# Potential energy **Potential energy** is the energy a system holds because of where its parts are rather than how fast they are moving: the [[Work_(physics)|work]] already invested in putting them there, available to be paid back when they are allowed to return. Lifting a mass m through a height h against gravity stores m·g·h; stretching a spring stores ½·k·x²; pulling two unlike charges apart stores work that the [[Coulomb's_law|Coulomb]] attraction will refund.[^murphy-mgh] The idea works only for [[Conservative_force|conservative forces]], those for which the work depends on the endpoints and not the route, and its content is one relation: F = −dU/dx, force is the downhill slope of the potential. In the microsim below the reader is given a hilly landscape U(x) = m·g·h(x) and two different paths between the same two points. The readout shows that both cost the same work, W = −ΔU, that the force at any point is the local slope, and that a path returning to its start costs nothing. Dragging the end point changes the cost; reshaping the hill between the end points does not — the fact that forbids the overbalanced wheel and every other [[Perpetual_motion|perpetual motion]] machine of the first kind. On the [[Energy]] flagship this article is the child of Part I — Forms, section *Potential energy* (row E3), sibling to [[Kinetic_energy|kinetic energy]] on the work-and-energy dial that [[Mechanical_energy|mechanical energy]] owns as its root. ## Overview Potential energy is not a substance located in a body but a property of a configuration. Strictly it belongs to the whole interacting system — the Earth and the raised stone together, the two charges together — and assigning it to one member is a shorthand that works only because the other is far heavier and effectively fixed. A survey of the forms energy takes lists m·g·h for hydro and tidal schemes beside c_p·m·ΔT for warmth and q·V for electricity; each is either a potential or something a potential converts into.[^murphy-table] The practical grip of the idea is that it turns a problem in dynamics into a problem in arithmetic. A falling apple carrying 7 J keeps a two-column ledger as it drops — 7 J of position and none of motion, then 5 and 2, then 3 and 4, then 1 and 6 — and the sum never moves, so the speed at any height follows without integrating any equation of motion.[^murphy-apple] Engineering thermodynamics keeps the same two columns, adding potential, kinetic and spring terms to internal energy in the closed-system balance.[^yan-ch4] ## History The quantity came before the name. In 1686 [[Gottfried_Wilhelm_Leibniz|Leibniz]] argued that a "living force" proportional to m·v² was conserved, and that it could be stored and recovered — the "dead force" of a compressed spring or a raised weight being the same thing held in reserve.[^leibniz1686] [[Thomas_Young_(scientist)|Thomas Young]] introduced *energy* in the modern sense in 1807.[^young1807] The mathematical apparatus arrived from a different direction: Lagrange's *Méchanique analitique* of 1788 built mechanics from a function of position alone, and George Green's 1828 essay on electricity and magnetism named such a function the *potential*.[^lagrange1788][^green1828] W. J. M. Rankine coined *potential energy* in 1853, at the moment when the new [[Thermodynamics|thermodynamics]] needed a vocabulary that distinguished stored energy from energy in transit.[^rankine1853] The order of events matters for how the subject is taught. The potential function was invented as a convenience for computing forces and only afterwards recognised as an energy that could be traded against motion and against [[Heat|heat]]. That is why two definitions coexist in textbooks — the work done against a force, and the scalar whose gradient is the force — and why the minus sign in F = −∇U is convention rather than physics, chosen so that systems run downhill in U. ## Feature of conservative systems A force is conservative when the work it does between two points is the same along every path, which is equivalent to saying that the work around any closed loop is zero: ∮F·dr = 0. Only then can a single number U(x) be attached to each position such that the work from a to b is U(a) − U(b). Gravity, the electrostatic force and an ideal spring qualify; [[Friction|friction]] and drag do not, because the work they do grows with path length and can never be recovered. The microsim's two-path readout is a direct test of this property. The reader gets a landscape and two routes between the same pair of points — one over the ridge, one around it — and a running integral of F·dr along each. The totals differ moment by moment and agree at the end, within the stated numerical tolerance. Closing the loop by dragging the end point onto the start drives the total to zero, which is why a wheel weighted to be permanently "heavier on one side" cannot turn forever: whatever torque it gains descending it loses ascending, and ∮τ·dθ = 0.[^murphy-mgh] Adding a friction slider breaks the agreement at once, and the two paths then differ by the extra length the longer one travels. ### Derivable from a potential A force field is conservative exactly when it is the gradient of a scalar function, F = −∇U; in one dimension, F = −dU/dx. The condition is checkable without doing any integrals: in a simply connected region a field is a gradient if and only if its curl vanishes, ∇ × F = 0, which for a two-dimensional landscape is the statement that the mixed partial derivatives agree. ### Computing potential energy Given the force, the potential is recovered by integrating along any convenient path: U(b) − U(a) = −∫F·dr. The freedom to pick the path is the benefit, and the usual choice makes the integral trivial — radially outward for a central force, straight up for near-Earth gravity, along the axis for a spring. ## Potential energy for near-Earth gravity Close to the Earth's surface the gravitational force on a mass is very nearly constant, so the integral collapses to U = m·g·h with g ≈ 9.81 m/s², and the landscape U(x) = m·g·h(x) that the microsim draws is a scaled copy of the terrain itself — the only rescaling being the factor m·g, which is why the same picture serves every mass.[^murphy-mgh] A 10 kg box, weighing about 100 N, lifted 2 m takes roughly 200 J; delivered in one second that is 200 W, and over four seconds 50 W, which is the distinction between energy and [[Power_(physics)|power]] in its simplest form.[^murphy-mgh] The linear store also explains why gravitational energy is cheap per joule and expensive per kilogram: raising one tonne by one metre buys only 9.8 kJ, about a thousandth of what the same mass of a chemical fuel holds. The approximation is ILLUSTRATIVE in that g is treated as constant. It falls by roughly 0.003 % per 100 m of altitude (derived from the inverse-square law at the Earth's radius), which is negligible for a hill or a dam and not negligible for a satellite, where the general form below is required instead. ## Potential energy for a linear spring A spring obeying [[Hooke's_law|Hooke's law]] pulls back with F = −k·x, so integrating from the relaxed position gives U = ½·k·x², a parabola symmetric in compression and extension. Doubling the deflection quadruples the [[Elastic_energy|elastic energy]], and the restoring force being linear in x is what makes the motion [[Simple_harmonic_motion|simple harmonic]] with a period independent of amplitude. Closed-system energy balances carry this spring term explicitly beside the kinetic and gravitational ones.[^yan-ch4] ## Potential energy for gravitational forces between two bodies Over distances where the inverse-square law matters, the constant-g approximation fails and the potential becomes U = −G·M·m/r, with the zero of energy placed at infinite separation. The negative sign is a consequence of that choice, not a physical oddity: bringing two masses together from far apart releases energy, so their shared store must fall below the value it had when they were apart. ### Derivation Integrating the attraction F = −G·M·m/r² inward from infinity gives U(r) = −G·M·m/r directly. The same integral fixes the [[Escape_velocity|escape speed]]: a body just able to reach infinity with nothing to spare has ½·m·v² = G·M·m/r, so v_esc = √(2·G·M/r), independent of the escaping mass. ## Potential energy for electrostatic forces between two bodies The electrostatic case has the same algebra with a different constant and a sign that depends on the charges. Coulomb's law gives F = k·|q₁·q₂|/r² with k = 1/(4·π·ε₀), so the [[Electric_potential_energy|electric potential energy]] is U = k·q₁·q₂/r — negative for unlike charges, which attract and so behave like the gravitational case, positive for like charges, which must be pushed together and fly apart if released.[^os-coulomb] Dividing by one charge gives the potential in volts, which is why a battery's terminal [[Voltage|voltage]] is an energy per unit charge and the energy delivered is q·V. ## Reference level Only differences in potential energy have physical consequences, so the zero may be placed anywhere convenient. Sea level, the floor, the workbench and infinity are all legitimate choices, and a problem can be solved with any of them provided the same choice is used throughout. The microsim exposes this by letting the reader drag the datum line up and down: every bar in the ledger shifts by the same amount, every difference stays put, and the motion of the particle on the landscape is unchanged. The arbitrariness is not a defect but a statement about the physics — the force is a derivative, and derivatives are blind to constants. The convention is effectively forced in one case. For the inverse-square potential, putting the zero at infinity is the only choice that leaves U finite everywhere else, and it is the reason bound systems are conventionally quoted with negative energies. In thermodynamic bookkeeping the same freedom appears as the arbitrary reference state of a property table, which is why enthalpies and internal energies are tabulated as differences from a stated datum rather than as absolute quantities.[^yan-ch4] ## Gravitational potential energy Gravitational potential energy is the largest-scale store in routine engineering use: a modest height over a large mass is a great many joules, and the store leaks only through evaporation and seepage. ### Local approximation For anything from a hammer to a hydroelectric reservoir, U = m·g·h is exact enough. The energy released by falling water is the same m·g·h, and per second of flow it becomes power: P = g·ṁ·h, or in the units the dam industry uses, P[W] = 84.6·Q[ft³/s]·h[ft].[^kerlin-hydro] ### General formula Away from a single surface the full −G·M·m/r form is needed and potential energy stops being proportional to height. The local form is the first term of the general one expanded about the Earth's radius, which is what separates a dam from an orbit. ### Negative gravitational energy With the zero at infinity, every bound system has negative gravitational potential energy, and a body in a closed [[Orbit|orbit]] has negative total energy, kinetic plus potential, because escaping means adding energy to reach zero. The depth of the well, not its sign, carries the physics: deeper means a larger escape speed and tighter binding. ### Uses Pumped storage is the direct industrial application. Water is lifted at night and dropped at peak demand, and the store is worth building where the head is large: Raccoon Mountain in Tennessee has a 528-acre reservoir behind a 230 ft dam with a 990 ft drop to its turbines, and delivers more than 1,500 MW.[^kerlin-raccoon] Because power is head times flow, that 990 ft does with one-fifth the water what a 200 ft river dam would need (derived) — which is why [[Pumped-storage_hydroelectricity|pumped storage]] looks for mountains and why [[Grid_energy_storage|grid storage]] is compared on head as much as volume. Weight-driven clocks and pile drivers are the same store at domestic scale. ## Chemical potential energy [[Chemical_energy|Chemical potential energy]] is held in the arrangement of [[Electron|electrons]] among nuclei, and released when weakly bound combinations are replaced by strongly bound ones. Breaking the bond in a hydrogen molecule costs 436 kJ per mole, a measure of how deep that well is.[^cboc-hh] Burning carbon releases 394 kJ for every 12 g; dividing by Avogadro's number makes that about 4 electronvolts per atom, roughly 1 eV per carbon–oxygen bond formed — the natural unit of chemistry, and the reason chemical fuels cluster within a factor of a few per atom.[^murphy-carbon] ## Electric potential energy Electric potential energy covers the energy of charges in each other's fields and the energy stored in the fields themselves, in a [[Capacitor|capacitor]] or an inductor. ### Electrostatic potential energy Assembling a set of charges costs the sum of k·qᵢ·qⱼ/rᵢⱼ over all pairs, and that is what a capacitor returns on discharge. A [[Dipole|dipole]] p = q·d in a uniform field feels no net force but does feel a torque τ = p × E, so it has an orientation energy lowest when aligned with the field.[^os-coulomb] ### Magnetic potential energy A magnetic moment in a [[Magnetic_field|magnetic field]] has the analogous orientation energy U = −m·B, minimised when moment and field agree, which is why compass needles settle. A current-carrying inductor stores ½·L·I² in its field, the magnetic counterpart of the spring's ½·k·x². ## Nuclear potential energy The nuclear store is the deepest of all, because the [[Nuclear_force|strong force]] is short-ranged and, within its range, enormously stronger than electrostatic repulsion. The mass of a nucleus is less than the sum of its parts, and the missing mass measures the depth of the well through ΔE = Δm·c²: the [[Nuclear_binding_energy|binding energy]] per nucleon rises from 1.11 MeV for deuterium through 7.07 for helium-4 and 7.68 for carbon-12 to a maximum of 8.79 MeV near iron-56, then falls again to 7.59 MeV for uranium-235.[^murphy-be] Because the curve has a peak, both directions pay: fusing light nuclei and splitting heavy ones each move nucleons toward the more tightly bound middle. Fusing four hydrogen nuclei into helium-4 releases 26.7 MeV, and the deuterium–tritium reaction releases 17.6 MeV, the second requiring around 4.5×10⁷ °C against the Sun's core temperature of 1.6×10⁷ °C.[^murphy-fusion] ## Forces and potential energy The relation F = −∇U runs in both directions, and reading it from the potential to the force is what makes a potential landscape a picture of the dynamics. Where the slope vanishes the force vanishes and the system is in equilibrium; where the curvature is positive the equilibrium is stable and small displacements oscillate; where it is negative the equilibrium is unstable and small displacements grow. A [[Pendulum_(mechanics)|pendulum]] shows all of this in one plot: its energy is E = ½·m·L²·(dθ/dt)² + m·g·L·(1 − cos θ), a well at θ = 0 and a hilltop at θ = π, and the value E = 2·m·g·L separates swinging back and forth from going over the top.[^cline-pendulum] Generalising that picture is what [[Lagrangian_mechanics|Lagrangian mechanics]] does. Writing L = T − U and requiring the action to be stationary yields the equations of motion in any coordinates, with constraint forces eliminated and conserved quantities appearing wherever U does not depend on a coordinate.[^cline-lagrangian] The [[Hamiltonian_mechanics|Hamiltonian]] form uses H = T + U, which for a conservative system is the total energy — the falling apple's ledger, written so it carries to fields, circuits and [[Quantum_mechanics|quantum mechanics]] unchanged. *See also:* [[Gravitational_energy]] · [[Elastic_energy]] · [[Perpetual_motion]] · [[Kinetic_energy]] · [[Mechanical_energy]] · [[Conservative_force]] ## Notes Explanatory material is carried in the body rather than in separate notes; every footnote definition on this page is collected under References below. Values marked "(derived)" were computed for this article from inputs the cited source prints, not read off the source's own printed answer. ## References [^murphy-mgh]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 5 "Energy and Fossil Fuels", pp. 88–91 (W = F·d and 1 J = 1 N·m; gravitational energy m·g·h; the 10 kg box lifted 2 m for ≈200 J, at 200 W in 1 s or 50 W in 4 s; some examples use g ≈ 10 m/s²). Portal Book 097, https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-apple]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, pp. 90–91 (the falling-apple ledger, total 7 J, stepping through (7, 0), (5, 2), (3, 4) and (1, 6) J). Portal Book 097. [^murphy-table]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, p. 90, Table 5.2 (the forms of energy and where each reappears: m·g·h, ½·m·v², h·ν, H − T·S, c_p·m·ΔT, q·V, m·c²). Portal Book 097. [^murphy-carbon]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 5, p. 98 (394 kJ per 12 g of carbon divided by 6×10²³ is 6.5×10⁻¹⁹ J, about 4 eV per atom and about 1 eV per C–O bond). Portal Book 097. [^murphy-be]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 15 "Nuclear Energy", pp. 266–269 (ΔE = Δm·c²; the mass defect; Table 15.5 binding energy per nucleon: ²H 1.11, ⁴He 7.07, ¹²C 7.68, ⁵⁶Fe 8.79, ²³⁵U 7.59 MeV, with the curve peaking near ⁵⁶Fe). Portal Book 097. [^murphy-fusion]: Murphy (2021), *Energy and Human Ambitions on a Finite Planet*, Chapter 15, p. 285 (4 ¹H → ⁴He + 26.7 MeV; ²H + ³H → ⁴He + n + 17.6 MeV at ≈4.5×10⁷ °C; the Sun's core at 1.6×10⁷ °C). Portal Book 097. [^yan-ch4]: Yan, Claire Yu (2022). *Introduction to Engineering Thermodynamics*. Chapter 4 "The First Law of Thermodynamics for Closed Systems", pp. 127–186 (the closed-system energy balance and its kinetic, gravitational and spring terms; page to pin — most equation displays were lost in extraction). Portal Book 115, https://open.umn.edu/opentextbooks/textbooks/introduction-to-engineering-thermodynamics [^kerlin-hydro]: Kerlin, Thomas (2013). *Future Energy: Opportunities & Challenges*. Chapter 6 "Hydro", pp. 236–239 (Eq. 9-1 P = g·ṁ·h; Eq. 9-2 P[W] = 84.6·Q[ft³/s]·h[ft]; Example 9.1, 200 ft at 50,000 ft³/s giving 846 MW; large turbine–generators at ≈90 %). Portal Book 048, https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges [^kerlin-raccoon]: Kerlin (2013), *Future Energy*, Chapter 6, pp. 240–241 (Raccoon Mountain pumped storage: a 230 ft dam, a 528-acre reservoir, a 990 ft drop and more than 1,500 MWe; pumped storage pairs well with plants that cannot throttle cheaply at night). The one-fifth water comparison against a 200 ft head is derived. Portal Book 048. [^cboc-hh]: Blackstock, Lindsay; Brewer, Sharon; Cinel, Bruno (2022). *Chemical Bonding and Organic Chemistry*. Chapter 4 "Chemical Bonding", p. 231 (H₂ → 2H, +436 kJ/mol). Portal Book 054, https://open.umn.edu/opentextbooks/textbooks/chemical-bonding-and-organic-chemistry [^os-coulomb]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*. OpenStax. Chapter 5 "Electric Charges and Fields", p. 191 (F = k·|q₁·q₂|/r² with k = 1/(4·π·ε₀)) and pp. 214–216 (dipole moment p = q·d, torque τ = p × E, no net force in a uniform field). The displayed equations were lost in extraction and are quoted here in standard form. Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^cline-pendulum]: Cline, Douglas (2018). *Variational Principles in Classical Mechanics*, revised 2nd ed., p. 79 (E = ½·m·L²·θ′² + m·g·L·(1 − cos θ); libration below E = 2·m·g·L, rotation above it, separatrix at that value). Portal Book 073, https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics [^cline-lagrangian]: Cline (2018), *Variational Principles in Classical Mechanics*, the Lagrangian dynamics chapters, pp. 155–198 (L = T − U, the Euler–Lagrange equations, constraint forces and cyclic coordinates; page to pin). Portal Book 073. [^leibniz1686]: Leibniz, G. W. (1686). "Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturalem." *Acta Eruditorum*, March 1686. [^young1807]: Young, Thomas (1807). *A Course of Lectures on Natural Philosophy and the Mechanical Arts*. London: Joseph Johnson. [^lagrange1788]: Lagrange, Joseph-Louis (1788). *Méchanique analitique*. Paris: Veuve Desaint. [^green1828]: Green, George (1828). *An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism*. Nottingham: printed for the author. [^rankine1853]: Rankine, W. J. M. (1853). "On the General Law of the Transformation of Energy." *Proceedings of the Philosophical Society of Glasgow*. ## External links - [Energy and Human Ambitions on a Finite Planet](https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet), Thomas Murphy — Portal Book 097 - [Variational Principles in Classical Mechanics](https://open.umn.edu/opentextbooks/textbooks/variational-principles-in-classical-mechanics), Douglas Cline — Portal Book 073 - [Future Energy: Opportunities & Challenges](https://open.umn.edu/opentextbooks/textbooks/future-energy-opportunities-challenges), Thomas Kerlin — Portal Book 048 - The Wikipedia pair's external links list further open resources <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Potential_energy.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Potential energy* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Potential_energy.html" data-title="Potential energy"></div> *Built from `MICROSIM_GUIDE/specs/sims/Potential_energy.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Potential_energy) : [Wikitube](https://en.wikitube.io/wiki/Potential_energy) · pinned revision [1356733354](https://en.wikipedia.org/w/index.php?oldid=1356733354) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Energy]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Energy row E3 · sim pending (matter/Potential_energy).*