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 ± 0.02) m from the slits. From a dark spot 8 further dark spots are counted and the distance is measured to be Z = (12.2 ± 0.4) 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 ± 0.02) m from the slits. From a dark spot 8 further dark spots are counted and the distance is measured to be Z = (12.2 ± 0.4) 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 that relates the slit spacing (d), the distance to the wall (L), the distance between dark spots (Z), and the wavelength (λ). This formula is given by:
λ = d * Z / (L * m)
where m is the number of dark spots counted. In this case, m = 8.
The uncertainty in a calculated quantity can be found by taking the square root of the sum of the squares of the relative uncertainties of the measured quantities. The relative uncertainty of a measured quantity is the absolute uncertainty divided by the measured value.
So, the relative uncertainties are:
- For d: 0.003 mm / 0.100 mm = 0.03
- For L: 0.02 m / 2.352 m = 0.0085
- For Z: 0.4 cm / 12.2 cm = 0.0328
The total relative uncertainty is then:
sqrt((0.03)^2 + (0.0085)^2 + (0.0328)^2) = 0.045
To find the absolute uncertainty in the wavelength, we multiply the relative uncertainty by the calculated wavelength:
0.045 * 648 nm = 29.16 nm
Rounding to the nearest whole number, the uncertainty in the wavelength is 29 nm.
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