In part A of the experiment a hair is illuminated with a laser and a diffraction pattern is observed. The wavelength λ = (648 ± 24) nm and the pattern is projected on to the wall a distance L = (2.226 ± 0.006) m from the hair. The distance across the central maximum is measured to be 2y= (0.048 ± 0.003) m.The width of the hair is calculated to be a= 60.1 µmCalculate the uncertainty in the width of the hair Δa in µm, to one decimal place.
Question
In part A of the experiment a hair is illuminated with a laser and a diffraction pattern is observed. The wavelength λ = (648 ± 24) nm and the pattern is projected on to the wall a distance L = (2.226 ± 0.006) m from the hair. The distance across the central maximum is measured to be 2y= (0.048 ± 0.003) m.The width of the hair is calculated to be a= 60.1 µmCalculate the uncertainty in the width of the hair Δa in µm, to one decimal place.
Solution
To calculate the uncertainty in the width of the hair, we need to use the formula for the propagation of uncertainty. The formula for the width of the hair is given by:
a = λL / 2y
The uncertainty in a, Δa, is given by:
Δa = √[(∂a/∂λ * Δλ)² + (∂a/∂L * ΔL)² + (∂a/∂y * Δy)²]
where ∂a/∂λ, ∂a/∂L, and ∂a/∂y are the partial derivatives of a with respect to λ, L, and y, respectively.
First, we calculate the partial derivatives:
∂a/∂λ = L / 2y ∂a/∂L = λ / 2y ∂a/∂y = - λL / 2y²
Then, we substitute the values into the formula for Δa:
Δa = √[(L / 2y * Δλ)² + (λ / 2y * ΔL)² + (- λL / 2y² * Δy)²]
Substituting the given values:
Δa = √[((2.226 m / 0.048 m) * 24 nm)² + ((648 nm / 0.048 m) * 0.006 m)² + ((- 648 nm * 2.226 m / 0.048 m²) * 0.003 m)²]
After calculating the above expression, we get the uncertainty in the width of the hair Δa in µm, to one decimal place.
Similar Questions
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