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

For two equal lenses with a virtual intermediate object, use positive distances for real objects on a lens's incident side and real images on its outgoing side. The first lens gives $1/v_1=1/f-1/u_1=1/30-1/60$, hence $v_1=60\mathrm{cm}$. At the second lens these rays are already converging toward a point $50\mathrm{cm}$ to its right: its object distance is therefore $u_2=-50\mathrm{cm}$.Thus the final image is real, $28.75\mathrm{cm}$ to the right of lens 1. Its transverse magnification is $(-60/60)(-18.75/[-50])=-3/8$: it is inverted. The intermediate focus is not a real object placed before lens 2.

For prism sweep under different rotation conventions, let the apex angle be $A=60^{\circ }$. Snell's law at both faces givesHere $e$ is measured from the second face's outward normal, whereas $D$ is the deviation from the incident direction. The two settings giveFor a fixed laser and rotating prism, the outgoing beam therefore sweepsIf instead the prism stays fixed and the input beam is steered, its exit-face normal stays fixed and the answer is $|\Delta e|=8.3325^{\circ }$. The source does not explicitly say which component moves, so the laboratory-angle convention matters. Both settings transmit through the second face; total internal reflection does not occur.

Apply Ampère's law on a concentric circle. Uniform current density is $J=I/(\pi a^2)$, so the enclosed current is $I{r}^2/a^2$ inside and $I$ outside. With permeability ${\mu }_0$,The field circulates azimuthally by the right-hand rule. Its magnitude rises linearly from zero, is continuous and maximal at the surface, then falls as $1/r$. End effects are neglected.

In SI base dimensions,Equating current, time, mass and length powers respectively yieldsThusDimensional analysis cannot determine the dimensionless proportionality constant.

For a proper lifetime inferred from a delayed decay flash, the measured interval ends when the light returns to the detector, not when the muon decays. Put passage at $(t,x)=(0,0)$ in the detector frame. If decay occurs at $t_d$, its distance is $v{t}_d$, and the backwards photon takes another $v{t}_d/c$ to arrive. ThereforeThe muon's proper elapsed time isApplying time dilation directly to the 10 ns reception interval would transform the wrong pair of events.

For background-subtracted radiative temperature uncertainty, assume the two quoted precisions are independent standard uncertainties, and treat the radiative proportionality factor as known. Write $P_B=B$ and $P_T=10B$. Then $P_S=9B$, while both absolute input uncertainties equal $0.5B$. ConsequentlyThus the temperature precision is about $1.96\%$, or $2\%$. Subtraction amplifies fractional uncertainty even though $P_T$ itself is precise.

Independence is a measurement assumption, not a consequence of the two percentages. For correlation coefficient $r_c$, the result is ${\sigma }_T/T=\sqrt{2(1-r_c)}/72$. If the percentages are strict error bounds combined pessimistically, the corresponding bound is $2.78\%$. Unknown calibration correlations cannot be inferred from the source.