A laser's wavelength is found from the angle of the first-order maximum of a…

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Uncertainty in measurement

Measurement and Method, Ch Type A, Type B and the uncertainty budget

Problem

A laser's wavelength is found from the angle θ\theta of the first-order maximum of a diffraction grating of spacing dd, using λ=dsin⁡θ/m\lambda = d\sin\theta/m with order m=1m=1 exact. The angular scale has a possible zero-offset δθ0\delta_{\theta_0}, best estimate zero, so the model is λ=d sin⁡(θ−δθ0)\lambda = d\,\sin(\theta - \delta_{\theta_0}). Six independent readings of θ\theta give a mean θˉ=19.47∘\bar\theta = 19.47^\circ with sample standard deviation s=0.05∘s = 0.05^\circ. Repeated checks of the scale against a level judge δθ0\delta_{\theta_0} to lie within ±0.03∘\pm0.03^\circ of zero, with nothing favouring any part of that range. The grating's certificate states its spacing is d=1666.7 nmd = 1666.7\,\mathrm{nm}, correct to within ±0.3 nm\pm0.3\,\mathrm{nm}, with nothing favouring any part of that range. ((a)) Evaluate the standard uncertainty of θ\theta from the two angle-related rows (state type and distribution for each) and combine them, converting to radians. ((b)) Evaluate u(d)u(d). ((c)) Take λ=dsin⁡θ\lambda = d\sin\theta (the offset's best estimate is zero, so it does not shift the central value) and combine the two rows into uc(λ)u_{\rm c}(\lambda), quoting λ\lambda with its standard uncertainty. ((d)) State which row dominates and how it could best be reduced.

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