At fixed electrical output, improving efficiency from ${\eta }_0$ to ${\eta }_1$ reduces fuel and associated combustion emissions by $1-{\eta }_0/{\eta }_1$. A change from 0.33 to 0.45 saves about 27%. If an illustrative large coal fleet emits 3.6 Gt annually, that is nearly 1 Gt, exceeding a hypothetical 0.5 Gt national total and far exceeding a 0.1 Gt electricity-sector total. Thus the comparison can be true on reasonable scale assumptions. It is not established by the words “state of the art”: fleet sizes, existing efficiencies, feasible retrofit fractions and whether UK electricity or all UK energy is meant must be specified. Continued growth and rebound can offset technical savings, and the options are not mutually exclusive.

A constant renewable contribution can come from geothermal resources, some hydro schemes and sustainably supplied biomass; variable resources can also contribute to firm supply through geographic diversity, storage, interconnection and demand management. These examples refute the absolute “never.” They do not show that every renewable mix can economically supply an unchanged constant demand without additional resources. Baseload is a load/system service rather than an intrinsic virtue of one generator. Long low-wind periods and seasonal storage require energy-capacity accounting, while dispatchable clean systems may make an inflexible minimum-output plant less desirable.

For a generous illustrative roof area 100 m${}^2$, annual rain depth one metre and five-metre usable fall, maximum potential energy is $\rho A{h}_{\mathrm{rain}}gH\simeq 4.9$ MJ, only 1.36 kWh per year before conversion losses. Even 25 million roofs give only about 34 GWh annually, around $10^{-4}$ of a 300 TWh electricity system. Tiny intermittent flows and installation embodied energy make the practical result worse. Household roof-rain generation is therefore a very small carbon-saving resource under realistic domestic heights; the small gravitational drop and water flux explain why, without needing to assume an inefficient turbine.

Electricity is a strong early decarbonization target because large stationary sources are comparatively accessible and a low-carbon grid enables cleaner vehicles, heat pumps and industry. For example a heat pump with coefficient of performance three divides electricity-related heat emissions by three per unit useful heat. But grid decarbonization does not automatically outrank every efficiency, methane, land-use or industrial intervention. Marginal abatement cost, cumulative emissions, construction time and available technology determine priorities. A very coal-intensive grid makes electrification alone less effective; a clean grid does not remove aviation or process emissions. The statement is a plausible strategic emphasis, not a deduction valid for every country and timescale.

“Never” cannot be established by one present price comparison. A useful comparison is levelized cost with common financing, capacity and system boundaries. For illustration, solar capital cost 1000 currency units/kW, annual capital-plus-maintenance charge 8%, and capacity factor 0.15 imply about 61 units/MWh. A coal plant with fuel, maintenance and capital costs of the same order is not separated by an immutable physical barrier. These are illustrative inputs, not current price quotations. Carbon costs and air pollution can worsen coal's comparison; intermittency, land and balancing costs can worsen solar's. Compare delivered services as well as panel-level energy. The defensible conclusion is contingent competition, not a universal impossibility.

Hydropower can convert a renewable water flux efficiently, has low operating fuel requirements, and reservoir schemes can provide rapid dispatch, inertia and storage. Long plant lifetimes can make capital expenditure worthwhile, while pumped storage shifts energy rather than producing a new primary resource.

Limitations include finite favorable sites, seasonal drought, sedimentation, habitat fragmentation, blocked fish migration, displaced populations and major civil-engineering/failure risks. Reservoir evaporation and methane emissions can matter, especially in warm flooded landscapes, so life-cycle emissions are not uniformly zero. Run-of-river plants avoid some reservoir impacts but offer less controllable output. A project assessment must compare useful annual energy and firm capacity, ecological and social costs, and the counterfactual electricity system.

A Pelton wheel splits and turns a free jet through nearly 180 degrees in moving buckets, transferring momentum to the runner. For ideal bucket speed $V$, the absolute exit speed is $2V-u$ and force is $2\rho Q(u-V)$. Power $2\rho QV(u-V)$ is maximized at $V=u/2$, giving $P_{\max }=\rho Q{u}^2/2$. If the same inflow must now pass a slit of height $d$, continuity gives $u=Q/d$. Using the preceding flow rate with $H=y_{\min }$,The head must actually be sufficient to supply this jet speed: $u^2/(2g)$ cannot exceed the available hydraulic drop. An arbitrarily small slit cannot produce infinite power from a fixed-height reservoir; it would reduce discharge or raise water until another outlet, including overtopping, operates. The printed scaling presupposes compatible head and fixed-flow conditions.

Neglect upstream speed and hydraulic losses. If $H$ is the upstream free-surface height above the crest, the specific energy at the crest is $H=y+v^2/(2g)$ and flow per unit width is $Q=yv$. At the critical broad-crested control section, extremizing $Q^2=2g{y}^2(H-y)$ gives $y=2H/3$ and $v^2=gy$. ThereforeFinite approach speed, friction and contraction introduce measured discharge coefficients; the ideal result assumes a long enough crest for the near-hydrostatic critical-flow approximation.

Mean kinetic energy $3k_{B}T/2=12.9$ keV means $T=8.6$ keV, not 12.9 keV. For fully ionized D-T with equal electron and ion temperatures, $p=2n_i{k}_{B}T$ gives $n_i\simeq 4.90\times 10^{19}$ m${}^{-3}$. Thusonly about one third of the stated requirement. The specified operating point therefore does not meet that criterion and requires improved confinement/pressure or continuing external heating. If the criterion's $n$ counts all thermal particles instead of ions, the product doubles but remains below its given threshold. A precise fusion gain or duty cycle cannot be inferred from the triple product alone.

Extraction energy is $6000\mathrm{kg}\times 2500\mathrm{kWh}\mathrm{k}{\mathrm{g}}^{-1}=15$ GWh per year. A continuously operating 2 GW electrical plant generates 17.5 TWh per year, so lithium extraction consumes about $\boxed{0.086\%}$ of its output under the stated estimate. This is energetically favorable even before claiming anything about capital costs or extraction feasibility.

For an ocean mass about $1.4\times 10^{21}$ kg, concentration 0.17 ppm implies about $2.4\times 10^{14}$ kg lithium. At the specified consumption/output ratio, a hypothetical 20 TW electric world would use $6\times 10^7$ kg annually, giving roughly $4\times 10^6$ years. For 3 TW the corresponding scale is $3\times 10^7$ years. State the assumed demand: “power the world” does not uniquely specify electric demand versus primary-energy use. These are inventory quotients ignoring growth, accessible fraction, ecological constraints and processing throughput.

Lithium in the blanket breeds tritium through neutron reactions, notably ${}^6\mathrm{Li}+n\to {}^4\mathrm{He}+T$. Tritium is not abundant enough in nature to supply a large reactor fleet without breeding. The blanket must also transfer heat and manage neutron damage and losses, and the breeding ratio must exceed consumption after accounting for processing and inventory. Lithium is therefore a fuel-cycle resource rather than the primary charged plasma reactant in the D-T core.

A tokamak confines a hot, approximately toroidal plasma using a strong toroidal magnetic field and a poloidal field associated with plasma current and external coils. Helical field lines and suitable magnetic surfaces reduce rapid transverse escape. D-T fusion releases 17.6 MeV, mostly in an uncharged neutron that escapes magnetic confinement and transfers energy to a surrounding blanket. A heat exchanger and thermal power cycle convert some of that energy into electricity.

Essential systems include the vacuum vessel, toroidal and poloidal magnets, current-drive/transformer and auxiliary heating equipment, fueling and exhaust systems, divertor for concentrated heat and particle removal, neutron-absorbing/breeding blanket and shielding. Alpha particles retained in the plasma contribute self-heating. Confinement and controlled exhaust are as important as reaching a high temperature; a magnetic field does not confine the neutrons or make material heat loads vanish.

The relative orbital speed is $Z{e}^2/(4\pi {ϵ}_0\hslash n)$. Divide the circumference by this speed to findPositronium has twice the hydrogen period at the same principal quantum number. The symbol $n$ here labels principal levels with $n=1$ the ground state; if “nth excited state” is read literally as excitation count, replace $n$ by $n+1$.

Coulomb centripetal balance and $\mu vr=n\hslash$ giveThe radius is the electron-nucleus separation. For positronium, $Z=1$ and $\mu =m_e/2$, so $r_n=2a_0{n}^2$ and $E_n=-\mathrm{Ry}/(2n^2)$. Each constituent's radius about the center of mass is half the separation; calling both that radius and $r_n$ the orbital radius without distinction would create another factor-two error.

A classically accelerating Coulomb electron radiates and loses orbital energy, so a stable nonradiating orbit is impossible within that unmodified theory. Bohr instead postulates stationary nonradiating orbits, quantized angular momentum $\mu vr=n\hslash$ and photon emission/absorption with $h\nu =E_i-E_f$. Here $\mu =m_{e}M/(m_e+M)$ describes relative motion. The angular-momentum condition is the standing-wave relation $\boxed{2\pi r=n{\lambda }_{\mathrm{dB}}}$ for the relative orbit. These are additional quantum postulates, not consequences of classical radiation theory.

A heated cathode emits electrons into low-pressure atomic vapor, and a variable accelerating voltage gives them kinetic energy. A grid and small retarding region precede the collecting anode. Below the first excitation threshold most collisions are elastic; above it, electrons can transfer a definite excitation energy to atoms and may then fail to overcome the collector's retarding field. Collector current develops repeated minima as accelerating voltage increases, separated by roughly the same excitation voltage, approximately 4.9 V for mercury. Contact potentials and energy spreads shift or smear the minima without removing their characteristic spacing. A smooth loss mechanism would not produce these repeated fixed-energy thresholds.^[ref]

Time evolution gives $e^{-i{E}_{0}t/\hslash }[{\varphi }_0+e^{-i\omega t}{\varphi }_1]/\sqrt{2}$. It becomes the opposite relative-sign superposition when $e^{-i\omega t}=-1$, first at $\boxed{t_{\min }=\pi /\omega }$. The overlap with the initial state is proportional to $1+e^{-i\omega t}$ and vanishes then; global phase is immaterial.

Both probability densities are even, so $⟨x⟩=0$. The supplied Gaussian moments give $⟨x^2{⟩}_0={\sigma }^2/2$ and $⟨x^2{⟩}_1=3{\sigma }^2/2$. Therefore $\boxed{\Delta x_0=\sigma /\sqrt{2},\Delta x_1=\sqrt{3/2}\sigma }$. The original plots mark these two symmetric rms intervals around the common mean.