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A potential barrier is a region of higher potential energy separating accessible regions of a physical system. A classical particle cannot pass through a barrier higher than its total energy along its trajectory, but a quantum particle can have nonzero transmission probability through a finite barrier by quantum tunnelling.

In a one-dimensional region with constant $V_0>E$, the Schrödinger equation gives exponential solutions with tunnelling decay constant $\kappa =\sqrt{2m(V_0-E)}/\hslash$. The wavefunction can therefore penetrate a finite barrier and match an outgoing wave on the other side. The rectangular potential barrier is an exactly solvable model; smooth barriers lead naturally to a semiclassical approximation. Even for $E>V_0$, quantum reflection can occur where the potential changes.

Potential energy assigns energy to a system's arrangement, so a change of arrangement accounts for the work done by a conservative force. For a time-independent potential energy $V(\mathbf{r})$,The gradient points toward increasing potential energy; the force points toward decreasing potential energy. Only differences matter: adding a constant to $V$ changes neither the force nor its work. This endpoint rule follows by substituting $-\nabla V$ into the line integral for work. ^[ref]

Near Earth's surface, stipulate constant gravitational acceleration $g=9.81m/{\mathrm{s}}^2$ and measure height $h$ upward from a chosen zero. Then gravitational potential energy is $V=mgh$. Raising a $2\mathrm{kg}$ mass by $0.5\mathrm{m}$ increases its gravitational potential energy by $9.81\mathrm{J}$, while gravity does $-9.81\mathrm{J}$ of work. These are computed model values, not measurements of a particular lift. An ideal spring obeying Hooke's law instead has $V=k{x}^2/2$, with displacement $x$ from its relaxed state and spring constant $k$: $k=100N/m$ and $x=0.2\mathrm{m}$ give $V=2\mathrm{J}$ relative to $x=0$. ^[ref]

A region of lower potential energy surrounded by higher potential energy; a particle must gain sufficient energy to escape classically.

A Potter-Elvehjem homogenizer disrupts biological tissue between a fitted pestle and a surrounding vessel, often using a rotating drive. Flow and mechanical deformation in the small clearance transfer energy from the drive into the sample. ^[ref]

For an idealized narrow gap containing a Newtonian fluid, the shear rate scale is $U/\delta$, with relative surface speed $U$ and clearance $\delta$; viscous stress scales as $\eta U/\delta$. Actual tissue is more complex, but the relation explains why clearance, motion and dynamic viscosity affect disruption.

Power describes how quickly energy is transferred. For mechanical work, the instantaneous power supplied by a force isThe SI unit is $\mathrm{W}=J/s=\mathrm{N}m/s$, distinct from the velocity unit $m/s$. The italic symbol $W$ for work and the upright unit $\mathrm{W}$ for power mean different things. Two processes can transfer the same energy while requiring different powers: doing $12\mathrm{J}$ of work in $2\mathrm{s}$ requires an average $6\mathrm{W}$, whereas doing it in $4\mathrm{s}$ requires an average $3\mathrm{W}$. ^[ref]

The singular equation $2x{w}^{\prime \prime }-w^{\prime }+\lambda x^{2}w=0$ becomes $W^{\prime \prime }+(2\lambda /9)W=0$ under $z=x^{3/2}$, $w(x)=W(z)$. Its zero-endpoint finite-energy quotient transforms as $2{\int }_0^1{x}^{-1/2}(w^{\prime }{)}^{2}dx/{\int }_0^1{x}^{1/2}{w}^{2}dx=(9/2){\int }_0^1(W^{\prime }{)}^{2}dz/{\int }_0^1{W}^{2}dz$. The Poincaré inequality therefore proves the sharp minimum $9{\pi }^2/2$, attained by $w=\sin (\pi x^{3/2})$. This coordinate change turns an apparently singular weighted variational problem into the usual interval energy problem. The admissible eigenfunction is $O(x^{3/2})$, making the integration-by-parts boundary term $w{x}^{-1/2}{w}^{\prime }$ vanish and both energy integrals converge.

The ratio of signal power delivered to a specified output load to signal power accepted at a specified input, with $P_{\mathrm{in}}>0$. An amplifier draws additional energy from its supply, so a power gain above one does not create energy.

A power inverter converts electrical power from a direct current input to an alternating current output. Its switches reverse or modulate the connection to the DC supply, while its control and filtering determine the output waveform. It transfers energy from its input; the switching operation itself is not an energy source. The DOE overview explains the switching-and-filtering principle and distinguishes equipment designed for independent operation from equipment relying on a grid reference.^[ref]

An ideal H-bridge uses four controlled switches around a two-terminal load. One diagonal pair applies $+V_d$ across the load, and the opposite pair applies $-V_d$. In a stipulated square-wave mode with equal half-periods,Its root mean square magnitude is $V_d$, but it is not sinusoidal. Its Fourier series isThe fundamental RMS magnitude is $2\sqrt{2}{V}_d/\pi$. Thus a $48\mathrm{V}$ input gives a $48\mathrm{V}$ RMS ideal square-wave output, but only about $43.22\mathrm{V}$ RMS in the fundamental. Equal total RMS does not imply equal waveform or equal response of every electrical load.

For a $12\Omega$ purely resistive load directly across this ideal square-wave output, mean active power is $V_d^2/R=192\mathrm{W}$. The fundamental contributes $8V_d^2/({\pi }^{2}R)\approx 155.63\mathrm{W}$ and the other harmonic components supply the remainder. The harmonic active power sum accounts for that difference. Adding a filter changes the circuit; the directly connected resistor calculation should not then be reused unchanged.

Rapid switching also permits averaged control. If a short switching interval applies $+V_d$ for fraction $d$ and $-V_d$ for the rest, its average output is $(2d-1)V_d$. Choosing $d(t)=(1+m\sin \omega t)/2$, with $0\le m\le 1$, makes that local average sinusoidal. The actual waveform still contains switching components that the load and filter must handle.

A proposed low-voltage demonstration records DC input electric potential difference and electric current, output waveform and load electric current, comparing average input and output power. Output frequency, total RMS, fundamental RMS and waveform distortion answer different questions. For islanded microgrid operation, control of a usable local supply is an additional capability, not a consequence of DC-to-AC conversion alone.

A radiation spectrum whose spectral flux density follows a power law over a stated frequency range. For $F_{\nu }\propto {\nu }^{-\alpha }$, the spectral energy distribution satisfies $\nu F_{\nu }\propto {\nu }^{1-\alpha }$. Thus $\alpha =1$ gives equal energy flux in equal logarithmic frequency intervals. A spectral break invalidates extrapolation with a single index.

Power-rating limit versus energy autonomy in a microgrid distinguishes how much electrical power can be delivered at an instant from how long a specified demand can be sustained. A storage output rating has units of watts; available output energy has units such as joules or watt-hours. Their quotient is a duration only when the output boundary and discharge assumptions agree.

For an original constant-deficit model, let $P_d$ be the remaining electrical load plus losses after other generation, and let $E_o$ be usable storage energy measured at the output. A necessary operating condition and the corresponding duration areA store with $P_{\max }=40\mathrm{kW}$ and $E_o=70\mathrm{kWh}$ supports a $35\mathrm{kW}$ deficit for two hours under these assumptions. A $65\mathrm{kW}$ deficit is infeasible immediately, even though dividing $70$ by $65$ gives a positive duration. That quotient cannot override the output constraint.

Variable demand adds a cumulative constraint. For discharge-only output $P_d(t)\ge 0$, the conditions becomeSuppose $P_{\max }=40\mathrm{kW}$ and $E_o=20\mathrm{kWh}$. A demand of $30\mathrm{kW}$ for ten minutes followed by $10\mathrm{kW}$ for ninety minutes consumes exactly $20\mathrm{kWh}$ and respects the rating. Replacing the first interval by $60\mathrm{kW}$ for five minutes leaves the same energy demand but violates the rating. Equal energy budgets need not imply equal feasibility.

If storage energy $E_s$ is specified before conversion and a constant discharge fraction $\eta$ reaches the output, then $E_o=\eta E_s$; do not apply that factor again when $E_o$ is already an output quantity. Charging, temperature, state-dependent ratings and reserve requirements require a more detailed model.

The DOE overview's controller and component diagram supplies the physical context: generation, storage and connected electrical loads must operate together.^[ref] A proposed islanded microgrid operation test therefore includes both a short high-demand interval and a sustained demand interval. Record unmet demand as well as discharged energy, so a long runtime obtained by disconnected loads is not mistaken for delivery of the originally requested service.

A small longitudinal electric field drives conductor current in a lossy coaxial line. Its cross product with the azimuthal magnetic field sends energy radially into both conductors, where it becomes heat; forward power decreases with distance.

The Poynting theorem is the local energy balance of classical electromagnetism. For vacuum field laws, with current density $\mathbf{J}$, the electromagnetic energy per unit volume isandThus a decrease in local electromagnetic energy can supply outward energy flux or work on electric charge. The term $\mathbf{J}\cdot \mathbf{E}$ is power transferred to matter per unit volume. Its sign matters: positive values remove field energy, while negative values describe matter supplying the field. The magnetic part of the Lorentz force supplies no work term.

For a fixed volume $V$, the divergence theorem gives the directly measurable balanceFor example, $2.0\mathrm{W}$ entering a region with $0.50\mathrm{W}$ leaving and $1.20\mathrm{W}$ transferred to matter means stored electromagnetic energy increases at $0.30\mathrm{W}$. In source-free vacuum only the boundary flux changes the stored energy; a nonzero local Poynting vector can still have zero net closed-surface flux.

A brief proof takes the dot product of the Ampère-Maxwell equation with $\mathbf{E}/{\mu }_0$ and uses Faraday's law with $\mathbf{B}/{\mu }_0$. The identity $\nabla \cdot (\mathbf{E}\times \mathbf{B})=\mathbf{B}\cdot (\nabla \times \mathbf{E})-\mathbf{E}\cdot (\nabla \times \mathbf{B})$ turns the remaining terms into the divergence of the Poynting vector. ^[ref].

In vacuum the Poynting vectoris the local energy flux, measured in watts per square metre. Its surface integral $P={\int }_A\mathbf{S}\cdot \mathbf{n}dA$ gives signed power crossing a surface with unit normal vector $\mathbf{n}$. Positive flux follows the chosen normal; for a closed surface the usual convention is outward. Power depends on the normal component and area, rather than just the field magnitude.

The Poynting vector applies to steady fields as well as electromagnetic waves. In a driven resistive electrical circuit, surrounding fields can carry energy into the conductor, where electrical work becomes Joule heating. This turns a circuit-level power balance into a spatial description of energy delivery. ^[ref].

The reflected-to-incident compressional pressure or normal-stress amplitude ratio at a planar interface. For normal incidence between lossless homogeneous media, the displayed expression uses acoustic impedances $Z_j={\rho }_j{V}_{pj}$. The reflected energy fraction is $r_p^2$, while a same-coordinate particle-velocity reflection coefficient has the opposite sign.

The transmitted-to-incident compressional pressure or normal-stress amplitude ratio. At normal incidence, continuity of normal stress gives $t_p=1+r_p$, with pressure-amplitude reflection coefficient $r_p$. The transmitted energy fraction is $(Z_1/Z_2)t_p^2$, not $t_p^2$ alone.

Collisions and fluctuating neighboring-particle electric fields alter atomic energy levels and coherence times, broadening a spectral line. In a simple impact treatment the coherence-decay rate gives a Lorentzian width of order $\Gamma /(2\pi )$; the precise definition of full or half width must be stated. Unlike Doppler broadening, this depends primarily on collision rates and perturber density. In stellar opacity calculations it alters line overlap and the low-opacity windows entering the Rosseland mean opacity.

A pressure difference compares the pressures at two stated locations or against two reference states. With $\Delta p=p_1-p_2$, a positive value means location one has higher pressure. A membrane or piston with equal effective face areas $A$ experiences a net pressure force of magnitude $|\Delta p|A$, even if both absolute pressures are large.

For example, pressures $120\mathrm{kPa}$ and $100\mathrm{kPa}$ across $10^{-3}{\mathrm{m}}^2$ give a net pressure force of $20\mathrm{N}$. A liquid-column manometer converts a pressure difference to a vertical level difference through hydrostatic pressure, while a differential pressure transducer senses it directly. Comparing ports at different heights requires accounting for the fluid between them. During flow, the relation between pressure difference and volumetric flow rate also depends on geometry, viscosity and other contributions to the energy balance.

Pressure work is mechanical work transferred by pressure forces acting through a moving boundary. For a piston of area $A$ moving through $dx$, $dV=Adx$ and the opposing applied force is $p_{\mathrm{ext}}A$. The work done by the fluid is thereforeThe familiar $\delta W=pdV$ uses the system pressure when it equals the applied boundary pressure, as in a reversible mechanical step. In a free expansion into vacuum, $p_{\mathrm{ext}}=0$ and the boundary work vanishes even if the gas initially has nonzero internal pressure.

In steady incompressible flow, the Bernoulli equation contains $p/\rho$ as flow work per unit mass. This accounts for pressure forces at a control volume's entrances and exits; it is not an independently conserved internal energy of an isolated parcel. The first law of thermodynamics keeps the complete work and energy balance.

Prey is an organism consumed by a predator in predation. The name describes the receiving end of that interaction: a small fish can be prey for a large fish while being a predator of zooplankton. In a food web drawn with resource-to-consumer arrows, a feeding arrow goes outward from the prey toward its predator. This convention makes the route of food and energy easy to follow.

A primary battery is designed to deliver stored chemical energy through discharge rather than to recover that energy repeatedly by electrical charging. The distinction concerns the practical cell design and reaction reversibility, not whether a balanced chemical equation can be written in both directions.

During discharge, coupled oxidation and reduction at the electrodes supply an external electric current. Reaction products, morphology changes and parasitic processes determine how much active material remains accessible. A rechargeable battery instead manages those changes sufficiently well to recover useful capacity over repeated cycles; nominal chemical reversibility alone does not establish that capability.