Past exam of the physics A course of the University of Cambridge / 2013 / Paper 1 / Section D / D13

A systematic error shifts results in a reproducible or correlated way relative to the measurand. Examples include zero offsets, gain miscalibration, nonlinearity, drift, geometrical misalignment, loading, background contamination and an inadequate response model. Repeated measurements under unchanged conditions can reduce independent noise without exposing such a shift; a small standard error is not proof of measurement accuracy.

Use traceable standards and independent measurement methods, test zero and span, reverse the physical signal where possible, and vary the conditions that could cause the bias. For common-offset cancellation by signal reversal, take measurements $y_{+}=S+b$ and $y_{-}=-S+b$. The half-difference estimates $S$ while the half-sum diagnoses an unchanged offset $b$. Drift between the two readings would violate that cancellation, so alternate or interleave measurements. Randomizing acquisition order separates slow drift from controlled parameter changes; blanks and calibration samples test backgrounds and instrument gain. Blinding helps reduce analysis choices that follow an expected answer.

Corrections have uncertainties and may be correlated across data. Include residual offsets and gains as nuisance parameters or propagate their calibration covariance, rather than treating a common calibration error as fresh independent noise for each datum. Document assumptions and check stability with controlled perturbations. These methods suppress or quantify specified systematic effects; they cannot guarantee elimination of every unknown bias.

The likelihood is $L(\theta ;d)=p(d\mid \theta )$ considered as a function of parameters at the observed data. It is not itself a probability distribution over $\theta$. Bayes' theorem givesThe prior represents parameter information before these data; the denominator normalizes the posterior and can compare fully specified models with compatible prior assumptions. Detector response, backgrounds, dependence and selection must be included in the data model.

For independent measurements $x_i\sim N(\theta ,{\sigma }_i^2)$ with known errors, $\log L=\mathrm{constant}-\frac{1}{2}{\sum }_i(x_i-\theta {)}^2/{\sigma }_i^2$. Maximum likelihood minimizes this chi-squared and gives $\hat{\theta }={\sum }_i{w}_i{x}_i/{\sum }_i{w}_i$, $w_i={\sigma }_i^{-2}$. A Gaussian prior $N({\mu }_0,s_0^2)$ instead yieldsThis explicitly combines data and prior precision, approaching the weighted mean when prior information is weak. Unknown calibration parameters should be jointly inferred and marginalized, not silently fixed. posterior intervals are conditional on the assumed model and prior; posterior predictive checks and sensitivity to plausible priors help expose poor fits and weakly identified parameters. The maximum-likelihood estimate and a Bayesian posterior answer different inferential questions even when their numerical centres agree.

temperature changes alter dimensions, resistances, refractive indices and offsets; gradients can generate thermoelectric voltages. Identify the dominant heat paths and the experiment's temperature coefficients before choosing insulation. Reduce convection with an enclosure or, where appropriate, vacuum; reduce conduction through supports and leads with suitable geometry and thermal anchoring; use thermal radiation shields and low-emissivity surfaces to reduce radiative exchange. vacuum alone does not remove heat conduction through wires or thermal radiation.

In first-order thermal isolation, a sample with heat capacity $C$ and thermal conductance $G$ to its surroundings obeysThermal mass and insulation suppress rapid fluctuations but only delay slow ambient drift: the zero-frequency gain remains one. Active temperature control, a stable reference environment, adequate equilibration time and an appropriate sensor location address long-term stability. Self-heating produces a steady offset $\delta T_s=\delta P/G$; increasing insulation can worsen that offset and settling time. Check temperature gradients rather than relying on a remote thermometer.

Match materials or use differential/reference measurements to reduce expansion and resistance drift; keep dissimilar-metal junctions isothermal to suppress thermoelectric offsets. Log temperature alongside the signal and deliberately vary environmental temperature to measure residual coupling, using a physically justified correction with uncertainty. Interleave calibrations and allow equilibration after changing power. Feedback must remain stable, and heater/sensor electronics should not introduce electrical interference into the measurement.