Past exam of the physics A course of the University of Cambridge / 2012 / Paper 1 / Section D / D12

A chi-squared test compares discrepancies with their expected random size. For independent Gaussian observations with known standard deviations, a fitted model givesUnder a correct regular model with $p$ fitted parameters, its reference statistical degrees of freedom are usually $\nu =n-p$. Correlated errors instead require ${\chi }^2={\mathbf{r}}^T{C}^{-1}\mathbf{r}$ with the correct covariance matrix. The reduced statistic has mean about one and standard deviation $\sqrt{2/\nu }$, so it should not be forced to equal exactly one.

For binned counts, Pearson's statistic is ${\sum }_i(O_i-E_i{)}^2/E_i$. With a fixed total and $k$ populated categories, fitting $p$ distribution parameters generally gives $\nu =k-1-p$. The approximation needs adequately populated expected bins and the appropriate dependence assumptions. Sparse data may require a Poisson likelihood, exact calculation or simulated reference distribution rather than a casual chi-squared approximation.

The upper-tail probability $p_{\mathrm{value}}=P({\chi }_{\nu }^2\ge {\chi }_{\mathrm{obs}}^2)$ measures how unusual the discrepancy would be under the null model; it is not the probability that the model is true. A large statistic can indicate a wrong model, underestimated noise or unmodelled systematic effects; an unusually small one can indicate overestimated errors or correlations. Residual plots and checks of the noise model are needed alongside the numerical test.

For an experimental-noise interpretation of Nyquist's theorem, a resistor $R$ in thermal equilibrium at temperature $T$ has one-sided open-circuit voltage-noise spectral densityin the classical range $hf\ll k_{B}T$, over an approximately rectangular bandwidth $\Delta f$. Across a matched load, half the open-circuit voltage appears, giving available noise power $k_{B}T\Delta f$. This is thermal agitation, present without a mean current, and is distinct from shot noise. A real readout weights the spectrum with its transfer function: $⟨v_{\mathrm{out}}^2⟩={\int }_0^{\infty }{S}_V(f)|H(f){|}^{2}df$. Cooling, reduced bandwidth and suitable impedance/noise matching reduce measured noise; the resistor's noise does not simply disappear in an ideal amplifier model.^[ref]

The bare name also has a sampling interpretation. The Nyquist–Shannon sampling theorem permits exact reconstruction of a bandlimited signal from sufficiently fast uniform samples. The practical condition is $f_s>2f_{\max }$, with an anti-aliasing filter before sampling; otherwise distinct continuous frequencies produce identical samples. Ideal reconstruction uses shifted sinc functions. Critical-rate equality needs an appropriate signal class and can fail for unrestricted edge-frequency sinusoids with unknown phase. Sampling fast enough prevents aliasing, but does not remove thermal noise. This distinction covers the source's unspecified use of “Nyquist's theorem”.^[ref]

A closed conducting enclosure redistributes surface charge so that an empty interior is protected from an external static electric field. Grounding fixes a reference and drains deposited charge, but an ideal isolated closed conductor also provides inward electrostatic shielding. Real seams, apertures and cables need careful bonding, shielding and feedthrough filtering. For time-varying fields the relevant wavelength, conductivity, geometry and penetration depth matter; a static argument alone is insufficient. Differential wiring and small pickup-loop areas can complement shielding, while careless multiple ground connections create unwanted current loops.

Ordinary copper or aluminium does not generally exclude a static magnetic field. High-permeability shells instead guide magnetic flux around the protected volume. Their effectiveness depends on shape, continuity, openings and saturation; magnetic history and mechanical stress can worsen performance. Nested shields and degaussing can reduce residual fields. Superconducting shielding can exploit the Meissner effect, but trapped flux, openings and critical-field limits require attention.

At nonzero frequency, eddy currents in conducting shields oppose changing magnetic flux. The skin depth isfor a good conductor in the usual diffusion approximation. Thickness of several skin depths suppresses penetration, but the very-low-frequency limit is weak for a normal metal. High-permeability and conducting layers can therefore complement each other. Active compensation coils are an additional method, not passive shielding. Test performance by applying known fields along relevant axes and measuring the residual field or induced signal over the apparatus bandwidth; room-temperature shielding materials and electronics can themselves add noise.