Double-Slit Wavelength Measurement That Disagrees with the Laser Label
An advanced lab group measures the wavelength of a diode laser labelled 650 nm ± 10 nm using a double slit with separation d = 0.250 mm (manufacturer tolerance ±2%) and a screen at L = 2.00 m. Using a steel rule, they mark the centre of the first bright fringe to one side of the central maximum and the centre of the tenth bright fringe on the same side; the separation is 47.5 mm. They divide 47.5 mm by 10, obtain Δy = 4.75 mm, and compute λ = dΔy/L = (2.50×10⁻⁴ m)(4.75×10⁻³ m)/(2.00 m) = 5.94×10⁻⁷ m, i.e. 594 nm. They conclude that the laser is defective and about 10% short in wavelength. The teaching assistant is sceptical: the pattern is clean and symmetric, the screen is square to the beam to within about 1°, the room is dark, and the slit-to-screen distance was measured along the bench with a steel tape. Before a replacement laser is ordered, the group must decide whether the measurement or the equipment is at fault.
1. Identify the likely source of the 10% discrepancy and recompute the wavelength. State whether the laser is out of specification.
Read the worked answer
The formula and the arithmetic are right for the numbers used; the fringe count is wrong. The centre of the first bright fringe is already one fringe spacing from the central maximum, so the first and tenth bright fringes are separated by nine spacings, not ten. Counting spacings rather than fringes is the whole error.
Δy = 47.5 mm / 9 = 5.28 mm λ = dΔy/L = (2.50×10⁻⁴)(5.28×10⁻³)/(2.00) = 6.60×10⁻⁷ m = 660 nm
That is within about 1.5% of the 650 nm label and inside the ±10 nm tolerance, so the laser is fine. Dividing by 10 instead of 9 multiplies the answer by 9/10 = 0.90, which is precisely the size and sign of the observed discrepancy - a strong sign that the error is procedural, not instrumental, because a genuinely off-spec laser would not land on exactly the factor produced by an off-by-one count.
Note that measuring from the central maximum to the tenth fringe and dividing by 10 would have been correct. The safeguard is to check the pattern's symmetry: measure the same order on both sides of the central maximum and confirm that the centre of the pattern falls midway between them.
2. The screen is tilted about 1° from perpendicular. Show quantitatively that this cannot account for the discrepancy, and identify which quantity actually limits the precision of the result.
Read the worked answer
A tilt φ about a vertical axis through the pattern centre scales every measured distance on the screen by 1/cos φ. At φ = 1°, 1/cos 1° = 1.00015 - a 0.015% effect, roughly a thousand times too small to explain a 10% gap. Even a 5° tilt gives only 0.4%. Tilt is not the problem, and neither is the tape measurement of L.
The limiting quantities are, in order of importance with this apparatus:
- d = 0.250 mm ± 2%. The tolerance propagates straight into λ and cannot be beaten by better technique with this slit: it alone contributes about 13 nm.
- Fringe position, ±0.2–0.3 mm at each endpoint by eye. Over the 47.5 mm span that is about 1.3%, or roughly 9 nm.
- L = 2.00 m ± 2 mm, i.e. 0.1%, or under 1 nm.
Combining in quadrature gives about 2.4%, so the honest result is λ = 660 ± 16 nm. The group should report that, and should say explicitly that the slit tolerance is the floor on this apparatus.
3. The group wants to cut the uncertainty by a factor of three. Evaluate three options - moving the screen to 6.00 m, measuring the same pattern with a travelling microscope, and switching to a 600 lines/mm diffraction grating - and recommend one.
Read the worked answer
Screen at 6.00 m. Fringe spacing triples to about 15.8 mm, so the fractional position uncertainty falls from 1.3% to about 0.4%, and L's relative contribution is unchanged. But d's 2% is untouched, so the total falls only from 2.4% to about 2.0%. No factor of three, and it needs a longer bench.
Travelling microscope. Fringe centres improve to perhaps ±0.02 mm, but again the 2% slit tolerance dominates: the total stays near 2%. Large effort, no useful gain.
600 lines/mm grating. d = (1/600) mm = 1.667 µm, and the spacing is set by the ruling engine and can be certified, or calibrated in situ against a known wavelength. The measurement becomes angular - sin θ = mλ/d - so distances on a screen and the value of L drop out of the analysis entirely, and a spectrometer reads angles to 0.01° or better. With d known to 0.1% and angles to 0.01°, the combined uncertainty falls to a few tenths of a percent, comfortably better than a factor of three.
Recommendation: switch to the grating and measure angles, preferably with the grating in autocollimation so the zero order is defined by the instrument rather than by eye. The double slit cannot deliver the requested precision because its stated slit separation is the dominant term and no amount of care in the measurement reduces it.