Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section A

Let the common tension be T, the linear mass densities be ${\mu }_1<{\mu }_2$ and the speeds be $c_j=\sqrt{T/{\mu }_j}$, so $c_2<c_1$. Displacement and transverse-force continuity give the displacement coefficients at a string junctionThe reflected triangular pulse is inverted, has smaller height $|r|H$, unchanged spatial width and unchanged speed, and travels left. Its left-to-right asymmetric profile is reversed as well as vertically inverted. The transmitted pulse remains upright, has smaller height tH, travels more slowly, and has width $W{c}_2/c_1$. The string-pulse width mapping at a junction leaves its temporal duration unchanged. The transmitted slopes are steeper by $2c_1/(c_1+c_2)$; the reflected slope magnitudes are smaller by $|r|$.

The density ratio is unspecified, so only these inequalities are determined. The original plot below illustrates ${\mu }_2/{\mu }_1=4$, giving $r=-1/3$, $t=2/3$ and half-width transmission; it is not a numerical reading of the examination figure.

For the adjacent reflected minima of an air-film-air layer in thin-film interference, reflection at the front surface acquires a phase reversal, while reflection at the rear does not. Near normal incidence the reflected minima therefore obey $2nt=m\lambda$ for integral m. A single intermediate maximum means that the two stated minima are consecutive orders:Thus $2nt=3000\mathrm{nm}$, with orders6 and5, andThe intermediate reflected maximum is at $\lambda =3000/5.5\simeq 545\mathrm{nm}$. This assumes negligible dispersion of the supplied refractive index and an air environment on both sides. ^[ref]

The angular separation is $\Delta \theta =10^{11}/(2\times 10^{17})=5.00\times 10^{-7}\mathrm{rad}$. Each star's angular diameter is below $5\times 10^{-9}\mathrm{rad}$, so treating it as a point source is justified at these diffraction scales.

At300nm the Airy disk first-zero radius is $1.22\lambda /D=1.525\times 10^{-7}\mathrm{rad}$. The separation is3.28 times this radius: two well-separated bright peaks occur. The stars are mutually incoherent, so their point spread functions add as intensities, not amplitudes:The value at u=0 is defined by continuity. Faint diffraction rings remain on either side of each principal peak. ^[ref]

At1000nm the first-zero radius is $5.083\times 10^{-7}\mathrm{rad}$, slightly larger than the $5.00\times 10^{-7}\mathrm{rad}$ separation. The images therefore overlap strongly and just fail the conventional Rayleigh criterion, rather than forming two clean isolated peaks.

The summed Airy profile can still have a shallow central dip and two local maxima: Rayleigh's criterion is a resolution convention, not the exact condition for a profile to become single-peaked. The two-panel original plot in part(a) shows the correct intensity sums at both wavelengths. ^[ref]

For hard spheres of radius r in a face-centered cubic lattice, contact occurs along a face diagonal: $4r=\sqrt{2}a$. Four spheres belong to a conventional cell, so the hard-sphere cubic packing fractions giveFor a body-centered cubic lattice, contact occurs along a body diagonal, $4r=\sqrt{3}a$, and there are two spheres per conventional cell:Their ratio $8/(3\sqrt{6})=1.0887>1$ proves that FCC packs more densely. Coordination alone would not establish the occupied volume fraction.

For a classical ideal monatomic gas, $C_{V,\mathrm{He}}=3N{k}_B/2$ and $C_{p,\mathrm{He}}=5N{k}_B/2$. Treat copper as having one conduction electron per atom. Its degenerate-electron contribution isConsequently the electron-only estimates at5K areThe constant-pressure gas versus electronic metal heat capacity convention makes the second numerator appropriate to a constant-pressure gas measurement. For the solid, the low-temperature difference between $C_p$ and $C_V$ is usually small, but its phonon contribution is not fixed by $E_F$. In a Debye model,The supplied $E_F$ alone therefore does not determine the ratio to copper's total heat capacity; the electronic-only result is conditional.^[ref] An estimate using additional material data is possible: the specific-heat fit in Material Properties: OFHC Copper by National Institute of Standards and Technology gives $c_{p,\mathrm{Cu}}(5\mathrm{K})\simeq 0.155\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. Using copper's rounded molar mass $0.0635\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ gives $C_{p,\mathrm{Cu}}\simeq 9.85\times 10^{-3}\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, and henceThis supplementary estimate includes copper's lattice contribution and retains the ideal-gas approximation for helium. The NIST fit reports a 10% deviation from its source data below 15 K; neither that fit nor the stated pressure alone supplies helium nonideality corrections.^[ref]