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For an incident beam uniform across its relevant impact parameters, an incident annulus has area $d\sigma =2\pi b|db|$. Rotational symmetry maps it to outgoing solid angle $d\Omega =2\pi \sin \theta d\theta$. The central-potential classical differential cross-section and the Rutherford impact-parameter relation therefore giveThis is a differential scattering cross-section, with units of area per steradian, conventionally ${\mathrm{m}}^2\mathrm{s}{\mathrm{r}}^{-1}$, not a detector count and not an area per degree. It applies for $0<\theta <\pi$, with a finite limiting value at $\pi$. Doubling incident kinetic energy divides it by four at fixed angle; doubling either electric charge number multiplies it by four.

A useful independent check integrates over all azimuths between two positive angles. For $0<{\theta }_1<{\theta }_2\le \pi$,This equals the incident annulus area $\pi (b_1^2-b_2^2)$ directly. A detector covering only a small patch at one angle does not collect this full azimuthal band.

The Rutherford model places a concentrated central electric charge inside a much larger chemical atom. In 1911, Ernest Rutherford argued that rare large deflections were predominantly single close encounters, rather than accumulated small deviations in a diffuse-charge atom. He treated the central electric charge as positive for calculation while noting that the scattering deductions did not establish its sign. He explicitly deferred atomic stability.^[ref]

The physical inference is a compact region generating a strong electric field, not a miniature hard wall. Most impact parameters miss that region, while a small subset give large deflections. The model supplies a way to infer interior structure from outgoing angles, a central use of scattering throughout atomic physics and nuclear physics. This inference does not by itself specify electron quantum states or explain spectral lines.

The repulsive model also sets a probing scale. At closest approach, conservation of energy with zero radial velocity givesFor the numerical model above, even a head-on projectile turns at $r_{\min }=2a=45.504\mathrm{fm}$. This is a calculated closest-approach distance, not a measured atomic nucleus radius. A point-charge fit can constrain how compact the scattering centre must be, but cannot directly resolve structure at distances the projectiles never reach.

The Rutherford differential cross-section must not be extrapolated to zero angle: the unscreened point model has a forward divergence, while real neutral chemical atoms exhibit the screening effect, finite beam acceptance, and accumulated small deflections. At very short separation, finite nuclear size and the nuclear force can change the interaction. Nuclear recoil changes the relation between laboratory and centre-of-mass angles for lighter targets; appreciable energy loss or repeated large deflections invalidates the simple thin-foil scattering counts. These limits delimit the calculation rather than supplying evidence for a nuclear contact collision in the early experiment.

Rutherford scattering is elastic scattering by a Coulomb potential. In the repulsive classical case, a positively charged projectile bends around a positively charged atomic nucleus. A close approach can turn it backwards without a contact collision. The calculation connects an invisible impact parameter with a measurable outgoing angle, then with a differential scattering cross-section and a detector count.

Assume a projectile of mass $m$ and electric charge $ze$, a stationary point atomic nucleus of electric charge $Ze$, and positive $z,Z$. Here $e$ is the elementary charge. Neglect nuclear recoil, electron screening effect, energy loss, and the nuclear force. With vacuum permittivity ${\epsilon }_0$, defineHere $E$ and $v$ are the incident kinetic energy and speed far from the atomic nucleus, $b\ge 0$ is the impact parameter, and $L$ is the magnitude of angular momentum. The nonrelativistic limit requires $v\ll c$, with $c$ the speed of light. The fixed-target approximation requires the target mass to be much greater than $m$. In SI, $\kappa$ has units $\mathrm{J}\mathrm{m}$, $E$ has units $\mathrm{J}$, and $b$ has units $\mathrm{m}$.

A salt bridge is an ionic connection between electrolyte compartments that limits bulk mixing while permitting electric charge transport. Its mobile ions compensate changes caused by the electrode half-reactions, keeping the bulk chemical solutions approximately electrically neutral during sustained electric current. It does not conduct electrons like the external wire, and its ions need not be consumed at either electrode. The ionic return path and compensating ion flows are explained in Purdue University, Electrochemical Reactions: salt bridge and charge balance.

In a discharging Daniell cell, anions moving toward the anode compartment and cations moving toward the cathode compartment both contribute to the internal conventional current from anode toward cathode. This charge-balance role is compatible with a negative anode: terminal signs do not imply that the whole anode chemical solution is negatively charged. The simple compensating-inventory example in Daniell cell does not fix all individual ion fluxes; diffusion and the cell's other mobile ions can contribute. A real bridge also introduces electrical resistance and may introduce a junction electric potential difference, so its presence does not guarantee an exact thermodynamic cell reading. See IUPAC Gold Book: salt bridge and liquid-junction potential.

A salt is an assembly of cations and anions with zero total electric charge. Its constituents can be single chemical atoms or groups joined internally by covalent bonds. Thus ionic bonding and covalent bonds can occur in the same chemical substance, at different structural levels.^[ref]

Saponification is hydrolysis of an ester under basic conditions. The conductometric saponification assay follows ethyl acetate consumption by hydroxide, producing ethanol and an acetate ion; its equal-starting-concentration case provides a practical second-order reaction example.

Saponification is ester hydrolysis promoted by a chemical base, with net consumption of hydroxide to form a carboxylate and an alcohol. For a carboxylic acid-derived ester, the net chemical equation isThe hydroxide is a reactant, rather than merely a regenerated catalyst.^[ref]

For ethyl acetate, the balanced chemical equation becomesBoth sides contain four carbon chemical atoms, nine hydrogen chemical atoms, three oxygen chemical atoms, and net electric charge $-1$. A sodium ion supplied by sodium hydroxide is a spectator ion; the net ionic equation omits it without losing electric charge conservation.

In the usual reaction mechanism, hydroxide adds to the carbonyl group, making a reaction intermediate. Departure of the alcohol's conjugate base gives a carboxylic acid; transfer of a proton then gives the carboxylate and the alcohol. Subsequent addition of a chemical acid gives the carboxylic acid by protonation:This separate acidification changes the reaction product's protonation state.^[ref]

The conductometric saponification assay makes hydroxide consumption observable. A university procedure equilibrates separate ethyl acetate and sodium hydroxide stocks in constant-temperature baths, then mixes them in a rinsed conductivity cell. Two platinum electrodes and an alternating-voltage Wheatstone bridge supply an electrical resistance readout. A matched sodium hydroxide–water dilution checks the initial signal; potassium chloride calibrates the cell constant. Replacing hydroxide with less mobile acetate ions lowers electrical conductance, supporting an inference about changing concentration. The manual supplies a procedure rather than measured outcomes. Temperature drift or additional ions can change the signal; it cannot alone identify the full reaction mechanism.^[ref]

A triglyceride contains three ester linkages joining glycerol to fatty acid-derived acyl groups. Complete saponification therefore consumes three hydroxide equivalents per triglyceride, yielding one glycerol and three fatty acid carboxylates.^[ref]

Writing the three hydrocarbon groups separately allows their compositions to differ:This is the preceding ester chemical equation applied three times: each linkage supplies one carboxylate, and each formerly esterified oxygen on the glycerol scaffold gains one hydrogen. The two sides retain the same three ${\mathrm{R}}_i$ groups and have net electric charge $-3$.

Soap consists of sodium or potassium salts of long-chain fatty acids. Its carboxylate head is hydrophilic, while its long hydrocarbon tail is hydrophobic. This makes it an amphiphile and a surfactant: the tails can associate with grease while the heads remain exposed to water, helping disperse grease for rinsing. An acetate ion is also a carboxylate, but lacks the long hydrophobic tail that gives soap this cleaning role.^[ref]

Electrons partially shield an atomic nucleus electric charge. At large distances a neutral chemical atom therefore does not generate the unscreened Coulomb potential of its bare atomic nucleus.

The Seebeck coefficient $S(T)$ quantifies the thermoelectric voltage response per small temperature difference at zero electric current. Use $S=-dV/dT$ along a homogeneous material, with voltage understood through electrochemical potential. Its units are volts per kelvin, commonly $\mu V/K$. The word thermopower is also used, despite this being a voltage response rather than power; some authors use that word for $|S|$, so check their convention. ^[ref].

For the A-minus-B terminal order in the Seebeck effect experiment,Thus a small temperature difference around $T_0$ gives $V_{AB}\approx [S_A(T_0)-S_B(T_0)]\Delta T$. The finite-interval ratio $V_{AB}/\Delta T$ is an average of the coefficient difference, not necessarily its value at either endpoint. Ordinary two-material measurements determine a relative Seebeck coefficient: to recover $S_A$ from the measured difference, add a separately known $S_B$. Neither the choice of a zero for voltage nor the use of a familiar reference material makes $S_B$ identically zero.

A concrete research example is Martin and colleagues' 2021 certification experiment. They measured small temperature differences and voltages on six silicon–germanium specimens at 13 base temperatures between $295$ and $900\mathrm{K}$. At each setting they acquired 40 voltage-and-temperature-difference pairs and used linear regression with a free intercept, so a constant voltage offset did not become part of the slope. Platinum voltage leads required a reference-material correction; their terminal order gave $S_{\mathrm{measured}}=S_{\mathrm{Pt}}-S_{\mathrm{specimen}}$, hence $S_{\mathrm{specimen}}=S_{\mathrm{Pt}}-S_{\mathrm{measured}}$. This is an actual measurement protocol, unlike the toy plot above. ^[ref].

Microscopically, a temperature gradient changes charge carrier transport until an opposing electrical response balances it at zero net electric current. In the usual simple semiconductor picture, electrons and electron holes give opposite Seebeck coefficient signs, and their contributions are weighted by their electrical conductivities. The Seebeck coefficient is a transport property: density of states alone does not determine it, since carrier motion and scattering also matter. Its usefulness extends beyond temperature sensing to diagnosing which carriers transport electric charge. ^[ref].

A semiconductor has an electronic band gap and a conductivity strongly affected by temperature, impurities and carrier injection. Optical absorption can create mobile electrons and holes, enabling a photodiode.

The useful feature of a semiconductor is that its charge carrier population can be controlled. Semiconductor doping changes the supply of mobile electrons or electron holes; carrier injection changes their population away from thermal equilibrium. A transistor turns that control into an electronic amplifier: a small input signal controls a larger flow of supplied energy.

For a uniform semiconductor in a weak electric field, a simple drift model writes its electrical conductivity asHere $q>0$ is the elementary charge, $n,p$ are the number densities of mobile electrons and electron holes, and ${\mu }_n,{\mu }_p$ their positive charge carrier mobilities. The two contributions add because opposite electric charges drift in opposite directions and therefore carry electric current in the same direction. This model assumes fixed temperature and mobilities and omits diffusion from density gradients.

An original dimensionless example isolates the effect of carrier injection. Suppose initially $p\ll n=n_0$ and ${\mu }_n=2{\mu }_p$. Adding equal excess populations $\Delta n=\Delta p=0.1n_0$ gives $\Delta \sigma /{\sigma }_0=0.15$, even though the added electron holes remain a minority carrier population. A semiconductor can respond strongly to a small injected population without changing its underlying band gap.

In a series RLC circuit, the same electric current $i=q^{\prime }$ flows through the resistor with electrical resistance $R$, the inductor, and the capacitor. Choose $q$ as the signed electric charge on the capacitor plate into which positive $i$ flows. With terminal drops measured in that direction, Kirchhoff's voltage law givesHere $V(t)$ is the applied electric potential difference; assume constant $L>0$, $C>0$, and $R>0$. With the applied source set to zero and the loop kept closed, the free response satisfiesThe angular frequency ${\omega }_0$ is the value for lossless free oscillation; $\alpha$ is the decay parameter. The three possibilities are underdamped RLC response for $\alpha <{\omega }_0$, critically damped RLC response for $\alpha ={\omega }_0$, and overdamped RLC response for $\alpha >{\omega }_0$. ^[ref].

For a fair comparison, start each circuit with the same electric charge $q(0)=q_0$ and zero electric current $q^{\prime }(0)=0$. Set $\tau ={\omega }_{0}t$ and $a=\alpha /{\omega }_0$; then $x=q/q_0$ obeys $d^{2}x/d{\tau }^2+2adx/d\tau +x=0$. The plot compares $a=0.5,1,2$. These dimensionless curves apply to any $L,C$ after rescaling time and electric charge.

The stored energy is the sum of magnetic energy and capacitor energy:Differentiating and using the circuit equation gives the complete power balance,In the free response, $E^{\prime }=-R{i}^2\le 0$: Joule heating accounts exactly for the lost energy. For the plotted initial conditions, $E(0)=q_0^2/(2C)$ and $E/E(0)=x^2+(dx/d\tau {)}^2$. Since every free solution tends to zero for $R>0$,In the lossless limit $R=0$, energy stays constant and shuttles between the capacitor and inductor. Thus an oscillation of electric charge and a decrease of total energy are compatible statements.

The analogy with a damped harmonic oscillator is exact within these ideal models: electric charge corresponds to displacement, electric current to velocity, inductance to mass, $1/C$ to spring constant, and $R$ to the coefficient of a resisting force proportional to velocity. The two stored energy terms play the roles of kinetic energy and elastic potential energy. ^[ref].

Differentiating the driven equation also gives $L{i}^{\prime \prime }+R{i}^{\prime }+i/C=V^{\prime }$. When using this equation to solve for electric current, retain the original constraint $L{i}^{\prime }(0)+Ri(0)+q(0)/C=V(0)$: the initial electric charge fixes the initial rate of change of electric current.

In 1986, Akira Tonomura and collaborators coated a ferromagnet shaped as a torus with superconducting niobium and copper. A cooled stage held it in a 150 kV field-emission electron microscope. An electron biprism overlapped object and reference waves; film holograms were reconstructed optically. Room-temperature leakage screening selected samples; varying copper thickness tested electron penetration.^[ref]

Below the superconducting transition, through-hole and outside electron waves differed in wave phase by zero or $\pi$ modulo $2\pi$. Warming through it changed the interference reversibly. Leakage and coherent-penetration bounds do not prove zero magnetic field.^[ref]

Why these two phases? The shield's superconducting flux quantum is $h/(2e)$, whereas the probing electron has electric charge $-e$. Substituting $\Phi =Nh/(2e)$ givesAn odd $N$ therefore moves a carrier fringe by half its spacing; an even $N$ leaves it unchanged modulo a full spacing. The phase quantization provides an internal check on the superconducting enclosure. It does not mean that zero wave phase proves zero trapped magnetic flux.^[ref]

The readout is electron holography in the off-axis holography geometry: a known tilted reference converts an unknown relative wave phase into a spatial carrier displacement. Reconstructing a magnetic wave phase assumes stable reference phase, adequate coherence and a calibrated carrier; electrostatic charging and material traversal can also shift wave phase. Optical wave phase amplification rescales the displayed fringe displacement. The experiment's shielding and comparison tests support the magnetic inference; an interferogram by itself measures the total relative wave phase, rather than separately certifying every contribution. The ideal field-free model describes the limiting geometry, while finite leakage and penetration remain questions of experimental bounds.

Shot noise is fluctuation associated with discrete transfer events. For an electric current, individual charge carriers contribute finite amounts of electric charge, even when a meter displays a smooth mean. The simplest model is a stream of equal, independently arriving charges with constant rate: its Poisson process predicts both the mean electric current and the variance around it. Discreteness alone does not imply a Poisson distribution; the timing and correlations matter.

The useful surprise is that noise can reveal something a mean hides. Two devices can carry the same mean electric current yet have different fluctuations because one transfers charges independently and the other groups or regulates them. A calibrated shot noise photocurrent measurement can connect a macroscopic voltage variance to the elementary charge. Thermal electrical noise instead exists in an unbiased resistor at thermal equilibrium; a measured background can contain both mechanisms.

Walter Schottky's 1918 paper Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern treated amplifier sensitivity limits. Part II considered independent discrete transfers, including thermionic emission through a vacuum. On p. 563 he proposed amplifying fluctuations from a separate discharge tube to determine the elementary charge; those experiments were prospective, not reported results.^[ref]

Let $q>0$ be the magnitude of each transferred electric charge, and let arrivals form a Homogeneous Poisson process of rate $\lambda$. Extend the stream over past and future times when considering stationary filtering. For electrons, $q=e$, the elementary charge. Choose the positive electric current direction to follow these transfers, so $I=q\lambda >0$ is its mean magnitude. If $N_T$ counts arrivals in a gate of duration $T>0$, thenThe gate's transferred electric charge is $Q_T=q{N}_T$, and its average electric current is ${\overline{i}}_T=Q_T/T$. Scaling the expected value and variance givesMore arrivals increase the absolute fluctuation but decrease its size relative to the mean. For an original count example with $\lambda T=4$, the mean count and its variance are both $4$, the standard deviation is $2$, and the relative standard deviation is $1/2$. The probability of no transfer is $e^{-4}\approx 0.01832$; the probability of four is $4^4{e}^{-4}/4!\approx 0.19537$. A constant arrival rate does not prescribe four arrivals in every gate.

These are fluctuations of a time average, not the variance of an ideal instantaneous electric current. A model of point transfers has arbitrarily sharp impulses; it acquires a finite measured variance only after specifying a gate or linear filter. Actual collection pulses have finite duration and electrical circuits have finite response times.

The Fano factor $F_T$ compares count variance with its mean over a specified gate. For a stationary unidirectional stream with fixed charge $q$, finite count second moments, and a generalized autocovariance with finite integral so that the long-gate limit exists,The zero-frequency relation concerns the continuous fluctuation spectrum, excluding the mean spectral line. It follows by writing $\mathrm{Var}(Q_T)$ as the double integral of the current autocovariance; for long $T$, its leading term is $T{S}_I^{(2)}(0)$. A finite-gate $F_T$, a frequency-dependent ratio $S_I^{(1)}(f)/(2qI)$, and a factor inferred from one filtered variance are not interchangeable. The last is weighted by the instrument's $|K(f){|}^2$.

Two original counting counterexamples show why electric charge discreteness is insufficient. In $n$ independent time slots, transfer one charge with probability $p$ in each slot: the binomial distribution gives mean $np$, variance $np(1-p)$ and Fano factor $1-p$. A random common origin for the slots removes a preferred time origin; the long-gate factor remains $1-p$. Alternatively, if independent Poisson process bursts each transfer exactly two charges, then $N_T=2M_T$ gives $\mathrm{Var}(N_T)=2\mathbb{E}[N_T]$ and factor $2$. Both count individual charges; only the first model allows at most one per slot, and the second groups them in pairs.

Consequently $2qI$ is a reference for independent unidirectional arrivals, not a universal noise law for all electrical conductors. Correlations, charge multiplication, and bidirectional transfer require a more specific model. For example, independent opposite streams with positive current magnitudes $I_{+}$ and $I_{-}$ have net mean $I=I_{+}-I_{-}$ but one-sided fluctuation spectrum $2q(I_{+}+I_{-})$. Using $2q|I|$ would miss their fluctuations when the net current is small. At thermal equilibrium, the full transport model must also respect thermal electrical noise; adding unrelated phenomenological formulas can double count the same underlying transfers.

Sodium chloride contains sodium cations ${\mathrm{Na}}^{+}$ and chlorine anions ${\mathrm{Cl}}^{-}$ in a $1:1$ ratio; its formula unit has zero net electric charge.

A solubility product constrains a product of thermodynamic activities; the available inventories determine which combinations of concentrations are possible. Their intersection determines the molar solubility. Consider an original, abstract AB solid, with the preceding one-to-one dissolution chemical equation. Stipulate $c^{\circ }=1\mathrm{M}$, ${\gamma }_{\mathrm{A}}={\gamma }_{\mathrm{B}}=1$ and $K_{\mathrm{sp},c}=10^{-4}{\mathrm{M}}^2$ at fixed temperature and pressure. Thus $K_{\mathrm{sp}}=10^{-4}$ is dimensionless. These are chosen model parameters, not data for an identified material.

Initially, a completely dissociated soluble XB salt supplies B anions at molar concentration $c_0\ge 0$ and X cations at the same molar concentration; A is absent. All three ions have unit electric charge magnitude. X remains a spectator ion, and only AB can form a solid. Assume constant volume, no additional chemical reactions, and negligible contributions from self-ionization of water. Dissolving AB raises both reacting-ion concentrations by the same $s$:Charge neutrality is satisfied: $c_{\mathrm{A}}+c_{\mathrm{X}}-c_{\mathrm{B}}=s+c_0-(s+c_0)=0$. Adding B alone without its accompanying cation would omit the initial electric charge balance.

In the plane of the two reacting-ion concentrations, this inventory constraint is the line $c_{\mathrm{B}}=c_{\mathrm{A}}+c_0$. Saturation supplies the hyperbola $c_{\mathrm{B}}=K_{\mathrm{sp},c}/c_{\mathrm{A}}$ for $c_{\mathrm{A}}>0$. Their positive intersection solves a quadratic equation:The quadratic formula supplies two roots of a polynomial; only the positive one is a physical concentration. Uniqueness also follows because $s(s+c_0)$ strictly increases for $s\ge 0$, starting at zero and growing without bound. The second expression, obtained by multiplying by the conjugate, avoids subtraction of nearly equal numbers at large $c_0$.

For pure water, $c_0=0$ gives $s_{\ast }=\sqrt{K_{\mathrm{sp},c}}=0.010\mathrm{M}$. For $c_0=0.030\mathrm{M}$,Their product is $10^{-4}{\mathrm{M}}^2$, although neither concentration equals the pure-water value. This is the useful geometrical distinction: an inventory line and an equilibrium constant specify different constraints.

Check the solid inventory before imposing saturation. If the initially added AB has amount of substance $n_0$ in volume $V$, the saturated branch requires $n_0>V{s}_{\ast }$ for excess solid to remain. For example, choose $n_0=0.0020\mathrm{mol}$ and $V=0.100\mathrm{L}$. In pure water, $0.0010\mathrm{mol}$ dissolves and the same amount of substance remains as solid. At $c_0=0.030\mathrm{M}$, approximately $0.0003027756\mathrm{mol}$ dissolves, leaving $0.0016972244\mathrm{mol}$ as solid. In this model, an insufficient supply instead givesIf $n_0/V<s_{\ast }$, all AB dissolves and the ion product is below the solubility product. At $n_0/V=s_{\ast }$, the last solid disappears exactly at saturation.

Solvation is stabilization through interactions between a solute and its solvent. It involves both the solute's surroundings and reorganization of the solvent: neighbors move, orientations change, and existing interactions compete with new ones. Electric forces, Van der Waals forces, and specific interactions such as hydrogen bonds can contribute. Solvation can also occur at groups on an insoluble material, so it does not require complete dissolution.^[ref]

Solvation by water is called hydration. Hydration of ions is the charged special case, but neutral molecules such as sucrose can also interact with water without becoming ions. A hydrated cation retains its positive net electric charge; orienting neutral water molecules around it does not supply an extra electron. Dissolution, desolvation, and ionization describe different changes, as the examples below show.^[ref][ref]

The useful question is therefore not just whether a solute attracts a solvent, but how much the whole system gains after paying for the rearrangements. Solvation free energy describes a specified transfer into a solvent; solvation thermodynamic balance explains why even cooling during dissolution can accompany favorable mixing. The existing solubility discussion concerns how much remains dissolved at chemical equilibrium.

Electric charge distributed through a region, whose own electric field can alter the motion of further charged particles.

A spark gap can rapidly discharge a charged capacitor through an inductor. After excitation, the stored energy runs down through Joule heating, radiation and other losses. A key held down can permit repeated discharges: each discharge starts a damped train of rapid oscillations. The spark repetition frequency, the electrical oscillation frequency within each train, and the rate of keyed symbols are different quantities. Repetition does not make each train a maintained sinusoid.

An original lumped model replaces the conducting spark gap and other losses by a constant total electrical resistance $R>0$ in an isolated series RLC circuit. For electric charge $q$ on the capacitor, the Kirchhoff cycle law givesFor $Q>1/2$, its characteristic roots are $-\alpha \pm \mathrm{i}{\omega }_d$, with ${\omega }_d=\sqrt{{\omega }_0^2-{\alpha }^2}$, proving the stated damped form. The electric current $\dot{q}$ has the same exponential decay scale. Its wave envelope falls by $\exp (-2\pi \alpha /{\omega }_d)$ per cycle; the energy envelope falls twice as fast in the exponent. A maintained sinusoid instead requires continuing input power balancing losses.

With $L=5\mathrm{mH}$, $C=2\mathrm{nF}$ and $R=100\Omega$, the model gives $f_0={\omega }_0/(2\pi )=50.329\mathrm{kHz}$, Q factor $Q=15.811$, amplitude time constant $1/\alpha =100\mu \mathrm{s}$, and damped frequency ${\omega }_d/(2\pi )=50.304\mathrm{kHz}$. These are chosen component values. A real spark gap switches nonlinearly, an antenna radiates, and coupling exchanges energy between circuits; one constant-$R$ free decay does not model the complete transmitter. Rapid damping also broadens the frequency spectrum, which helps explain why short trains make sharp receiver selection difficult.