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A vertical field linking a conducting loop on an incline contributes flux $Bl\cos \theta s$. For zero resistance and zero initial current, flux conservation gives $LI=Bl\cos \theta s$. The uphill magnetic force component then makes the downhill coordinate oscillate about $s_e=mgL\sin \theta /(B^2{l}^2{\cos }^2\theta )$. Release from $s=0$ first stops at $2s_e$, not at the force-free point $s_e$. The energy is stored in the inductor, not dissipated.

An inductor is an electrical circuit component designed to provide self-inductance, often a coil such as a solenoid. An ideal linear inductor has constant $L>0$ and zero electrical resistance. Its stored magnetic energy, with zero chosen at zero electric current, isFor self-inductance $0.5\mathrm{H}$ and electric current $3\mathrm{A}$, this gives $U=2.25\mathrm{J}$. Unlike Joule heating, ideal storage is recoverable: the inductor can return energy to an electrical circuit when its electric current decreases. ^[ref].

The short derivation uses the terminal convention in self-inductance. The power absorbed by an ideal inductor isStarting from zero electric current, the integral of this power givesPositive $P_L$ increases stored magnetic energy; negative $P_L$ returns energy. At constant electric current $3\mathrm{A}$ in the example, $P_L=0$ and induced electromotive force is zero, but $U=2.25\mathrm{J}$ still resides in the magnetic field. The battery in the steady RL circuit supplies Joule heating in the electrical resistance, rather than further increasing the inductor's magnetic energy.

During the closed-loop release in RL circuit, stored magnetic energy plus accumulated Joule heating stays $2.25\mathrm{J}$. A real coil also has electrical resistance, so its Joule heating must be included separately from ideal inductor storage. The constant-$L$ formulas assume a fixed linear magnetic response.

An inelastic collision changes total kinetic energy. In the dissipative case it decreases, transferring energy to internal energy, including deformation and microscopic motion. Isolated colliding bodies still obey momentum conservation; conservation of energy requires accounting for the transferred energy. Bodies need not stick in every inelastic collision. The decrease in translational kinetic energy drives dissipative spreading of a planetary ring.

For the incoming masses and velocities in the elastic collision example, stipulate that the bodies stick. This perfectly inelastic collision has a shared outgoing velocityThe outgoing momenta are $4$ and $2\mathrm{kg}m/s$, totaling $6\mathrm{kg}m/s$. The outgoing kinetic energies are $4$ and $2\mathrm{J}$, totaling $6\mathrm{J}$. Thus $9-6=3\mathrm{J}$ of kinetic energy becomes internal energy within the isolated pair. Nothing in momentum conservation requires this $3\mathrm{J}$ to remain as kinetic energy. ^[ref]

The inelasticity in deep inelastic scattering is $y=P\cdot q/(P\cdot p)$, where $p$ is the incoming lepton momentum, $P$ the target momentum and $q$ the transfer. In the target rest frame it is the fraction of the incoming lepton energy transferred to the target. It is unchanged by replacing $P$ by a collinear parton momentum $\xi P$, and $Q^2=xyS$ in massless kinematics.

Inelastic neutron scattering resolves $E_i-E_f\ne 0$. Through conservation of energy, a peak at positive $\hslash \omega$ identifies an allowed excitation energy; through momentum conservation, its location in $\mathbf{Q}$ constrains the spatial variation of that excitation. A phonon is a quantum of lattice vibration; a magnon is a quantum of a spin wave. Their different couplings make energy transfer alone insufficient to identify which was created.

Bertram Neville Brockhouse's aluminium experiment rotated a large single crystal, scanning analyzer angle for neutron groups. His vanadium comparison exposed the elastic resolution. Later magnetite measurements used a magnetic field intensity test to identify magnons.^[ref]

Successive orientations sample different wave vectors; analyzer settings distinguish outgoing energies. This supplies points on a dispersion relation, rather than only an angular diffraction pattern. Finite resolution can merge neighboring excitations; background, multiple scattering and processes involving several phonons can complicate interpretation. A lattice model is a testable explanation of selected features, not a guarantee that every spectral peak has the same origin.

The experiment follows phonon peaks as both momentum and energy transfer are varied. In inelastic neutron scattering, $\mathbf{q}={\mathbf{k}}_i-{\mathbf{k}}_f$ locates the lattice modulation and $\hslash \omega =E_i-E_f$ measures the mode energy, so a softened mode can be distinguished from static diffraction alone. Measurements on deuterated TTF-TCNQ between $60$ and $300\mathrm{K}$ found a growing Kohn anomaly near $2k_F$ below about $150\mathrm{K}$ as the $54\mathrm{K}$ transition was approached.^[ref]

In inelastic scattering, the participants exchange energy with internal degrees of freedom, so their total translational kinetic energy need not be conserved.

Inelastic X-ray scattering measures energy and momentum transferred from X-rays to matter. With sufficiently high energy resolution, it can map a phonon dispersion relation by measuring inelastic peaks at different momentum transfers. Crystal momentum is determined modulo a reciprocal lattice vector.

For a bipartite spin-one-half Heisenberg antiferromagnet with coordination $z\to \infty$, the two-sublattice mean-field state becomes exact. Antiparallel pure Bloch vectors give bond energy $-J/4$ for the convention $J{\mathbf{S}}_i\cdot {\mathbf{S}}_j$.

An inflaton is a hypothetical scalar field whose energy density drives cosmic inflation. In the canonical single-field description, a spatially homogeneous value $\varphi (t)$ haswhere $V$ is the inflaton potential. In units $c=\hslash =1$, its inflaton equation of motion iswith Hubble parameter $H$. Potential domination makes pressure negative. When this scalar field dominates the Friedmann equations, accelerating expansion requires $V>{\dot{\varphi }}^2$. Stronger potential domination gives $P_{\varphi }\simeq -{\rho }_{\varphi }$ and nearly exponential expansion.^[ref]

The slow-roll approximation supplies a tractable evolution in this regime. Couplings to other fields can transfer the inflaton's energy into particles through reheating. The identity of the inflaton and the form of its inflaton potential are model assumptions to be constrained by primordial perturbations, rather than an identification of a presently known particle. Its quantum fluctuations supply the scalar seeds in canonical single-field models of cosmic inflation.

Interference involving uncertainty, source confusion or selection beyond a simple account of overlapping acoustic energy. Demonstrating it requires suitable controls for audibility and peripheral overlap; merely finding a difficult listening task is insufficient.

Infrared is electromagnetic radiation with wavelengths longer than the red end of ordinarily visible light. A useful named convention is the CIE range $780\mathrm{nm}$ to $1\mathrm{mm}$; the approximate visible range $400$–$700\mathrm{nm}$ used in some teaching texts puts the boundary nearer $700\mathrm{nm}$. ^[ref]. Neither number is a sharp physical cutoff of human vision. The CIE divisions IR-A, IR-B and IR-C are respectively $0.78$–$1.4\mu \mathrm{m}$, $1.4$–$3\mu \mathrm{m}$ and $3$–$1000\mu \mathrm{m}$; the boundaries of near-, middle- and far-infrared vary between applications. ^[ref].

Use vacuum wavelength $\lambda$ when comparing these ranges. With frequency $\nu$, speed of light $c$, Planck constant $h$ and photon energy $E$,Thus the CIE infrared interval corresponds approximately to $3.00\times 10^{11}$–$3.84\times 10^{14}\mathrm{Hz}$ and $1.99\times 10^{-22}$–$2.55\times 10^{-19}\mathrm{J}$ per photon, with the smaller energy at the longer wavelength. A $940\mathrm{nm}$ infrared photon has $E\simeq 2.11\times 10^{-19}\mathrm{J}$, whereas a $254\mathrm{nm}$ ultraviolet photon has $E\simeq 7.82\times 10^{-19}\mathrm{J}$. These are calculations from $hc/\lambda$, not measurements of a particular source. A television remote is a familiar approximately $940\mathrm{nm}$ source; its information is carried by variations in the emitted radiant flux. ^[ref].

Detection and visual perception answer different questions. A photodetector can respond to infrared even when an observer perceives no light; a thermometer detects an increase in temperature after photon absorption. Conversely, a small readout can reflect a small spectral responsivity, rather than absence of incident electromagnetic radiation. Images made from infrared readouts use display colors chosen by the imaging system; those colors are not the colors of the incoming infrared.

The important microscopic mechanisms also differ. In infrared spectroscopy, photon absorption often increases the energy of a normal mode in which chemical bonds stretch or bend. The corresponding optical spectrum helps distinguish molecules by their motions. For the usual electric dipole mechanism, the motion must change the electric dipole moment; matching the frequency alone is not sufficient. ^[ref]. Electronic processes can also emit or absorb infrared, so it is not exclusively radiation from molecular motion. ^[ref]; ^[ref].

Infrared is especially useful for observing thermal radiation from objects near everyday temperatures. For an ideal $300\mathrm{K}$ blackbody, Wien's displacement law gives a wavelength peak near $9.66\mu \mathrm{m}$. A thermal camera's readout nevertheless depends on emissivity, reflected surroundings and transmission along the path, as well as temperature. ^[ref]. A source emitting infrared need not be hot: an infrared remote is useful precisely because a photodetector can distinguish its signal from a background. Visible light and ultraviolet can also cause heating when absorbed; “infrared” names a wavelength range, not a unique form of heat.

Herschel's 1800 discovery made the distinction between sight and detection experimentally concrete. His report describes a optical prism spreading sunlight across a white, ruled receiving surface, with three thermometers at different positions near the end of the visible optical spectrum. He observed heating beyond the red edge where no visible color appeared. The result established that sunlight contained invisible radiation less strongly refracted by that optical prism than the visible red; it did not measure a photon energy or establish today's infrared boundaries. ^[ref]; ^[ref].

A documented modern version by Caltech/IPAC uses a glass optical prism, white paper in a shaded box and three blackened liquid-filled thermometers. Initial readings in shade provide each thermometer's baseline; subsequent readings compare blue, yellow and just beyond red after the bulbs have had time to warm. The measured quantity is the rise in temperature, not a “temperature of a color”. The lesson reports the expected largest rise beyond visible red and specifically discusses matching the bulb coatings and thermometer calibration. Clouds and changes in the projected optical spectrum affect the comparison. ^[ref].

For a stronger modern interpretation, an additional shaded thermometer would monitor ambient drift, exchanging bulbs between positions would test differences between thermometers, and blocking the incident beam would distinguish beam-dependent heating from warmed supports or air. These are suggested controls, not claims about Herschel's historical procedure. Thermal conduction through the mounting and convection can change the readout. A broad bulb also samples a range of wavelengths; the optical prism's transmission and optical dispersion, the bulb's absorption and heat loss all affect the position of the largest rise. In particular, the result does not show that an infrared photon carries more energy than a visible photon.

A simple analytical illustration makes those limitations explicit. Let $\theta =T-T_{\mathrm{amb}}$ be the bulb's excess temperature, $C$ its heat capacity, $G$ its effective heat-loss coefficient in $\mathrm{W}{\mathrm{K}}^{-1}$, and $P_{\mathrm{abs}}$ the absorbed power. Assume one uniform bulb temperature, constant ambient temperature, constant $C,G$, and losses linear in $\theta$. Thenfor a beam switched on at $t=0$ with $\theta (0)=0$. After switching the beam off at $t=t_s$, $\theta (t)=\theta (t_s)e^{-(t-t_s)/\tau }$. The exponential function follows by solving this linear differential equation. The steady excess temperature measures $P_{\mathrm{abs}}/G$; it is not a direct measurement of the optical spectrum. With illustrative $C=0.020\mathrm{J}{\mathrm{K}}^{-1}$ and $G=0.0010\mathrm{W}{\mathrm{K}}^{-1}$, $\tau =20\mathrm{s}$. Absorbed powers of $1.0$ and $0.5\mathrm{mW}$ give steady rises of $1.0$ and $0.5\mathrm{K}$, regardless of which wavelengths supplied that absorbed power.

The upper schematic represents the logic of the optical prism experiment; its light ray positions are not a calibrated wavelength scale. The lower curves use the illustrative heat-balance model, with the beam blocked at $60\mathrm{s}$; they are not historical or modern measured data. The beauty of the experiment is that a simple change of detector extends observation beyond what the eye can perceive.

Application of infrared energy to produce a localized tissue effect, including in treatment of internal haemorrhoids. It differs from cutting away tissue or applying a mechanical ligature.

An infrared divergence arises from contributions at arbitrarily low energy or long wavelength. A massless mode can produce such a problem because many soft excitations are available at very small energy, or because a long-distance fluctuation integral fails to converge.

Its interpretation depends on the observable. In scattering, an exclusive amplitude with no additional arbitrarily soft radiation need not be separately measurable; combining appropriately inclusive real-emission and virtual contributions can remove the relevant divergence. In equilibrium field theory, an infrared-divergent order parameter fluctuation can instead diagnose failure of an assumed ordered-state approximation. A finite size or detector resolution may regulate the calculation, but the dependence on that regulator must be interpreted rather than silently discarded.

An observable unchanged by adding a particle with vanishing energy or replacing one massless particle by exactly collinear particles with the same total momentum. This property permits cancellation of real and virtual infrared divergences and collinear divergences in its perturbative cross-section. Fully inclusive electron-positron hadronic rates and suitably defined jet observables are examples.

A prescribed initial beam profile may vanish at a pivot while having nonzero endpoint curvature. If the post-release pivot is moment-free, its bending moment condition is $EI{y}_{xx}=0$. The data then cannot have a classical solution with continuous curvature at the initial boundary corners. A finite-energy weak solution can still represent release from preparing end moments. For example $y_0=ax(x-L)$ has odd sine coefficients proportional to $n^{-3}$; $\sum n^4|A_n{|}^2$ converges, giving finite bending energy. The displacement series takes the prescribed initial shape, but termwise endpoint second derivatives cannot be identified with its interior curvature limit. This conservative beam evolution does not have the immediate smoothing of a heat equation.

The inner mitochondrial membrane separates mitochondrial matrix from intermembrane space. Its selective permeability permits distinct chemical conditions, while embedded mitochondrial respiratory chain proteins and ATP synthase support oxidative phosphorylation. Cristae increase its surface area.^[ref]

A selective boundary and a productive enzyme have different jobs. Even active respiratory enzymes cannot sustain a useful electrochemical gradient if protons return freely across the boundary. Conversely, an intact boundary without an electron donor does not supply the energy for continued pumping. The isolated mitochondria respiration experiment separates some of these dependencies by additions to the same preparation.

If a nonrelativistic relic decays entirely into promptly thermalized photons, energy conservation at fixed scale factor gives ${\rho }_{\gamma }^{+}={\rho }_{\gamma }^{-}+{\rho }_{\chi }^{-}$. Since thermal photons have ${\rho }_{\gamma }\propto T_{\gamma }^4$, their temperature rises by the displayed factor while an already decoupled Cosmic neutrino background is not heated. Before decay the relic-to-photon density ratio grows as $a$.

Instantaneous power is power at a particular moment. Consider a point mass with mass $m=2\mathrm{kg}$, initially at rest, acted on by a constant net force $F=3\mathrm{N}$ along a straight path. Newton's second law gives constant acceleration $a=F/m=1.5m/{\mathrm{s}}^2$, hence velocity $v(t)=at$ andThe instantaneous power starts at zero even though the force is nonzero, because the initial velocity is zero. At distance $4\mathrm{m}$, the kinetic energy example below gives speed $\sqrt{12}m/s$, so the final instantaneous power is $3\sqrt{12}\mathrm{W}\approx 10.3923\mathrm{W}$. Negative instantaneous power means the selected force is removing energy at that moment.

Start with an inflationary patch of physical radius $H_i^{-1}$ in units $c=1$. After $N$ e-folds and subsequent expansion, its present radius is $H_i^{-1}{e}^N{a}_0/a_{\mathrm{reh}}$. Requiring it to cover the present Hubble radius gives the displayed inequality. With instantaneous conversion to radiation, fixed relativistic degrees of freedom and a fixed ratio $H_i/H_{\mathrm{end}}$, energy conservation gives $H_i\propto T_{\mathrm{reh}}^2$, whereas adiabatic subsequent expansion gives $a_0/a_{\mathrm{reh}}\propto T_{\mathrm{reh}}$. Thus the required $N$ increases approximately as $\log T_{\mathrm{reh}}$. Holding $H_i$ fixed while varying the end energy independently is a different comparison and reverses that simple temperature-only trend. The inflationary energy history and post-inflationary entropy history must be specified.^[ref]