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Use transverse coordinates $(x,x^{\prime })$ and define a positive signed focal length as focusing in that plane. A thin quadrupole magnet of length $\ell$ and signed gradient $G$ has integrated strength $1/f=|e|G\ell /p_0$; the electron charge and chosen field orientation are absorbed into the sign of $G$. Its beam transfer matrix isFor total electron energy $E_0$, $p_0=\sqrt{E_0^2-m_e^2{c}^4}/c$, so $f\simeq E_0/(|e|cG\ell )$ in the ultrarelativistic limit. The other transverse plane has the opposite signed focal length. A quadrupole magnet therefore focuses one plane while defocusing the other; the longitudinal magnetic field component is negligible in this paraxial idealization.

At fixed gradient and magnet length, $f\propto p$. If $E$ is total energy,Here ${\beta }_0=v_0/c$, not one of the Courant-Snyder parameters. For ultrarelativistic electrons this becomes $f\simeq f_0(1+\delta )$, or $1/f\simeq (1-\delta )/f_0$: higher-energy particles are bent and focused less strongly. If the symbol $E$ instead denotes kinetic energy $K$, the coefficient is $(K+m{c}^2)/(K+2m{c}^2)$, approaching $1/2$ in the nonrelativistic limit. Stating the energy convention avoids applying the ultrarelativistic result to a slow beam.

For an infinitesimal continuous symmetry $\delta q_i=\epsilon {\xi }_i(q,t)$, allow the Lagrangian to change by a total derivative, $\delta L=\epsilon dF/dt$. Direct variation givesOn a solution of the Euler-Lagrange equations, the first term vanishes. This proves Noether's theorem in the fixed-time form:Spatial translations yield total momentum, and rotations yield angular momentum. Time translations yield energy when the Lagrangian has no explicit time dependence; this follows either from the extended time-transformation form of the theorem or by differentiating $H={\sum }_i{p}_i{\dot{q}}_i-L$ to obtain $\dot{H}=-\partial L/\partial t$.

The central pair potential is invariant under simultaneous spatial translations, simultaneous rotations and time translations. The corresponding time-independent Noether charges are total momentum $P=p_1+p_2$, total angular momentum about a fixed origin, and total energy $H$.

In one dimension this supplies $P$ and $H$: two charges. In two dimensions it supplies $P_x,P_y,L_{xy},H$: four charges. In three dimensions it supplies three momentum components, three angular-momentum components and $H$: seven charges. For generic states these listed charges are independent in those dimensions. Galilean boosts can also be expressed by explicitly time-dependent conserved combinations, but are not part of the requested translation-rotation-energy count.

Source paper: www.whatdotheyknow.com/request/physics_past_exam_papers_2015_16/response/973997/attach/3/Cowell%20Attachment.zip?cookie_passthrough=1. The dated cover identifies Trinity Term 2016, 14 June. Unlettered marked requests are numbered in source order; explicit letters are retained. All optional Energy Studies alternatives are addressed. S18 is intentionally excluded from new solutions because complete official answers match that section.

Take electrostatic perturbations proportional to $e^{i(kz-\omega t)}$, with ions immobile on the fast electron time scale. Linearized Vlasov gives $f_1=-i(q{E}_1/m)(\partial f_0/\partial v_z)/(\omega -k{v}_z)$. Integrating over transverse velocities gives $F_0(v_z)=n_0\sqrt{m/(2\pi k_{B}T)}{e}^{-m{v}_z^2/(2k_{B}T)}$. Combine the perturbed charge density with Poisson's equation:Since $F_0^{\prime }=-m{v}_z{F}_0/(k_{B}T)$, this is preciselyThe pole selects particles moving at phase speed. A causal initial-value prescription specifies how the contour passes it and produces resonant energy exchange, including Landau damping for a decreasing Maxwellian. A bare real improper integral without that prescription is incomplete. The unclosed brace in the printed Maxwellian exponent is typographical; the normalized Gaussian above fixes its meaning.

At zero dispersion and $\alpha =0$, $\sigma =\sqrt{ϵ\beta }$ and ${\sigma }^{\prime }=\sqrt{ϵ/\beta }$. The results areThe repeated $D_x,D_y$ entries in the source are interpreted as intended vanishing dispersion and its slope. If dispersion slopes were not zero, energy spread would contribute to divergence and further data would be needed.

Lower energy changes the source spectrum as well as emittance. At fixed bending geometry, bending-radiation critical energy scales as ${\gamma }^3$, so it falls by $10^{3/2}$; for fixed undulator period and deflection parameter the fundamental wavelength grows as ${\gamma }^{-2}$, by ten. A smaller electron emittance therefore does not automatically improve brightness at the original 10 nm wavelength: the device may no longer radiate there on its fundamental, flux and harmonic operation change, and diffraction remains. Lattice redesign can retain beam energy and target spectrum while reducing emittance, which is why energy reduction alone is not a generally satisfactory substitute.

TODO — the source does not supply the longitudinal position-energy correlation, imposed chirp, or chicane dispersion/geometry needed for a unique compressed bunch length. A four-bend drawing and an uncorrelated energy spread are insufficient. In linear optics,If the given spread is uncorrelated, the minimum is the unchanged 0.2 m, not a compressed submillimetre bunch. If a chirp is supplied, optimal $R_{56}$ depends on it and on the uncorrelated residual spread.

A sometimes intended phase-space lower-bound estimate conserves longitudinal emittance and permits final relative energy spread of order unity, giving ${\sigma }_{zf}\gtrsim {\sigma }_z{\sigma }_{\delta }\sim 20\mu$m. This is an additional extreme energy-spread allowance, not a consequence of an unspecified magnetic chicane. If one illustratively matches a plasma scale $k_p{\sigma }_{zf}\sim 1$, then $n_e\sim {ϵ}_0{m}_e{c}^2/(e^2{\sigma }_{zf}^2)\simeq 7\times 10^{22}$ m${}^{-3}$. Matching to a whole wavelength instead changes this by $4{\pi }^2$. The parametric density relation is valid; the unique requested numerical compression is not established by the source.

The required energy increment is 9.9 TeV. If the usable longitudinal gradient is $\eta E_0$, with $E_0=m_{e}c{\omega }_p/e$, the channel length isUnder the illustrative preceding ${\sigma }_{zf}=20\mu$m and $k_p{\sigma }_{zf}=1$ choices, $E_0\simeq 25.6$ GV m${}^{-1}$ and $L_p\simeq 390/\eta$ m. TODO — a unique numerical length inherits the missing compression/density specification and additionally requires a usable wake amplitude/loading factor $\eta$ or driver information. A driver energy per proton alone does not fix its bunch population or accelerating gradient.

For a 1 micrometre laser, $n_c\simeq 1.1\times 10^{27}$ m${}^{-3}$ and a cold underdense wave has ${\gamma }_{\varphi }^2\simeq n_c/n_e$. At peak wave scale $E_0$, optimal passage through an accelerating half-cycle gives ideal stage gain $\Delta \mathcal{E}\sim 4(n_c/n_e)m_e{c}^2$; crest-to-zero injection gives half that. With the illustrative $n_e\simeq 7\times 10^{22}$ m${}^{-3}$, the optimal ideal gain is about 32 GeV and a 9.9 TeV increment needs order 300 stages, or order 600 with the crest convention.

TODO — no unique stage count follows without the missing density and laser/wake amplitude assumptions. Depletion, diffraction, guiding, beam loading, injection, energy spread, driver evolution and interstage losses can all increase or otherwise alter the requirement. The parametric count is $N\gtrsim \Delta {\mathcal{E}}_{\mathrm{total}}/[4\eta (n_c/n_e)m_e{c}^2]$, with the phase and field conventions explicitly retained.

Hydrostatic balance is $dP/dr=-GM(r)\rho /r^2$. A free-fall estimate gives $t_{\mathrm{dyn}}\sim \sqrt{R_{\odot }^3/(G{M}_{\odot })}\simeq 1.6\times 10^3$ s; a test particle falling in a fixed point-mass field adds factor $\pi /(2\sqrt{2})$, giving about 30 minutes.

For an ideal nonrelativistic gas with negligible boundary pressure, the virial theorem is $2U+\Omega =0$, so total energy is $\Omega /2$. Since $\Omega =-{\int }_0^{M}Gmdm/r(m)$ and $r\le R$, $|\Omega |\ge G{M}^2/(2R)$. This is a minimum magnitude, not a lower bound on the negative energy: unrestricted central concentration can make the latter arbitrarily negative. A uniform-density estimate gives $\Omega =-3G{M}^2/(5R)$ and available contraction energy $3G{M}^2/(10R)$. Its release at present solar luminosity lasts $\boxed{t\sim 10^7\mathrm{yr}}$ (order $10^7$–$10^8$ years for alternative structural estimates), much less than a nuclear-burning lifetime.

Using the supplied atomic masses consistently includes the net electron bookkeeping after positron annihilation. The released fraction is $\eta =(4m_H-m_{He})/(4m_H)\simeq 0.0071$, corresponding to about 26.7 MeV per helium nucleus. If 10% of the solar mass is processed and luminosity stays constant, $t\simeq 0.1\eta M_{\odot }{c}^2/L_{\odot }\simeq 10^{10}$ years.

At separation one femtometre the proton Coulomb energy is 1.44 MeV. At $2\times 10^7$ K, mean thermal kinetic energy $3k_{B}T/2\simeq 2.59$ keV, making their ratio about $5.6\times 10^2$. Classical over-barrier collisions are therefore extremely improbable. Quantum tunnelling through the Coulomb barrier and the high-energy distribution tail allow fusion, with the slow weak-interaction step regulating the proton-proton chain. The reaction summary is a net energy/matter balance, not one simultaneous four-proton collision.

Radiative equilibrium balances absorbed solar power against emitted thermal power, with no net energy accumulation. A spherical body's projected area absorbs $\pi r^2{F}_S(1-A)/R_{AU}^2$; its full area emits $4\pi r^2ϵ\sigma T^4$. HenceHere $A$ is bolometric Bond albedo, not a geometric albedo measured at one phase or wavelength. The radius cancels because the absorption and emission areas scale together.

Jupiter emits at about 130 K rather than the solar-only estimate around 111 K. Intrinsic heat contributes additional outgoing flux, so $\sigma T_{\mathrm{emit}}^4=F_{\mathrm{absorbed}}+F_{\mathrm{internal}}$. The difference of fourth powers is the relevant energy excess, not their direct temperature difference. Cooling and contraction of the interior provide the main physical motivation for including this extra term.

Io's volcanic structures and young resurfaced regions indicate strong ongoing geological activity, despite a low sunlight-set equilibrium temperature. Tidal dissipation forced by its orbital interactions supplies interior energy. Europa's fractured relatively lightly cratered ice indicates resurfacing and a mechanically active ice shell on geological timescales, with tidal heating a plausible contributor; surface appearance alone does not prove every fracture is currently opening. Ganymede has both older cratered regions and grooved younger terrain, indicating a more varied resurfacing history; primordial/radiogenic heat and past tides can matter. Callisto's heavily cratered surface suggests much less extensive recent resurfacing, without proving that all interior activity or subsurface liquid is absent.

The comparison is between surface ages and possible energy sources, not simply “warmer means active.” Surface irradiation is too cold to explain silicate volcanism, while internal heat, tidal work and material-dependent melting temperatures can sustain processes below a cold surface. The NASA images in the source are evidence to interpret; no exam illustration is reproduced.

In a gravity-dominated excavation model, displaced mass scales as $\rho D^3$ and lifting it through a distance of order $D$ costs order $\rho g{D}^4$. If a comparable fraction of impact energy supplies this work across similar impacts, $\boxed{D\propto E_{\mathrm{impact}}^{1/4}}$. This is a scaling justification, not a universal crater law: strength-dominated small craters or varying coupling efficiencies can have different exponents.

At fixed projectile mass and the assumed common quarter-power law, changing diameter from 30 to 75 m increases energy by $(75/30{)}^4$ and speed by $(75/30{)}^2=6.25$. The corresponding primary impact speed is about 15.6 km s${}^{-1}$. Five percent transferred to secondary-forming ejecta gives $v_{ej}\simeq 781$ m s${}^{-1}$. ThereforeThis assumes a 45-degree launch and similar target properties/coupling, as well as the source's fixed mass and velocity-transfer rules. Such a range is a noticeable fraction of the lunar radius, so curvature would produce a correction; the supplied flat-surface formula is an estimate rather than exact lunar orbit dynamics.

Blackbody radiation defeated classical equipartition because it predicts excessive short-wavelength emission. Quantized oscillator energy exchange $h\nu$, and ultimately the photon occupation law, gives the Planck spectrum. In the photoelectric effect, electron energy follows light frequency with a threshold, rather than simply increasing with beam intensity; individual photons supply $h\nu$ and the maximum kinetic energy is $h\nu -W$. Intensity chiefly changes the event rate at a fixed frequency. These explain two distinct failures of a purely continuous classical wave-plus-matter model, while retaining wave propagation and interference.

To distinguish $2^b$ equal transverse bins across width $w$, $\boxed{d=w/2^b}$. Six bits gives one-nanometre pixels; ten gives 0.0625 nm, below the helium wavelength. In the stipulated fixed-energy freely propagating beam, the latter entails transverse diffraction momenta comparable to or exceeding the available total momentum and lies outside the paraxial aperture model. Six-bit localization is less demanding, although fine detector resolution and losses remain practical constraints. This is not a universal theorem that an atom can carry at most six bits in every encoding: accessible modes, energy, aperture, measurements and success probabilities define the channel.