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In part B of the experiment a Michelson Interferometer is used to determine the wavelength λ of the laser. An interference pattern is observed due to the light being split into two paths with slightly different lengths.A vacuum chamber (of length d) is placed in one of the paths and the air pumped out. Then as air leaks back into the chamber the number of fringes, N, that pass a certain point are counted as the interference pattern changes.The following measurements were taken:N= 64 ± 5d= (0.077 ± 0.006) mThe wavelength λ of the laser is calculated to be 650 nm (using the value of the refractive index of air n given in the lab manual). Calculate the uncertainty of the wavelength λ of the laser in nm to the nearest whole number.

Question

In part B of the experiment a Michelson Interferometer is used to determine the wavelength λ of the laser. An interference pattern is observed due to the light being split into two paths with slightly different lengths.A vacuum chamber (of length d) is placed in one of the paths and the air pumped out. Then as air leaks back into the chamber the number of fringes, N, that pass a certain point are counted as the interference pattern changes.The following measurements were taken:N= 64 ± 5d= (0.077 ± 0.006) mThe wavelength λ of the laser is calculated to be 650 nm (using the value of the refractive index of air n given in the lab manual). Calculate the uncertainty of the wavelength λ of the laser in nm to the nearest whole number.

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Solution

To calculate the uncertainty of the wavelength λ of the laser, we need to use the formula for the propagation of uncertainty. The formula we will use is:

Δλ/λ = sqrt[(ΔN/N)² + (Δd/d)²]

where Δλ is the uncertainty in the wavelength, λ is the wavelength, ΔN is the uncertainty in the number of fringes, N is the number of fringes, Δd is the uncertainty in the length of the vacuum chamber, and d is the length of the vacuum chamber.

Substituting the given values into the formula, we get:

Δλ/λ = sqrt[(5/64)² + (0.006/0.077)²]

Calculate the values inside the square root:

Δλ/λ = sqrt[0.0061 + 0.0061]

Add the values:

Δλ/λ = sqrt[0.0122]

Take the square root:

Δλ/λ = 0.1105

To find the uncertainty in the wavelength, multiply this result by the wavelength:

Δλ = λ * 0.1105 = 650 nm * 0.1105 = 71.825 nm

Rounding to the nearest whole number, the uncertainty in the wavelength of the laser is 72 nm.

This problem has been solved

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