Environment
114-558: Energy Smart Cities First Mile is a UK environmental and waste management business based in London. It works with businesses to reduce their climate impact through services such as waste, recycling, and renewable energy. First Mile was founded in 2004 by Bruce Bratley in Islington, North London. In the early days, trucks were parked near Bruce's house and he would often drive them himself to complete weekend collections. The name
228-529: A ) = F a b , {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial _{b}A_{a})}}=F^{ab}\,,} obtains Maxwell's equations in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are ∂ b F a b = μ 0 j a . {\displaystyle \partial _{b}F^{ab}=\mu _{0}j^{a}\,.} while
342-462: A . {\displaystyle F_{ab}=\partial _{a}A_{b}-\partial _{b}A_{a}.} To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have L = − 1 4 μ 0 F a b F a b . {\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}F^{ab}F_{ab}\,.} We can use gauge field theory to get
456-548: A basal metabolic rate of 80 watts. For example, if our bodies run (on average) at 80 watts, then a light bulb running at 100 watts is running at 1.25 human equivalents (100 ÷ 80) i.e. 1.25 H-e. For a difficult task of only a few seconds' duration, a person can put out thousands of watts, many times the 746 watts in one official horsepower. For tasks lasting a few minutes, a fit human can generate perhaps 1,000 watts. For an activity that must be sustained for an hour, output drops to around 300; for an activity kept up all day, 150 watts
570-462: A battery (from chemical energy to electric energy ), a dam (from gravitational potential energy to kinetic energy of moving water (and the blades of a turbine ) and ultimately to electric energy through an electric generator ), and a heat engine (from heat to work). Examples of energy transformation include generating electric energy from heat energy via a steam turbine, or lifting an object against gravity using electrical energy driving
684-457: A field ), the elastic energy stored in a solid object, chemical energy associated with chemical reactions , the radiant energy carried by electromagnetic radiation , the internal energy contained within a thermodynamic system , and rest energy associated with an object's rest mass . All living organisms constantly take in and release energy. The Earth's climate and ecosystems processes are driven primarily by radiant energy from
798-435: A Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law. In the context of chemistry , energy is an attribute of a substance as a consequence of its atomic, molecular, or aggregate structure. Since a chemical transformation is accompanied by a change in one or more of these kinds of structure, it is usually accompanied by a decrease, and sometimes an increase, of
912-474: A biological cell or organelle of a biological organism. Energy used in respiration is stored in substances such as carbohydrates (including sugars), lipids , and proteins stored by cells . In human terms, the human equivalent (H-e) (Human energy conversion) indicates, for a given amount of energy expenditure, the relative quantity of energy needed for human metabolism , using as a standard an average human energy expenditure of 12,500 kJ per day and
1026-515: A bound system is discrete (a set of permitted states, each characterized by an energy level ) which results in the concept of quanta . In the solution of the Schrödinger equation for any oscillator (vibrator) and for electromagnetic waves in a vacuum, the resulting energy states are related to the frequency by Planck's relation : E = h ν {\displaystyle E=h\nu } (where h {\displaystyle h}
1140-413: A charge density ρ ( r , t ) and current density J ( r , t ), there will be both an electric and a magnetic field, and both will vary in time. They are determined by Maxwell's equations , a set of differential equations which directly relate E and B to the electric charge density (charge per unit volume) ρ and current density (electric current per unit area) J . Alternatively, one can describe
1254-541: A continuous mass distribution ρ instead, the sum is replaced by an integral, g ( r ) = − G ∭ V ρ ( x ) d 3 x ( r − x ) | r − x | 3 , {\displaystyle \mathbf {g} (\mathbf {r} )=-G\iiint _{V}{\frac {\rho (\mathbf {x} )d^{3}\mathbf {x} (\mathbf {r} -\mathbf {x} )}{|\mathbf {r} -\mathbf {x} |^{3}}}\,,} Note that
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#17327719551551368-417: A core concept. Work , a function of energy, is force times distance. This says that the work ( W {\displaystyle W} ) is equal to the line integral of the force F along a path C ; for details see the mechanical work article. Work and thus energy is frame dependent . For example, consider a ball being hit by a bat. In the center-of-mass reference frame, the bat does no work on
1482-492: A crane motor. Lifting against gravity performs mechanical work on the object and stores gravitational potential energy in the object. If the object falls to the ground, gravity does mechanical work on the object which transforms the potential energy in the gravitational field to the kinetic energy released as heat on impact with the ground. The Sun transforms nuclear potential energy to other forms of energy; its total mass does not decrease due to that itself (since it still contains
1596-413: A distribution of mass (or charge), the potential can be expanded in a series of spherical harmonics , and the n th term in the series can be viewed as a potential arising from the 2 -moments (see multipole expansion ). For many purposes only the monopole, dipole, and quadrupole terms are needed in calculations. Modern formulations of classical field theories generally require Lorentz covariance as this
1710-556: A few exceptions, like those generated by volcanic events for example. An example of a solar-mediated weather event is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, suddenly give up some of their thermal energy to power a few days of violent air movement. In a slower process, radioactive decay of atoms in the core of the Earth releases heat. This thermal energy drives plate tectonics and may lift mountains, via orogenesis . This slow lifting represents
1824-474: A given temperature T . This exponential dependence of a reaction rate on temperature is known as the Arrhenius equation . The activation energy necessary for a chemical reaction can be provided in the form of thermal energy. In biology , energy is an attribute of all biological systems, from the biosphere to the smallest living organism. Within an organism it is responsible for growth and development of
1938-477: A kind of gravitational potential energy storage of the thermal energy, which may later be transformed into active kinetic energy during landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store that has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy that has been stored as potential energy in
2052-427: A path ℓ will exert a force on nearby charged particles that is quantitatively different from the electric field force described above. The force exerted by I on a nearby charge q with velocity v is F ( r ) = q v × B ( r ) , {\displaystyle \mathbf {F} (\mathbf {r} )=q\mathbf {v} \times \mathbf {B} (\mathbf {r} ),} where B ( r )
2166-508: A plant start-up to deliver plants destined for waste to UK homes. First Mile has acquired a number of other environmental businesses; - Cory Environmental's London waste collection business in 2012 - Revolution Recycling in 2014 - All Clear Recycling in 2013 - Giraffe Recycling in 2020 By 2017, First Mile had over 10,000 clients. It has attracted investments from private equity company Growth Capital Partners. Zero emissions electric vans and cargo bikes were also introduced in 2017. Today,
2280-739: A scalar potential, V ( r ) E ( r ) = − ∇ V ( r ) . {\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla V(\mathbf {r} )\,.} Gauss's law for electricity is in integral form ∬ E ⋅ d S = Q ε 0 {\displaystyle \iint \mathbf {E} \cdot d\mathbf {S} ={\frac {Q}{\varepsilon _{0}}}} while in differential form ∇ ⋅ E = ρ e ε 0 . {\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{e}}{\varepsilon _{0}}}\,.} A steady current I flowing along
2394-450: A small test mass m located at r , and then dividing by m : g ( r ) = F ( r ) m . {\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}.} Stipulating that m is much smaller than M ensures that the presence of m has a negligible influence on the behavior of M . According to Newton's law of universal gravitation , F ( r )
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#17327719551552508-425: A system. This property is responsible for the inertia and strength of gravitational interaction of the system ("mass manifestations"), and is also responsible for the potential ability of the system to perform work or heating ("energy manifestations"), subject to the limitations of other physical laws. In classical physics , energy is a scalar quantity, the canonical conjugate to time. In special relativity energy
2622-426: A tiny fraction of the original chemical energy is used for work : It would appear that living organisms are remarkably inefficient (in the physical sense) in their use of the energy they receive (chemical or radiant energy); most machines manage higher efficiencies. In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism tissue to be highly ordered with regard to
2736-572: Is ∇ 2 ϕ = σ {\displaystyle \nabla ^{2}\phi =\sigma } where σ is a source function (as a density, a quantity per unit volume) and ø the scalar potential to solve for. In Newtonian gravitation, masses are the sources of the field so that field lines terminate at objects that have mass. Similarly, charges are the sources and sinks of electrostatic fields: positive charges emanate electric field lines, and field lines terminate at negative charges. These field concepts are also illustrated in
2850-506: Is ∇ ⋅ g = − 4 π G ρ m {\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho _{m}} Therefore, the gravitational field g can be written in terms of the gradient of a gravitational potential φ ( r ) : g ( r ) = − ∇ ϕ ( r ) . {\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla \phi (\mathbf {r} ).} This
2964-402: Is ∬ B ⋅ d S = 0 , {\displaystyle \iint \mathbf {B} \cdot d\mathbf {S} =0,} while in differential form it is ∇ ⋅ B = 0. {\displaystyle \nabla \cdot \mathbf {B} =0.} The physical interpretation is that there are no magnetic monopoles . In general, in the presence of both
3078-534: Is Newton's gravitational constant . Therefore, the gravitational field of M is g ( r ) = F ( r ) m = − G M r 2 r ^ . {\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}=-{\frac {GM}{r^{2}}}{\hat {\mathbf {r} }}.} The experimental observation that inertial mass and gravitational mass are equal to unprecedented levels of accuracy leads to
3192-515: Is a conserved quantity —the law of conservation of energy states that energy can be converted in form, but not created or destroyed; matter and energy may also be converted to one another. The unit of measurement for energy in the International System of Units (SI) is the joule (J). Forms of energy include the kinetic energy of a moving object, the potential energy stored by an object (for instance due to its position in
3306-481: Is a derived unit that is equal to the energy expended, or work done, in applying a force of one newton through a distance of one metre. However energy can also be expressed in many other units not part of the SI, such as ergs , calories , British thermal units , kilowatt-hours and kilocalories , which require a conversion factor when expressed in SI units. The SI unit of power , defined as energy per unit of time,
3420-496: Is a consequence of the gravitational force F being conservative . A charged test particle with charge q experiences a force F based solely on its charge. We can similarly describe the electric field E generated by the source charge Q so that F = q E : E ( r ) = F ( r ) q . {\displaystyle \mathbf {E} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{q}}.} Using this and Coulomb's law
3534-484: Is a continuity equation, representing the conservation of mass ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0} and the Navier–Stokes equations represent the conservation of momentum in the fluid, found from Newton's laws applied to
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3648-460: Is about the maximum. The human equivalent assists understanding of energy flows in physical and biological systems by expressing energy units in human terms: it provides a "feel" for the use of a given amount of energy. Sunlight's radiant energy is also captured by plants as chemical potential energy in photosynthesis , when carbon dioxide and water (two low-energy compounds) are converted into carbohydrates, lipids, proteins and oxygen. Release of
3762-434: Is also a scalar (although not a Lorentz scalar but a time component of the energy–momentum 4-vector ). In other words, energy is invariant with respect to rotations of space , but not invariant with respect to rotations of spacetime (= boosts ). Energy may be transformed between different forms at various efficiencies . Items that transform between these forms are called transducers . Examples of transducers include
3876-448: Is an inversion of the “Last Mile” concept associated with the last leg of telecommunications delivery. “First Mile" focuses on the first leg of recycling, collecting it from the end user in as convenient a way as possible, and delivering to local recycling facilities. The company has acquired multiple businesses including part of Cory Environmental , Revolution Recycling, All Clear, Paperchasers Birmingham, and Giraffe Recycling. In 2015
3990-575: Is called the Lagrangian , after Joseph-Louis Lagrange . This formalism is as fundamental as the Hamiltonian, and both can be used to derive the equations of motion or be derived from them. It was invented in the context of classical mechanics , but is generally useful in modern physics. The Lagrangian is defined as the kinetic energy minus the potential energy. Usually, the Lagrange formalism
4104-473: Is defined in terms of the energy operator (Hamiltonian) as a time derivative of the wave function . The Schrödinger equation equates the energy operator to the full energy of a particle or a system. Its results can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of a slowly changing (non-relativistic) wave function of quantum systems. The solution of this equation for
4218-418: Is directly proportional to the mass of the body: E 0 = m 0 c 2 , {\displaystyle E_{0}=m_{0}c^{2},} where For example, consider electron – positron annihilation, in which the rest energy of these two individual particles (equivalent to their rest mass) is converted to the radiant energy of the photons produced in the process. In this system
4332-516: Is either from gravitational collapse of matter (usually molecular hydrogen) into various classes of astronomical objects (stars, black holes, etc.), or from nuclear fusion (of lighter elements, primarily hydrogen). The nuclear fusion of hydrogen in the Sun also releases another store of potential energy which was created at the time of the Big Bang . At that time, according to theory, space expanded and
4446-510: Is extremely large relative to ordinary human scales, the conversion of an everyday amount of rest mass (for example, 1 kg) from rest energy to other forms of energy (such as kinetic energy, thermal energy, or the radiant energy carried by light and other radiation) can liberate tremendous amounts of energy (~ 9 × 10 16 {\displaystyle 9\times 10^{16}} joules = 21 megatons of TNT), as can be seen in nuclear reactors and nuclear weapons. Conversely,
4560-419: Is given by F ( r ) = − G M m r 2 r ^ , {\displaystyle \mathbf {F} (\mathbf {r} )=-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }},} where r ^ {\displaystyle {\hat {\mathbf {r} }}} is a unit vector pointing along the line from M to m , and G
4674-623: Is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction). Noether's theorem (1918) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalisation of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with
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4788-402: Is no friction or other losses, the conversion of energy between these processes would be perfect, and the pendulum would continue swinging forever. Energy is also transferred from potential energy ( E p {\displaystyle E_{p}} ) to kinetic energy ( E k {\displaystyle E_{k}} ) and then back to potential energy constantly. This
4902-410: Is not conservative in general, and hence cannot usually be written in terms of a scalar potential. However, it can be written in terms of a vector potential , A ( r ): B ( r ) = ∇ × A ( r ) {\displaystyle \mathbf {B} (\mathbf {r} )=\nabla \times \mathbf {A} (\mathbf {r} )} Gauss's law for magnetism in integral form
5016-411: Is now recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by using Lagrangians . This is a function that, when subjected to an action principle , gives rise to the field equations and a conservation law for the theory. The action is a Lorentz scalar, from which the field equations and symmetries can be readily derived. Throughout we use units such that
5130-616: Is processed at the facility every day. In its first year of operations, the Sacktory has prevented 250 tonnes of CO 2 emissions and helped First Mile reach the Mayor's 65% recycling target for London. The company continues to invest in recycling strategies. It is currently developing its RecycleLab, which will be completed in 2019. The RecycleLab analyses non-traditional materials to find recycling or sustainable end-of-life solutions. It also aims to help brands design products that can remain in
5244-425: Is produced. Newtonian gravitation is now superseded by Einstein's theory of general relativity , in which gravitation is thought of as being due to a curved spacetime , caused by masses. The Einstein field equations, G a b = κ T a b {\displaystyle G_{ab}=\kappa T_{ab}} describe how this curvature is produced by matter and radiation, where G ab
5358-415: Is referred to as conservation of energy. In this isolated system , energy cannot be created or destroyed; therefore, the initial energy and the final energy will be equal to each other. This can be demonstrated by the following: The equation can then be simplified further since E p = m g h {\displaystyle E_{p}=mgh} (mass times acceleration due to gravity times
5472-400: Is specifically intended to describe electromagnetism and gravitation , two of the fundamental forces of nature. A physical field can be thought of as the assignment of a physical quantity at each point of space and time . For example, in a weather forecast, the wind velocity during a day over a country is described by assigning a vector to each point in space. Each vector represents
5586-547: Is the Einstein tensor , G a b = R a b − 1 2 R g a b {\displaystyle G_{ab}\,=R_{ab}-{\frac {1}{2}}Rg_{ab}} written in terms of the Ricci tensor R ab and Ricci scalar R = R ab g , T ab is the stress–energy tensor and κ = 8 πG / c is a constant. In the absence of matter and radiation (including sources)
5700-696: Is the Planck constant and ν {\displaystyle \nu } the frequency). In the case of an electromagnetic wave these energy states are called quanta of light or photons . When calculating kinetic energy ( work to accelerate a massive body from zero speed to some finite speed) relativistically – using Lorentz transformations instead of Newtonian mechanics – Einstein discovered an unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest energy : energy which every massive body must possess even when being at rest. The amount of energy
5814-539: Is the magnetic field , which is determined from I by the Biot–Savart law : B ( r ) = μ 0 I 4 π ∫ d ℓ × d r ^ r 2 . {\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}I}{4\pi }}\int {\frac {d{\boldsymbol {\ell }}\times d{\hat {\mathbf {r} }}}{r^{2}}}.} The magnetic field
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#17327719551555928-493: Is the mass density , ρ e the charge density , G the gravitational constant and k e = 1/4πε 0 the electric force constant. Incidentally, this similarity arises from the similarity between Newton's law of gravitation and Coulomb's law . In the case where there is no source term (e.g. vacuum, or paired charges), these potentials obey Laplace's equation : ∇ 2 ϕ = 0. {\displaystyle \nabla ^{2}\phi =0.} For
6042-496: Is the watt , which is a joule per second. Thus, one joule is one watt-second, and 3600 joules equal one watt-hour. The CGS energy unit is the erg and the imperial and US customary unit is the foot pound . Other energy units such as the electronvolt , food calorie or thermodynamic kcal (based on the temperature change of water in a heating process), and BTU are used in specific areas of science and commerce. In 1843, French physicist James Prescott Joule , namesake of
6156-445: Is the main input to Earth's energy budget which accounts for its temperature and climate stability. Sunlight may be stored as gravitational potential energy after it strikes the Earth, as (for example when) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbines or generators to produce electricity). Sunlight also drives most weather phenomena, save
6270-1860: Is the volume form in curved spacetime. ( g ≡ det ( g μ ν ) ) {\displaystyle (g\equiv \det(g_{\mu \nu }))} Therefore, the Lagrangian itself is equal to the integral of the Lagrangian density over all space. Then by enforcing the action principle , the Euler–Lagrange equations are obtained δ S δ ϕ = ∂ L ∂ ϕ − ∂ μ ( ∂ L ∂ ( ∂ μ ϕ ) ) + ⋯ + ( − 1 ) m ∂ μ 1 ∂ μ 2 ⋯ ∂ μ m − 1 ∂ μ m ( ∂ L ∂ ( ∂ μ 1 ∂ μ 2 ⋯ ∂ μ m − 1 ∂ μ m ϕ ) ) = 0. {\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \phi }}={\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\right)+\cdots +(-1)^{m}\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\phi )}}\right)=0.} Two of
6384-424: Is transformed to what other kind) is often determined by entropy (equal energy spread among all available degrees of freedom ) considerations. In practice all energy transformations are permitted on a small scale, but certain larger transformations are not permitted because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces. Energy transformations in
6498-433: Is trapped in a system with zero momentum, where it can be weighed. It is also equivalent to mass, and this mass is always associated with it. Mass is also equivalent to a certain amount of energy, and likewise always appears associated with it, as described in mass–energy equivalence . The formula E = mc ², derived by Albert Einstein (1905) quantifies the relationship between relativistic mass and energy within
6612-408: Is used. The electromagnetic four-potential is defined to be A a = (− φ , A ) , and the electromagnetic four-current j a = (− ρ , j ) . The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor F a b = ∂ a A b − ∂ b A
6726-489: The circular economy , following the waste management hierarchy developed by the Waste Framework Directive (WFD) . Energy Energy (from Ancient Greek ἐνέργεια ( enérgeia ) 'activity') is the quantitative property that is transferred to a body or to a physical system , recognizable in the performance of work and in the form of heat and light . Energy
6840-436: The composite motion of the object's components – while potential energy reflects the potential of an object to have motion, generally being based upon the object's position within a field or what is stored within the field itself. While these two categories are sufficient to describe all forms of energy, it is often convenient to refer to particular combinations of potential and kinetic energy as its own form. For example,
6954-487: The gravitational collapse of supernovae to "store" energy in the creation of heavy isotopes (such as uranium and thorium ), and nuclear decay , a process in which energy is released that was originally stored in these heavy elements, before they were incorporated into the Solar System and the Earth. This energy is triggered and released in nuclear fission bombs or in civil nuclear power generation. Similarly, in
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#17327719551557068-416: The matter and antimatter (electrons and positrons) are destroyed and changed to non-matter (the photons). However, the total mass and total energy do not change during this interaction. The photons each have no rest mass but nonetheless have radiant energy which exhibits the same inertia as did the two original particles. This is a reversible process – the inverse process is called pair creation – in which
7182-474: The ' vacuum field equations , G a b = 0 {\displaystyle G_{ab}=0} can be derived by varying the Einstein–Hilbert action , S = ∫ R − g d 4 x {\displaystyle S=\int R{\sqrt {-g}}\,d^{4}x} with respect to the metric, where g is the determinant of the metric tensor g . Solutions of
7296-710: The 4-potential A , and it's this potential which enters the Euler-Lagrange equations. The EM field F is not varied in the EL equations. Therefore, ∂ b ( ∂ L ∂ ( ∂ b A a ) ) = ∂ L ∂ A a . {\displaystyle \partial _{b}\left({\frac {\partial {\mathcal {L}}}{\partial \left(\partial _{b}A_{a}\right)}}\right)={\frac {\partial {\mathcal {L}}}{\partial A_{a}}}\,.} Evaluating
7410-571: The Earth's gravitational field or elastic strain (mechanical potential energy) in rocks. Prior to this, they represent release of energy that has been stored in heavy atoms since the collapse of long-destroyed supernova stars (which created these atoms). In cosmology and astronomy the phenomena of stars , nova , supernova , quasars and gamma-ray bursts are the universe's highest-output energy transformations of matter. All stellar phenomena (including solar activity) are driven by various kinds of energy transformations. Energy in such transformations
7524-412: The action functional can be constructed by integrating over spacetime, S = ∫ L − g d 4 x . {\displaystyle {\mathcal {S}}=\int {{\mathcal {L}}{\sqrt {-g}}\,\mathrm {d} ^{4}x}.} Where − g d 4 x {\displaystyle {\sqrt {-g}}\,\mathrm {d} ^{4}x}
7638-457: The advent of relativity theory in 1905, and had to be revised to be consistent with that theory. Consequently, classical field theories are usually categorized as non-relativistic and relativistic . Modern field theories are usually expressed using the mathematics of tensor calculus . A more recent alternative mathematical formalism describes classical fields as sections of mathematical objects called fiber bundles . Michael Faraday coined
7752-510: The ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball. The total energy of a system is sometimes called the Hamiltonian , after William Rowan Hamilton . The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems. These classical equations have direct analogs in nonrelativistic quantum mechanics. Another energy-related concept
7866-447: The case of a chemical explosion , chemical potential energy is transformed to kinetic and thermal energy in a very short time. Yet another example is that of a pendulum . At its highest points the kinetic energy is zero and the gravitational potential energy is at its maximum. At its lowest point the kinetic energy is at its maximum and is equal to the decrease in potential energy . If one (unrealistically) assumes that there
7980-965: The case of animals. The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as food molecules, mostly carbohydrates and fats, of which glucose (C 6 H 12 O 6 ) and stearin (C 57 H 110 O 6 ) are convenient examples. The food molecules are oxidized to carbon dioxide and water in the mitochondria C 6 H 12 O 6 + 6 O 2 ⟶ 6 CO 2 + 6 H 2 O {\displaystyle {\ce {C6H12O6 + 6O2 -> 6CO2 + 6H2O}}} C 57 H 110 O 6 + ( 81 1 2 ) O 2 ⟶ 57 CO 2 + 55 H 2 O {\displaystyle {\ce {C57H110O6 + (81 1/2) O2 -> 57CO2 + 55H2O}}} and some of
8094-470: The comma indicates a partial derivative . After Newtonian gravitation was found to be inconsistent with special relativity , Albert Einstein formulated a new theory of gravitation called general relativity . This treats gravitation as a geometric phenomenon ('curved spacetime ') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor . The Einstein field equations describe how this curvature
8208-471: The complete conversion of matter (such as atoms) to non-matter (such as photons) is forbidden by conservation laws . Classical field theory A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations , without considering effects of quantization ; theories that incorporate quantum mechanics are called quantum field theories . In most contexts, 'classical field theory'
8322-451: The complex organisms can occupy ecological niches that are not available to their simpler brethren. The conversion of a portion of the chemical energy to heat at each step in a metabolic pathway is the physical reason behind the pyramid of biomass observed in ecology . As an example, to take just the first step in the food chain : of the estimated 124.7 Pg/a of carbon that is fixed by photosynthesis , 64.3 Pg/a (52%) are used for
8436-553: The concept of special relativity. In different theoretical frameworks, similar formulas were derived by J.J. Thomson (1881), Henri Poincaré (1900), Friedrich Hasenöhrl (1904) and others (see Mass–energy equivalence#History for further information). Part of the rest energy (equivalent to rest mass) of matter may be converted to other forms of energy (still exhibiting mass), but neither energy nor mass can be destroyed; rather, both remain constant during any process. However, since c 2 {\displaystyle c^{2}}
8550-399: The conservation of energy is a consequence of the fact that the laws of physics do not change over time. Thus, since 1918, theorists have understood that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time. In the International System of Units (SI), the unit of energy is the joule . It
8664-413: The density ρ , pressure p , deviatoric stress tensor τ of the fluid, as well as external body forces b , are all given. The velocity field u is the vector field to solve for. In 1839, James MacCullagh presented field equations to describe reflection and refraction in "An essay toward a dynamical theory of crystalline reflection and refraction". The term " potential theory " arises from
8778-449: The derivative of the Lagrangian density with respect to the field components ∂ L ∂ A a = μ 0 j a , {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{a}}}=\mu _{0}j^{a}\,,} and the derivatives of the field components ∂ L ∂ ( ∂ b A
8892-440: The direction of the field points from the position r to the position of the masses r i ; this is ensured by the minus sign. In a nutshell, this means all masses attract. In the integral form Gauss's law for gravity is ∬ g ⋅ d S = − 4 π G M {\displaystyle \iint \mathbf {g} \cdot d\mathbf {S} =-4\pi GM} while in differential form it
9006-405: The direction of the movement of air at that point, so the set of all wind vectors in an area at a given point in time constitutes a vector field . As the day progresses, the directions in which the vectors point change as the directions of the wind change. The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields were developed in classical physics before
9120-407: The electric field due to a single charged particle is E = 1 4 π ε 0 Q r 2 r ^ . {\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }}\,.} The electric field is conservative , and hence is given by the gradient of
9234-473: The energy is used to convert ADP into ATP : The rest of the chemical energy of the carbohydrate or fat are converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains is used for other metabolism when ATP reacts with OH groups and eventually splits into ADP and phosphate (at each stage of a metabolic pathway , some chemical energy is converted into heat). Only
9348-457: The energy stored during photosynthesis as heat or light may be triggered suddenly by a spark in a forest fire, or it may be made available more slowly for animal or human metabolism when organic molecules are ingested and catabolism is triggered by enzyme action. All living creatures rely on an external source of energy to be able to grow and reproduce – radiant energy from the Sun in the case of green plants and chemical energy (in some form) in
9462-524: The entire fleet of over 70 vehicles is either zero or low-emissions, and the company was ready for ULEZ ahead of its implementation in April 2019. In 2018 the construction of the pre-sorting facility named the Sacktory was completed. Located at the depot in Park Royal, the Sacktory ensures a higher recycling rate by separating mixed recycling materials from general waste. An average of 100 tonnes of waste
9576-491: The fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from scalar potentials which satisfied Laplace's equation . Poisson addressed the question of the stability of the planetary orbits , which had already been settled by Lagrange to the first degree of approximation from the perturbation forces, and derived the Poisson's equation , named after him. The general form of this equation
9690-493: The first time in the work of Aristotle in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure. In the late 17th century, Gottfried Leibniz proposed the idea of the Latin : vis viva , or living force, which defined as the product of the mass of an object and its velocity squared; he believed that total vis viva
9804-476: The fluid, ∂ ∂ t ( ρ u ) + ∇ ⋅ ( ρ u ⊗ u + p I ) = ∇ ⋅ τ + ρ b {\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} )=\nabla \cdot {\boldsymbol {\tau }}+\rho \mathbf {b} } if
9918-1002: The general divergence theorem , specifically Gauss's law's for gravity and electricity. For the cases of time-independent gravity and electromagnetism, the fields are gradients of corresponding potentials g = − ∇ ϕ g , E = − ∇ ϕ e {\displaystyle \mathbf {g} =-\nabla \phi _{g}\,,\quad \mathbf {E} =-\nabla \phi _{e}} so substituting these into Gauss' law for each case obtains ∇ 2 ϕ g = 4 π G ρ g , ∇ 2 ϕ e = 4 π k e ρ e = − ρ e ε 0 {\displaystyle \nabla ^{2}\phi _{g}=4\pi G\rho _{g}\,,\quad \nabla ^{2}\phi _{e}=4\pi k_{e}\rho _{e}=-{\rho _{e} \over \varepsilon _{0}}} where ρ g
10032-718: The head office moved from Essex Road railway station to the workspace Screenworks in Islington , North London. The main depot is located in Park Royal , northwest of the city. The company also has another office and depot in Birmingham . The company today serves over 27,000 businesses across the UK. In 2020, during the Coronavirus pandemic, First Mile provided its vans to deliver goods to hospitals, charities, and to support
10146-413: The height) and E k = 1 2 m v 2 {\textstyle E_{k}={\frac {1}{2}}mv^{2}} (half mass times velocity squared). Then the total amount of energy can be found by adding E p + E k = E total {\displaystyle E_{p}+E_{k}=E_{\text{total}}} . Energy gives rise to weight when it
10260-773: The identification of the gravitational field strength as identical to the acceleration experienced by a particle. This is the starting point of the equivalence principle , which leads to general relativity . For a discrete collection of masses, M i , located at points, r i , the gravitational field at a point r due to the masses is g ( r ) = − G ∑ i M i ( r − r i ) | r − r i | 3 , {\displaystyle \mathbf {g} (\mathbf {r} )=-G\sum _{i}{\frac {M_{i}(\mathbf {r} -\mathbf {r_{i}} )}{|\mathbf {r} -\mathbf {r} _{i}|^{3}}}\,,} If we have
10374-411: The interaction of charged matter with the electromagnetic field. The first formulation of this field theory used vector fields to describe the electric and magnetic fields. With the advent of special relativity, a more complete formulation using tensor fields was found. Instead of using two vector fields describing the electric and magnetic fields, a tensor field representing these two fields together
10488-460: The interaction term, and this gives us L = − 1 4 μ 0 F a b F a b − j a A a . {\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu _{0}}}F^{ab}F_{ab}-j^{a}A_{a}\,.} To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of
10602-482: The marginalia of her French language translation of Newton's Principia Mathematica , which represented the first formulation of a conserved measurable quantity that was distinct from momentum , and which would later be called "energy". In 1807, Thomas Young was possibly the first to use the term "energy" instead of vis viva , in its modern sense. Gustave-Gaspard Coriolis described " kinetic energy " in 1829 in its modern sense, and in 1853, William Rankine coined
10716-424: The mass equivalent of an everyday amount energy is minuscule, which is why a loss of energy (loss of mass) from most systems is difficult to measure on a weighing scale, unless the energy loss is very large. Examples of large transformations between rest energy (of matter) and other forms of energy (e.g., kinetic energy into particles with rest mass) are found in nuclear physics and particle physics . Often, however,
10830-474: The metabolism of green plants, i.e. reconverted into carbon dioxide and heat. In geology , continental drift , mountain ranges , volcanoes , and earthquakes are phenomena that can be explained in terms of energy transformations in the Earth's interior, while meteorological phenomena like wind, rain, hail , snow, lightning, tornadoes and hurricanes are all a result of energy transformations in our atmosphere brought about by solar energy . Sunlight
10944-427: The molecules it is built from. The second law of thermodynamics states that energy (and matter) tends to become more evenly spread out across the universe: to concentrate energy (or matter) in one specific place, it is necessary to spread out a greater amount of energy (as heat) across the remainder of the universe ("the surroundings"). Simpler organisms can achieve higher energy efficiencies than more complex ones, but
11058-404: The most well-known Lorentz-covariant classical field theories are now described. Historically, the first (classical) field theories were those describing the electric and magnetic fields (separately). After numerous experiments, it was found that these two fields were related, or, in fact, two aspects of the same field: the electromagnetic field . Maxwell 's theory of electromagnetism describes
11172-542: The other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that F is the 4-curl of A , or, in other words, from the fact that the Bianchi identity holds for the electromagnetic field tensor. 6 F [ a b , c ] = F a b , c + F c a , b + F b c , a = 0. {\displaystyle 6F_{[ab,c]}\,=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.} where
11286-405: The rest mass of particles is created from the radiant energy of two (or more) annihilating photons. In general relativity, the stress–energy tensor serves as the source term for the gravitational field, in rough analogy to the way mass serves as the source term in the non-relativistic Newtonian approximation. Energy and mass are manifestations of one and the same underlying physical property of
11400-474: The same total energy even in different forms) but its mass does decrease when the energy escapes out to its surroundings, largely as radiant energy . There are strict limits to how efficiently heat can be converted into work in a cyclic process, e.g. in a heat engine, as described by Carnot's theorem and the second law of thermodynamics . However, some energy transformations can be quite efficient. The direction of transformations in energy (what kind of energy
11514-467: The situation is the reverse. Chemical reactions are usually not possible unless the reactants surmount an energy barrier known as the activation energy . The speed of a chemical reaction (at a given temperature T ) is related to the activation energy E by the Boltzmann's population factor e ; that is, the probability of a molecule to have energy greater than or equal to E at
11628-543: The speed of light in vacuum is 1, i.e. c = 1. Given a field tensor ϕ {\displaystyle \phi } , a scalar called the Lagrangian density L ( ϕ , ∂ ϕ , ∂ ∂ ϕ , … , x ) {\displaystyle {\mathcal {L}}(\phi ,\partial \phi ,\partial \partial \phi ,\ldots ,x)} can be constructed from ϕ {\displaystyle \phi } and its derivatives. From this density,
11742-526: The sum of translational and rotational kinetic and potential energy within a system is referred to as mechanical energy , whereas nuclear energy refers to the combined potentials within an atomic nucleus from either the nuclear force or the weak force , among other examples. The word energy derives from the Ancient Greek : ἐνέργεια , romanized : energeia , lit. 'activity, operation', which possibly appears for
11856-420: The sun . The energy industry provides the energy required for human civilization to function, which it obtains from energy resources such as fossil fuels , nuclear fuel , and renewable energy . The total energy of a system can be subdivided and classified into potential energy , kinetic energy , or combinations of the two in various ways. Kinetic energy is determined by the movement of an object – or
11970-773: The system in terms of its scalar and vector potentials V and A . A set of integral equations known as retarded potentials allow one to calculate V and A from ρ and J , and from there the electric and magnetic fields are determined via the relations E = − ∇ V − ∂ A ∂ t {\displaystyle \mathbf {E} =-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}} B = ∇ × A . {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} .} Fluid dynamics has fields of pressure, density, and flow rate that are connected by conservation laws for energy and momentum. The mass continuity equation
12084-429: The term " potential energy ". The law of conservation of energy was also first postulated in the early 19th century, and applies to any isolated system . It was argued for some years whether heat was a physical substance, dubbed the caloric , or merely a physical quantity, such as momentum . In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat. These developments led to
12198-461: The term "field" and lines of forces to explain electric and magnetic phenomena. Lord Kelvin in 1851 formalized the concept of field in different areas of physics. Some of the simplest physical fields are vector force fields. Historically, the first time that fields were taken seriously was with Faraday's lines of force when describing the electric field . The gravitational field was then similarly described. The first field theory of gravity
12312-461: The theory of conservation of energy, formalized largely by William Thomson ( Lord Kelvin ) as the field of thermodynamics . Thermodynamics aided the rapid development of explanations of chemical processes by Rudolf Clausius , Josiah Willard Gibbs , and Walther Nernst . It also led to a mathematical formulation of the concept of entropy by Clausius and to the introduction of laws of radiant energy by Jožef Stefan . According to Noether's theorem ,
12426-420: The total energy of the substances involved. Some energy may be transferred between the surroundings and the reactants in the form of heat or light; thus the products of a reaction have sometimes more but usually less energy than the reactants. A reaction is said to be exothermic or exergonic if the final state is lower on the energy scale than the initial state; in the less common case of endothermic reactions
12540-406: The unit of measure, discovered that the gravitational potential energy lost by a descending weight attached via a string was equal to the internal energy gained by the water through friction with the paddle. In classical mechanics, energy is a conceptually and mathematically useful property, as it is a conserved quantity . Several formulations of mechanics have been developed using energy as
12654-421: The universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store of potential energy that can be released by fusion. Such a fusion process is triggered by heat and pressure generated from gravitational collapse of hydrogen clouds when they produce stars, and some of the fusion energy is then transformed into sunlight. In quantum mechanics , energy
12768-401: The universe over time are characterized by various kinds of potential energy, that has been available since the Big Bang , being "released" (transformed to more active types of energy such as kinetic or radiant energy) when a triggering mechanism is available. Familiar examples of such processes include nucleosynthesis , a process ultimately using the gravitational potential energy released from
12882-416: Was Newton's theory of gravitation in which the mutual interaction between two masses obeys an inverse square law . This was very useful for predicting the motion of planets around the Sun. Any massive body M has a gravitational field g which describes its influence on other massive bodies. The gravitational field of M at a point r in space is found by determining the force F that M exerts on
12996-440: Was conserved. To account for slowing due to friction, Leibniz theorized that thermal energy consisted of the motions of the constituent parts of matter, although it would be more than a century until this was generally accepted. The modern analog of this property, kinetic energy , differs from vis viva only by a factor of two. Writing in the early 18th century, Émilie du Châtelet proposed the concept of conservation of energy in
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