroth-order (m = 0): Δy = 3β.

3) Calculations (all intermediate steps)
- Convert β to meters: β = 2.15 cm = 0.0215 m.
- Multiply by 3: Δy = 3 × 0.0215 m = 0.0645 m.
- Convert back to cm: 0.0645 m = 6.45 cm.
(As a check, the wavelength implied by the given β would be λ = β d / L = (0.0215 m)(30.0×10^−6 m)/(1.20 m) = 5.375×10^−7 m ≈ 537.5 nm, a visible wavelength — consistent.)

4) Why this is correct
- The fringe spacing β is constant for the setup; the third bright is three spacings away from the central maximum, so the separation is exactly three times β.

5) Why other options are incorrect
- 8.60 cm = 4 × 2.15 cm (would be the 4th-order separation, not 3rd).
- 7.35 cm is not an integer multiple of 2.15 cm, so it does not correspond to any integer-order bright fringe separation.
- 4.30 cm = 2 × 2.15 cm (would be the 2nd-order separation, not 3rd).

6) Conclusion
- The correct separation is 6.45 cm, which matches the provided answer.

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