In part A of the experiment a pair of slits are illuminated with a laser and an interference pattern is observed. The slit spacing is d = (0.100 ± 0.003) mm and the pattern is projected on to the wall a distance L= (2.352 ± 0.019) m from the slits. From a dark spot 8 further dark spots are counted and the distance is measured to be Z = (12.2 ± 0.1) cm.The wavelength of the laser is calculated to be λ = 648 nm.Calculate the uncertainty in the wavelength. Express answer in nm (1x10-9m) to the nearest whole number.
Question
In part A of the experiment a pair of slits are illuminated with a laser and an interference pattern is observed. The slit spacing is d = (0.100 ± 0.003) mm and the pattern is projected on to the wall a distance L= (2.352 ± 0.019) m from the slits. From a dark spot 8 further dark spots are counted and the distance is measured to be Z = (12.2 ± 0.1) cm.The wavelength of the laser is calculated to be λ = 648 nm.Calculate the uncertainty in the wavelength. Express answer in nm (1x10-9m) to the nearest whole number.
Solution
To calculate the uncertainty in the wavelength, we first need to understand the formula used to calculate the wavelength in the double-slit experiment. The formula is:
λ = (d * Z) / (L * m)
where: λ is the wavelength, d is the slit spacing, Z is the distance from the central maximum to the mth dark spot, L is the distance from the slits to the screen, m is the order of the dark spot.
The uncertainty in a product or quotient is given by the sum of the relative uncertainties of the quantities involved. The relative uncertainty of a quantity X with uncertainty ΔX is given by (ΔX / X).
So, the relative uncertainty in λ is given by:
Δλ/λ = Δd/d + ΔZ/Z + ΔL/L
Substituting the given values:
Δλ/λ = (0.003 mm / 0.100 mm) + (0.1 cm / 12.2 cm) + (0.019 m / 2.352 m)
Calculate each of these fractions to get:
Δλ/λ = 0.03 + 0.00819672131 + 0.00807754443
Adding these up gives:
Δλ/λ = 0.04627426574
To find the absolute uncertainty Δλ, multiply this by the calculated wavelength:
Δλ = λ * Δλ/λ = 648 nm * 0.04627426574 = 29.969758 nm
Rounding to the nearest whole number gives an uncertainty in the wavelength of approximately 30 nm.
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