Definitions

electromagnetic energy

Energy

[en-er-jee]
In physics and other sciences, energy (from the Greek ἐνέργεια - energeia, "activity, operation", from ἐνεργός - energos, "active, working) is a scalar physical quantity, an attribute of objects and systems that is conserved in nature. In physics textbooks energy is often defined as the ability to do work.

Several different forms of energy, including, but not limited to, kinetic, potential, thermal, gravitational, sound energy, light energy, elastic, electromagnetic, chemical, nuclear, and mass have been defined to explain all known natural phenomena.

While one form of energy may be transformed to another, the total energy remains the same. This principle, the conservation of energy, was first postulated in the early 19th century, and applies to any isolated system. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.

Although the total energy of a system does not change with time, its value may depend on the frame of reference. For example, a seated passenger in a moving airplane has zero kinetic energy relative to the airplane, but non-zero kinetic energy relative to the earth.

History

The word "energy" derives from Greek ἐνέργεια (energeia), which appears for the first time in the work Nicomachean Ethics of Aristotle in the 4th century BC. In 1021 AD, the Arabian physicist, Alhazen, in the Book of Optics, held light rays to be streams of minute energy particles, stating that "the smallest parts of light" retain "only properties that can be treated by geometry and verified by experiment" and that "they lack all sensible qualities except energy. In 1121, Al-Khazini, in The Book of the Balance of Wisdom, proposed that the gravitational potential energy of a body varies depending on its distance from the centre of the Earth.

The concept of energy emerged out of the idea of vis viva, which Leibniz defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved. To account for slowing due to friction, Leibniz claimed that heat consisted of the random motion of the constituent parts of matter — a view shared by Isaac Newton, although it would be more than a century until this was generally accepted. In 1807, Thomas Young was 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 the term "potential energy." It was argued for some years whether energy was a substance (the caloric) or merely a physical quantity, such as momentum.

He[who?] amalgamated all of these laws into the laws of thermodynamics, which aided in the rapid development of explanations of chemical processes using the concept of energy 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.

During a 1961 lecture for undergraduate students at the California Institute of Technology, Richard Feynman, a celebrated physics teacher and Nobel Laureate, said this about the concept of energy:

There is a fact, or if you wish, a law, governing natural phenomena that are known to date. There is no known exception to this law; it is exact, so far we know. The law is called conservation of energy; it states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity, which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same.| | |The Feynman Lectures on Physics

Since 1918 it has been known that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time. That is, energy is conserved because the laws of physics do not distinguish between different moments of time (see Noether's theorem).

Energy in various contexts since the beginning of the universe

The concept of energy and its transformations is useful in explaining and predicting most natural phenomena. The direction of transformations in energy (what kind of energy is transformed to what other kind) is often described by entropy (equal energy spread among all available degrees of freedom) considerations, since 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.

The concept of energy is widespread in all sciences.

  • In biology, energy is an attribute of the biological structures that is responsible for growth and development of a biological cell or an organelle of a biological organism. Energy is thus often said to be stored by cells in the structures of molecules of substances such as carbohydrates (including sugars) and lipids, which release energy when reacted with oxygen.
  • In 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 invariably accompanied by an increase or decrease of energy of the substances involved.
  • In geology and meteorology, continental drift, mountain ranges, volcanos, 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 brought about by solar energy on the planet Earth.
  • 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 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).

Energy transformations in the universe over time are characterized by various kinds of potential energy which has been available since the Big Bang, later 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 nuclear decay, in which energy is released which was originally "stored" in heavy isotopes (such as uranium and thorium), by nucleosynthesis, a process which ultimately uses the gravitational potential energy released from the gravitational collapse of supernovae, to store energy in the creation of these heavy elements before they were incorporated into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs. In a slower process, heat from nuclear decay of these atoms in the core of the Earth releases heat, which in turn may lift mountains, via orogenesis. This slow lifting represents a kind of gravitational potential energy storage of the heat energy, which may be released to active kinetic energy in landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store which has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy which has been stored as potential energy in the Earth's gravitational field or elastic strain (mechanical potential energy) in rocks; but prior to this, represents energy that has been stored in heavy atoms since the collapse of long-destroyed stars created these atoms.

In another similar chain of transformations beginning at the dawn of the universe, nuclear fusion of hydrogen in the Sun 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 the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store of potential energy which 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. Such sunlight from our Sun may again be stored as gravitational potential energy after it strikes the Earth, as (for example) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbine/generators to produce electricity). Sunlight also drives many weather phenomena, save those generated by volcanic events. An example of a solar-mediated weather event is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, give up some of their thermal energy suddenly to power a few days of violent air movement. Sunlight is also captured by plants as chemical potential energy, when carbon dioxide and water are converted into a combustible combination of carbohydrates, lipids, and oxygen. Release of this energy as heat and light may be triggered suddenly by a spark, in a forest fire; or it may be available more slowly for animal or human metabolism, when these molecules are ingested, and catabolism is triggered by enzyme action. Through all of these transformation chains, potential energy stored at the time of the Big Bang is later released by intermediate events, sometimes being stored in a number of ways over time between releases, as more active energy. In all these events, one kind of energy is converted to other types of energy, including heat.

Regarding applications of the concept of energy

Energy is subject to a strict global conservation law; that is, whenever one measures (or calculates) the total energy of a system of particles whose interactions do not depend explicitly on time, it is found that the total energy of the system always remains constant.

  • The total energy of a system can be subdivided and classified in various ways. For example, it is sometimes convenient to distinguish potential energy (which is a function of coordinates only) from kinetic energy (which is a function of coordinate time derivatives only). It may also be convenient to distinguish gravitational energy, electric energy, thermal energy, and other forms. These classifications overlap; for instance thermal energy usually consists partly of kinetic and partly of potential energy.
  • The transfer of energy can take various forms; familiar examples include work, heat flow, and advection, as discussed below.
  • The word "energy" is also used outside of physics in many ways, which can lead to ambiguity and inconsistency. The vernacular terminology is not consistent with technical terminology. For example, the important public-service announcement, "Please conserve energy" uses vernacular notions of "conservation" and "energy" which make sense in their own context but are utterly incompatible with the technical notions of "conservation" and "energy" (such as are used in the law of conservation of energy).

In classical physics energy is considered a scalar quantity, the canonical conjugate to time. In special relativity energy 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 space-time (= boosts).

Energy transfer

Because energy is strictly conserved and is also locally conserved (wherever it can be defined), it is important to remember that by definition of energy the transfer of energy between the "system" and adjacent regions is work. A familiar example is mechanical work. In simple cases this is written as:

Delta{}E = W             (1)

if there are no other energy-transfer processes involved. Here Delta{}E  is the amount of energy transferred, and W  represents the work done on the system.

More generally, the energy transfer can be split into two categories:

Delta{}E = W + Q             (2)

where Q  represents the heat flow into the system.

There are other ways in which an open system can gain or lose energy. In chemical systems, energy can be added to a system by means of adding substances with different chemical potentials, which potentials are then extracted (both of these process are illustrated by fueling an auto, a system which gains in energy thereby, without addition of either work or heat). Winding a clock would be adding energy to a mechanical system. These terms may be added to the above equation, or they can generally be subsumed into a quantity called "energy addition term E" which refers to any type of energy carried over the surface of a control volume or system volume. Examples may be seen above, and many others can be imagined (for example, the kinetic energy of a stream of particles entering a system, or energy from a laser beam adds to system energy, without either being either work-done or heat-added, in the classic senses).

Delta{}E = W + Q + E             (3)

Where E in this general equation represents other additional advected energy terms not covered by work done on a system, or heat added to it.

Energy is also transferred from potential energy (E_p) to kinetic energy (E_k) and then back to potential energy constantly. This is referred to as conservation of energy. In this closed system, energy can not be created or destroyed, so the initial energy and the final energy will be equal to each other. This can be demonstrated by the following:

E_{pi} + E_{ki} = E_{pF} + E_{kF}

The equation can then be simplified further since E_p = mgh (mass times acceleration due to gravity times the height) and E_k = frac{1}{2} mv^2 (half times mass times velocity squared). Then the total amount of energy can be found by adding E_p + E_k = E_{total}.

Energy and the laws of motion

In classical mechanics, energy is a conceptually and mathematically useful property since it is a conserved quantity.

The Hamiltonian

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 remarkably direct analogs in nonrelativistic quantum mechanics.

The Lagrangian

Another energy-related concept is called the Lagrangian, after Joseph Louis Lagrange. This is even more fundamental than the Hamiltonian, and can be used to derive the equations of motion. In non-relativistic physics, the Lagrangian is the kinetic energy minus potential energy.

Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (like systems with friction).

Energy and thermodynamics

Internal energy

Internal energy – the sum of all microscopic forms of energy of a system. It is related to the molecular structure and the degree of molecular activity and may be viewed as the sum of kinetic and potential energies of the molecules; it comprises the following types of energy:

Type Composition of Internal Energy (U)
Sensible energy the portion of the internal energy of a system associated with kinetic energies (molecular translation, rotation, and vibration; electron translation and spin; and nuclear spin) of the molecules.
Latent energy the internal energy associated with the phase of a system.
Chemical energy the internal energy associated with the different kinds of aggregation of atoms in matter.
Nuclear energy the tremendous amount of energy associated with the strong bonds within the nucleus of the atom itself.
Energy interactions those types of energies not stored in the system (e.g. heat transfer, mass transfer, and work), but which are recognized at the system boundary as they cross it, which represent gains or losses by a system during a process.
Thermal energy the sum of sensible and latent forms of internal energy.

The laws of thermodynamics

According to the second law of thermodynamics, work can be totally converted into heat, but not vice versa.This is a mathematical consequence of statistical mechanics. The first law of thermodynamics simply asserts that energy is conserved, and that heat is included as a form of energy transfer. A commonly-used corollary of the first law is that for a "system" subject only to pressure forces and heat transfer (e.g. a cylinder-full of gas), the differential change in energy of the system (with a gain in energy signified by a positive quantity) is given by:

mathrm{d}E = Tmathrm{d}S - Pmathrm{d}V,,

where the first term on the right is the heat transfer into the system, defined in terms of temperature T and entropy S (in which entropy increases and the change dS is positive when the system is heated); and the last term on the right hand side is identified as "work" done on the system, where pressure is P and volume V (the negative sign results since compression of the system requires work to be done on it and so the volume change, dV, is negative when work is done on the system). Although this equation is the standard text-book example of energy conservation in classical thermodynamics, it is highly specific, ignoring all chemical, electric, nuclear, and gravitational forces, effects such as advection of any form of energy other than heat, and because it contains a term that depends on temperature. The most general statement of the first law (i.e., conservation of energy) is valid even in situations in which temperature is undefinable.

Energy is sometimes expressed as:

mathrm{d}E=delta Q+delta W,,

which is unsatisfactory because there cannot exist any thermodynamic state functions W or Q that are meaningful on the right hand side of this equation, except perhaps in trivial cases.

Equipartition of energy

The energy of a mechanical harmonic oscillator (a mass on a spring) is alternatively kinetic and potential. At two points in the oscillation cycle it is entirely kinetic, and alternatively at two other points it is entirely potential. Over the whole cycle, or over many cycles net energy is thus equally split between kinetic and potential. This is called equipartition principle - total energy of a system with many degrees of freedom is equally split among all available degrees of freedom.

This principle is vitally important to understanding the behavior of a quantity closely related to energy, called entropy. Entropy is a measure of evenness of a distribution of energy between parts of a system. When an isolated system is given more degrees of freedom (= is given new available energy states which are the same as existing states), then total energy spreads over all available degrees equally without distinction between "new" and "old" degrees. This mathematical result is called the second law of thermodynamics.

Oscillators, phonons, and photons

In an ensemble (connected collection) of unsynchronized oscillators, the average energy is spread equally between kinetic and potential types.

In a solid, thermal energy (often referred to loosely as heat content) can be accurately described by an ensemble of thermal phonons that act as mechanical oscillators. In this model, thermal energy is equally kinetic and potential.

In an ideal gas, the interaction potential between particles is essentially the delta function which stores no energy: thus, all of the thermal energy is kinetic.

Because an electric oscillator (LC circuit) is analogous to a mechanical oscillator, its energy must be, on average, equally kinetic and potential. It is entirely arbitrary whether the magnetic energy is considered kinetic and the electric energy considered potential, or vice versa. That is, either the inductor is analogous to the mass while the capacitor is analogous to the spring, or vice versa.

  1. By extension of the previous line of thought, in free space the electromagnetic field can be considered an ensemble of oscillators, meaning that radiation energy can be considered equally potential and kinetic. This model is useful, for example, when the electromagnetic Lagrangian is of primary interest and is interpreted in terms of potential and kinetic energy.
  2. On the other hand, in the key equation m^2 c^4 = E^2 - p^2 c^2, the contribution mc^2 is called the rest energy, and all other contributions to the energy are called kinetic energy. For a particle that has mass, this implies that the kinetic energy is 0.5 p^2/m at speeds much smaller than c, as can be proved by writing E = mc^2  √(1 + p^2 m^{-2}c^{-2}) and expanding the square root to lowest order. By this line of reasoning, the energy of a photon is entirely kinetic, because the photon is massless and has no rest energy. This expression is useful, for example, when the energy-versus-momentum relationship is of primary interest.

The two analyses are entirely consistent. The electric and magnetic degrees of freedom in item 1 are transverse to the direction of motion, while the speed in item 2 is along the direction of motion. For non-relativistic particles these two notions of potential versus kinetic energy are numerically equal, so the ambiguity is harmless, but not so for relativistic particles.

Work and virtual work

Work is roughly force times distance. But more precisely, it is

W = int mathbf{F} cdot mathrm{d}mathbf{s}

This says that the work (W) is equal to the integral (along a certain path) of the force; 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 the ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball.

Quantum mechanics

In quantum mechanics energy is defined in terms of the energy operator 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. It thus can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of slow changing (non-relativistic) wave function of quantum systems. The solution of this equation for 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 wave in vacuum, the resulting energy states are related to the frequency by the Planck equation E = hnu (where h is the Planck's constant and nu the frequency). In the case of electromagnetic wave these energy states are called quanta of light or photons.

Relativity

When calculating kinetic energy (= work to accelerate a mass from zero speed to some finite speed) relativistically - using Lorentz transformations instead of Newtonian mechanics, Einstein discovered unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest mass energy - energy which every mass must possess even when being at rest. The amount of energy is directly proportional to the mass of body:

E = m c^2 ,
where
m is the mass,
c is the speed of light in vacuum,
E is the rest mass energy.

For example, consider electron-positron annihilation, in which the rest mass of individual particles is destroyed, but the inertia equivalent of the system of the two particles (its invariant mass) remains (since all energy is associated with mass), and this inertia and invariant mass is carried off by photons which individually are massless, but as a system retain their mass. This is a reversible process - the inverse process is called pair creation - in which the rest mass of particles is created from 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.

It is not uncommon to hear that energy is "equivalent" to mass. It would be more accurate to state that every energy has inertia and gravity equivalent, and because mass is a form of energy, then mass too has inertia and gravity associated with it.

Measurement

There is no absolute measure of energy, because energy is defined as the work that one system does (or can do) on another. Thus, only of the transition of a system from one state into another can be defined and thus measured.

Methods

The methods for the measurement of energy often deploy methods for the measurement of still more fundamental concepts of science, namely mass, distance, radiation, temperature, time, electric charge and electric current. Conventionally the technique most often employed is calorimetry, a thermodynamic technique that relies on the measurement of temperature using a thermometer or of intensity of radiation using a bolometer.

Units

Throughout the history of science, energy has been expressed in several different units such as ergs and calories. At present, the accepted unit of measurement for energy is the SI unit of energy, the joule.

Forms of energy

Classical mechanics distinguishes between potential energy, which is a function of the position of an object, and kinetic energy, which is a function of its movement. Both position and movement are relative to a frame of reference, which must be specified: this is often (and originally) an arbitrary fixed point on the surface of the Earth, the terrestrial frame of reference. It has been attempted to categorize all forms of energy as either kinetic or potential: this is not incorrect, but neither is it clear that it is a real simplification, as Feynman points out:

Examples of the interconversion of energy
Mechanical energy is converted
into by
Mechanical energy Lever
Thermal energy Brakes
Electric energy Dynamo
Electromagnetic radiation Synchrotron
Chemical energy Matches
Nuclear energy Particle accelerator

Potential energy

Potential energy, symbols Ep, V or Φ, is defined as the work done against a given force (= work of given force with minus sign) in changing the position of an object with respect to a reference position (often taken to be infinite separation). If F is the force and s is the displacement,
E_{rm p} = -int mathbf{F}cdot{rm d}mathbf{s}
with the dot representing the scalar product of the two vectors.

The name "potential" energy originally signified the idea that the energy could readily be transferred as work—at least in an idealized system (reversible process, see below). This is not completely true for any real system, but is often a reasonable first approximation in classical mechanics.

The general equation above can be simplified in a number of common cases, notably when dealing with gravity or with elastic forces.

Gravitational potential energy

The gravitational force near the Earth's surface varies very little with the height, h, and is equal to the mass, m, multiplied by the gravitational acceleration, g = 9.81 m/s². In these cases, the gravitational potential energy is given by
E_{rm p,g} = mgh

A more general expression for the potential energy due to Newtonian gravitation between two bodies of masses m1 and m2, useful in astronomy, is

E_{rm p,g} = -G{{m_1m_2}over{r}},
where r is the separation between the two bodies and G is the gravitational constant, 6.6742(10)×10−11 m³kg−1s−2. In this case, the reference point is the infinite separation of the two bodies.

Elastic potential energy

Elastic potential energy is defined as a work needed to compress (or expand) a spring. The force, F, in a spring or any other system which obeys Hooke's law is proportional to the extension or compression, x,

F = -kx
where k is the force constant of the particular spring (or system). In this case, the calculated work becomes
E_{rm p,e} = {1over 2}kx^2.
Hooke's law is a good approximation for behaviour of chemical bonds under normal conditions, i.e. when they are not being broken or formed.

Kinetic energy

Kinetic energy, symbols Ek, T or K, is the work required to accelerate an object to a given speed. Indeed, calculating this work one easily obtains the following:
E_{rm k} = int mathbf{F} cdot d mathbf{x} = int mathbf{v} cdot d mathbf{p}= {1over 2}mv^2
At speeds approaching the speed of light, c, this work must be calculated using Lorentz transformations, which results in the following:
E_{rm k} = m c^2left(frac{1}{sqrt{1 - (v/c)^2}} - 1right)

This equation reduces to the one above it, at small (compared to c) speed. A mathematical by-product of this work (which is immediately seen in the last equation) is that even at rest a mass has the amount of energy equal to:

E_{rm rest} = mc^2

This energy is thus called rest mass energy.

Thermal energy

Examples of the interconversion of energy
Thermal energy is converted
into by
Mechanical energy Steam turbine
Thermal energy Heat exchanger
Electric energy Thermocouple
Electromagnetic radiation Hot objects
Chemical energy Blast furnace
Nuclear energy Supernova

Thermal energy (of some media - gas, plasma, solid, etc) is the energy associated with the microscopical random motion of particles constituting the media. For example, in case of monoatomic gas it is just a kinetic energy of motion of atoms of gas as measured in the reference frame of the center of mass of gas. In case of many-atomic gas rotational and vibrational energy is involved. In the case of liquids and solids there is also potential energy (of interaction of atoms) involved, and so on.

A heat is defined as a transfer (flow) of thermal energy across certain boundary (for example, from a hot body to cold via the area of their contact. A practical definition for small transfers of heat is

Delta q = int C_{rm v}{rm d}T
where Cv is the heat capacity of the system. This definition will fail if the system undergoes a phase transition—e.g. if ice is melting to water—as in these cases the system can absorb heat without increasing its temperature. In more complex systems, it is preferable to use the concept of internal energy rather than that of thermal energy (see Chemical energy below).

Despite the theoretical problems, the above definition is useful in the experimental measurement of energy changes. In a wide variety of situations, it is possible to use the energy released by a system to raise the temperature of another object, e.g. a bath of water. It is also possible to measure the amount of electric energy required to raise the temperature of the object by the same amount. The calorie was originally defined as the amount of energy required to raise the temperature of one gram of water by 1 °C (approximately 4.1855 J, although the definition later changed), and the British thermal unit was defined as the energy required to heat one pound of water by 1 °F (later fixed as 1055.06 J).

Electric energy

Examples of the interconversion of energy
Electric energy is converted
into by
Mechanical energy Electric motor
Thermal energy Resistor
Electric energy Transformer
Electromagnetic radiation Light-emitting diode
Chemical energy Electrolysis
Nuclear energy Synchrotron

The electric potential energy of given configuration of charges is defined as the work which must be done against the Coulomb force to rearrange charges from infinite separation to this configuration (or the work done by the Coulomb force separating the charges from this configuration to infinity). For two point-like charges Q1 and Q2 at a distance r this work, and hence electric potential energy is equal to:

E_{rm p,e} = {1over {4piepsilon_0}}{{Q_1Q_2}over{r}}
where ε0 is the electric constant of a vacuum, 107/4πc0² or 8.854188…×10−12 F/m. If the charge is accumulated in a capacitor (of capacitance C), the reference configuration is usually selected not to be infinite separation of charges, but vice versa - charges at an extremely close proximity to each other (so there is zero net charge on each plate of a capacitor). The justification for this choice is purely practical - it is easier to measure both voltage difference and magnitude of charges on a capacitor plates not versus infinite separation of charges but rather versus discharged capacitor where charges return to close proximity to each other (electrons and ions recombine making the plates neutral). In this case the work and thus the electric potential energy becomes
E_{rm p,e} = {{Q^2}over{2C}}

If an electric current passes through a resistor, electric energy is converted to heat; if the current passes through an electric appliance, some of the electric energy will be converted into other forms of energy (although some will always be lost as heat). The amount of electric energy due to an electric current can be expressed in a number of different ways:

E = UQ = UIt = Pt = U^2t/R = I^2Rt
where U is the electric potential difference (in volts), Q is the charge (in coulombs), I is the current (in amperes), t is the time for which the current flows (in seconds), P is the power (in watts) and R is the electric resistance (in ohms). The last of these expressions is important in the practical measurement of energy, as potential difference, resistance and time can all be measured with considerable accuracy.

Magnetic energy

There is no fundamental difference between magnetic energy and electric energy: the two phenomena are related by Maxwell's equations. The potential energy of a magnet of magnetic moment m in a magnetic field B is defined as the work of magnetic force (actually of magnetic torque) on re-alignment of the vector of the magnetic dipole moment, and is equal:
E_{rm p,m} = -mcdot B
while the energy stored in a inductor (of inductance L) when current I is passing via it is

E_{rm p,m} = {1over 2}LI^2.
This second expression forms the basis for superconducting magnetic energy storage.

Electromagnetic fields

Examples of the interconversion of energy
Electromagnetic radiation is converted
into by
Mechanical energy Solar sail
Thermal energy Solar collector
Electric energy Solar cell
Electromagnetic radiation Non-linear optics
Chemical energy Photosynthesis
Nuclear energy Mössbauer spectroscopy

Calculating work needed to create an electric or magnetic field in unit volume (say, in a capacitor or an inductor) results in the electric and magnetic fields energy densities:

u_e=frac{epsilon_0}{2} E^2
and
u_m=frac{1}{2mu_0} B^2 ,
in SI units.

Electromagnetic radiation, such as microwaves, visible light or gamma rays, represents a flow of electromagnetic energy. Applying the above expressions to magnetic and electric components of electromagnetic field both the volumetric density and the flow of energy in e/m field can be calculated. The resulting Poynting vector, which is expressed as

mathbf{S} = frac{1}{mu} mathbf{E} times mathbf{B},
in SI units, gives the density of the flow of energy and its direction.

The energy of electromagnetic radiation is quantized (has discrete energy levels). The spacing between these levels is equal to

E = hnu

where h is the Planck constant, 6.6260693(11)×10−34 Js, and ν is the frequency of the radiation. This quantity of electromagnetic energy is usually called a photon. The photons which make up visible light have energies of 270–520 yJ, equivalent to 160–310 kJ/mol, the strength of weaker chemical bonds.

Chemical energy

Examples of the interconversion of energy
Chemical energy is converted
into by
Mechanical energy Muscle
Thermal energy Fire
Electric energy Fuel cell
Electromagnetic radiation Glowworms
Chemical energy Chemical reaction

Chemical energy is the energy due to associations of atoms in molecules and various other kinds of aggregates of matter. It may be defined as a work done by electric forces during re-arrangement of electric charges, electrons and protons, in the process of aggregation. If the chemical energy of a system decreases during a chemical reaction, the difference is transferred to the surroundings in some form (often heat or light); on the other hand if the chemical energy of a system increases as a result of a chemical reaction - the difference then is supplied by the surroundings (usually again in form of heat or light). For example,

when two hydrogen atoms react to form a dihydrogen molecule, the chemical energy decreases by 724 zJ (the bond energy of the H–H bond);
when the electron is completely removed from a hydrogen atom, forming a hydrogen ion (in the gas phase), the chemical energy increases by 2.18 aJ (the ionization energy of hydrogen).
It is common to quote the changes in chemical energy for one mole of the substance in question: typical values for the change in molar chemical energy during a chemical reaction range from tens to hundreds of kJ/mol.

The chemical energy as defined above is also referred to by chemists as the internal energy, U: technically, this is measured by keeping the volume of the system constant. However, most practical chemistry is performed at constant pressure and, if the volume changes during the reaction (e.g. a gas is given off), a correction must be applied to take account of the work done by or on the atmosphere to obtain the enthalpy, H:

ΔH = ΔU + pΔV
A second correction, for the change in entropy, S, must also be performed to determine whether a chemical reaction will take place or not, giving the Gibbs free energy, G:
ΔG = ΔHTΔS
These corrections are sometimes negligible, but often not (especially in reactions involving gases).

Since the industrial revolution, the burning of coal, oil, natural gas or products derived from them has been a socially significant transformation of chemical energy into other forms of energy. the energy "consumption" (one should really speak of "energy transformation") of a society or country is often quoted in reference to the average energy released by the combustion of these fossil fuels:

1  tonne of coal equivalent (TCE) = 29 GJ
tonne of oil equivalent (TOE) = 41.87 GJ
On the same basis, a tank-full of gasoline (45 litres, 12 gallons) is equivalent to about 1.6 GJ of chemical energy. Another chemically-based unit of measurement for energy is the "tonne of TNT", taken as 4.184 GJ. Hence, burning a tonne of oil releases about ten times as much energy as the explosion of one tonne of TNT: fortunately, the energy is usually released in a slower, more controlled manner.

Simple examples of chemical energy are batteries and food. When you eat the food is digested and turned into chemical energy which can be transformed to kinetic energy.

Nuclear energy

Examples of the interconversion of energy
Nuclear binding energy is converted
into by
Mechanical energy Alpha radiation
Thermal energy Sun
Electrical energy Beta radiation
Electromagnetic radiation Gamma radiation
Chemical energy Radioactive decay
Nuclear energy Nuclear isomerism

Nuclear potential energy, along with electric potential energy, provides the energy released from nuclear fission and nuclear fusion processes. The result of both these processes are nuclei in which strong nuclear forces bind nuclear particles more strongly and closely. Weak nuclear forces (different from strong forces) provide the potential energy for certain kinds of radioactive decay, such as beta decay. The energy released in nuclear processes is so large that the relativistic change in mass (after the energy has been removed) can be as much as several parts per thousand.

Nuclear particles (nucleons) like protons and neutrons are not destroyed (law of conservation of baryon number) in fission and fusion processes. A few lighter particles may be created or destroyed (example: beta minus and beta plus decay, or electron capture decay), but these minor processes are not important to the immediate energy release in fission and fusion. Rather, fission and fusion release energy when collections of baryons become more tightly bound, and it is the energy associated with a fraction of the mass of the nucleons (but not the whole particles) which appears as the heat and electromagnetic radiation generated by nuclear reactions. This heat and radiation retains the "missing" mass, but the mass is missing only because it escapes in the form of heat and light, which retain the mass and conduct it out of the system where it is not measured. The energy from the Sun, also called solar energy, is an example of this form of energy conversion. In the Sun, the process of hydrogen fusion converts about 4 million metric tons of solar matter per second into light, which is radiated into space, but during this process, the number of total protons and neutrons in the sun does not change. In this system, the light itself retains the inertial equivalent of this mass, and indeed the mass itself (as a system), which represents 4 million tons per second of electromagnetic radiation, moving into space. Each of the helium nuclei which are formed in the process are less massive than the four protons from they were formed, but (to a good approximation), no particles or atoms are destroyed in the process of turning the sun's nuclear potential energy into light.

Surface energy

If there is any kind of tension in a surface, such as a stretched sheet of rubber or material interfaces, it is possible to define surface energy. In particular, any meeting of dissimilar materials that don't mix will result in some kind of surface tension, if there is freedom for the surfaces to move then, as seen in capillary surfaces for example, the minimum energy will as usual be sought.

A minimal surface, for example, represents the smallest possible energy that a surface can have if its energy is proportional to the area of the surface. For this reason, (open) soap films of small size are minimal surfaces (small size reduces gravity effects, and openness prevents pressure from building up. Note that a bubble is a minimum energy surface but not a minimal surface by definition).

Transformations of energy

One form of energy can often be readily transformed into another with the help of a device- for instance, a battery, from chemical energy to electric energy; a dam: gravitational potential energy to kinetic energy of moving water (and the blades of a turbine) and ultimately to electric energy through an electric generator. Similarly, in the case of a chemical explosion, chemical potential energy is transformed to kinetic energy 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 maximum. At its lowest point the kinetic energy is at maximum and is equal to the decrease of potential energy. If one (unrealistically) assumes that there is no friction, the conversion of energy between these processes is perfect, and the pendulum will continue swinging forever.

Energy can be converted into matter and vice versa. The mass-energy equivalence formula E = mc², derived by several authors: Olinto de Pretto, Albert Einstein, Friedrich Hasenöhrl, Max Planck and Henri Poincaré, quantifies the relationship between mass and rest energy. Since c^2 is extremely large relative to ordinary human scales, the conversion of ordinary amount of mass (say, 1 kg) to other forms of energy can liberate tremendous amounts of energy (~9x10^{16} Joules), as can be seen in nuclear reactors and nuclear weapons. Conversely, the mass equivalent of a unit of energy is minuscule, which is why a loss of energy from most systems is difficult to measure by weight, unless the energy loss is very large. Examples of energy transformation into matter (particles) are found in high energy nuclear physics.

In nature, transformations of energy can be fundamentally classed into two kinds: those that are thermodynamically reversible, and those that are thermodynamically irreversible. A reversible process in thermodynamics is one in which no energy is dissipated (spread) into empty energy states available in a volume, from which it cannot be recovered into more concentrated forms (fewer quantum states), without degradation of even more energy. A reversible process is one in which this sort of dissipation does not happen. For example, conversion of energy from one type of potential field to another, is reversible, as in the pendulum system described above. In processes where heat is generated, however, quantum states of lower energy, present as possible exitations in fields between atoms, act as a reservoir for part of the energy, from which it cannot be recovered, in order to be converted with 100% efficiency into other forms of energy. In this case, the energy must partly stay as heat, and cannot be completely recovered as usable energy, except at the price of an increase in some other kind of heat-like increase in disorder in quantum states, in the universe (such as an expansion of matter, or a randomization in a crystal).

As the universe evolves in time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or other kinds of increases in disorder). This has been referred to as the inevitable thermodynamic heat death of the universe. In this heat death the energy of the universe does not change, but the fraction of energy which is available to do produce work through a heat engine, or be transformed to other usable forms of energy (through the use of generators attached to heat engines), grows less and less.

Law of conservation of energy

Energy is subject to the law of conservation of energy. According to this law, energy can neither be created (produced) nor destroyed by itself. It can only be transformed.

Most kinds of energy (with gravitational energy being a notable exception) are also subject to strict local conservation laws, as well. In this case, energy can only be exchanged between adjacent regions of space, and all observers agree as to the volumetric density of energy in any given space. There is also a global law of conservation of energy, stating that the total energy of the universe cannot change; this is a corollary of the local law, but not vice versa. Conservation of energy is the mathematical consequence of translational symmetry of time (that is, the indistinguishability of time intervals taken at different time) - see Noether's theorem.

According to energy conservation law the total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system.

This law is a fundamental principle of physics. It follows from the translational symmetry of time, a property of most phenomena below the cosmic scale that makes them independent of their locations on the time coordinate. Put differently, yesterday, today, and tomorrow are physically indistinguishable.

Thus is because energy is the quantity which is canonical conjugate to time. This mathematical entanglement of energy and time also results in the uncertainty principle - it is impossible to define the exact amount of energy during any definite time interval. The uncertainty principle should not be confused with energy conservation - rather it provides mathematical limits to which energy can in principle be defined and measured.

In quantum mechanics energy is expressed using the Hamiltonian operator. On any time scales, the uncertainty in the energy is by

Delta E Delta t ge frac { hbar } {2 }

which is similar in form to the Heisenberg uncertainty principle (but not really mathematically equivalent thereto, since H and t are not dynamically conjugate variables, neither in classical nor in quantum mechanics).

In particle physics, this inequality permits a qualitative understanding of virtual particles which carry momentum, exchange by which and with real particles, is responsible for the creation of all known fundamental forces (more accurately known as fundamental interactions). Virtual photons (which are simply lowest quantum mechanical energy state of photons) are also responsible for electrostatic interaction between electric charges (which results in Coulomb law), for spontaneous radiative decay of exited atomic and nuclear states, for the Casimir force, for van der Waals bond forces and some other observable phenomena.

Energy and life

Any living organism relies on an external source of energy—radiation from the Sun in the case of green plants; chemical energy in some form in the case of animals—to be able to grow and reproduce. The daily 1500–2000 Calories (6–8 MJ) recommended for a human adult are taken as a combination of oxygen and food molecules, the latter mostly carbohydrates and fats, of which glucose (C6H12O6) and stearin (C57H110O6) are convenient examples. The food molecules are oxidised to carbon dioxide and water in the mitochondria
C6H12O6 + 6O2 → 6CO2 + 6H2O
C57H110O6 + 81.5O2 → 57CO2 + 55H2O
and some of the energy is used to convert ADP into ATP
ADP + HPO42− → ATP + H2O
The rest of the chemical energy in the carbohydrate or fat is converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains when split and reacted with water, is used for other metabolism (at each stage of a metabolic pathway, some chemical energy is converted into heat). Only a tiny fraction of the original chemical energy is used for work:
gain in kinetic energy of a sprinter during a 100 m race: 4 kJ
gain in gravitational potential energy of a 150 kg weight lifted through 2 metres: 3kJ
Daily food intake of a normal adult: 6–8 MJ

It would appear that living organisms are remarkably inefficient (in the physical sense) in their use of the energy they receive (chemical energy or radiation), and it is true that most real machines manage higher efficiencies. However, 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 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 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: 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 the metabolism of green plants, i.e. reconverted into carbon dioxide and heat.

See also

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