A resistor obeys exactly

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Data analysis and model fitting in the laboratory · Data analysis and model fitting in the laboratory

Mathematical Methods for Physics, Ch Estimation, fitting and hypothesis testing

Problem

A resistor obeys V=IRV = IR exactly, with no offset. Its resistance RR is found from five voltage readings VjV_j taken at five known currents IjI_j. Each reading carries an independent Gaussian error of the same known width σ=0.2 V\sigma = 0.2\,\mathrm{V}. The data are IjI_j (A): 1.0,2.0,3.0,4.0,5.01.0, 2.0, 3.0, 4.0, 5.0; VjV_j (V): 4.9,10.3,14.8,20.1,24.94.9, 10.3, 14.8, 20.1, 24.9. ((a)) Write the likelihood of the five voltage measurements as a function of RR, and show that maximising ln⁡L\ln\mathcal{L} is equivalent to minimising χ2=∑j(Vj−IjR)2/σ2\chi^2 = \sum_j (V_j - I_jR)^2/\sigma^2. Derive the maximum-likelihood estimator R^\hat{R}. ((b)) Evaluate R^\hat{R} for the data above and quote it with its standard uncertainty. ((c)) State the condition on the errors under which the derivation in part (a) fails, and give one concrete measurement situation where it does.

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