Lightning is the electric breakdown of air by strong electric fields, which produce a force on charges. When these charges move through a distance, a flow of energy occurs. The electric
potential energy in the atmosphere then is transformed into thermal energy, light, and sound, which are other forms of energy.
In
physics,
energy (
Ancient Greek:
ἐνέργεια energeia "activity, operation"
[1]) is a quantity that is often understood as the ability a
physical system has to do
work on other physical systems.
[2][3] Since work is defined as a
force acting through a distance (a length of space), energy is always equivalent to the ability to exert pulls or pushes against the basic forces of nature, along a path of a certain length.
The total energy contained in an object is identified with its
mass, and energy (like mass), cannot be created or destroyed. When
matter (ordinary material particles) is changed into energy (such as energy of motion, or into radiation), the
mass of the system does not change through the transformation process. However, there may be mechanistic limits as to how much of the matter in an object may be changed into other types of energy and thus into
work, on other systems. Energy, like mass, is a
scalar physical quantity. In the
International System of Units (SI), energy is measured in
joules, but in many fields other units, such as
kilowatt-hours and
kilocalories, are customary. All of these units translate to units of work, which is always defined in terms of forces and the distances that the forces act through.
A system can transfer energy to another system by simply transferring matter to it (since matter is equivalent to energy, in accordance with its mass). However, when energy is transferred by means other than matter-transfer, the transfer produces changes in the second system, as a result of work done on it. This work manifests itself as the effect of force(s) applied through distances within the target system. For example, a system can emit energy to another by transferring (radiating)
electromagnetic energy, but this creates forces upon the particles that absorb the radiation. Similarly, a system may transfer energy to another by physically impacting it, but that case the energy of motion in an object, called
kinetic energy, results in forces acting over distances (new energy) to appear in another object that is struck. Transfer of
thermal energy by
heat occurs by both of these mechanisms: heat can be transferred by electromagnetic radiation, or by physical contact in which direct particle-particle impacts transfer kinetic energy.
Energy may be stored in systems without being present as matter, or as kinetic or electromagnetic energy. Stored energy is created whenever a particle has been moved through a field it interacts with (requiring a force to do so), but the energy to accomplish this is stored as a new position of the particles in the field—a configuration that must be "held" or fixed by a different type of force (otherwise, the new configuration would resolve itself by the field pushing or pulling the particle back toward its previous position). This type of energy "stored" by force-fields and particles that have been forced into a new physical configuration in the field by doing work on them by another system, is referred to as
potential energy. A simple example of potential energy is the work needed to lift an object in a gravity field, up to a support. Each of the basic forces of nature is associated with a different type of potential energy, and all types of potential energy (like all other types of energy) appears as system
mass, whenever present. For example, a compressed spring will be slightly more massive than before it was compressed. Likewise, whenever energy is transferred between systems by any mechanism, an associated mass is transferred with it.
Any form of energy may be
transformed into another form. For example, all types of potential energy are converted into kinetic energy when the objects are given freedom to move to different position (as for example, when an object falls off a support). When energy is in a form other than thermal energy, it may be transformed with good or even perfect efficiency, to any other type of energy, including
electricity or production of new particles of matter. With thermal energy, however, there are often limits to the efficiency of the conversion to other forms of energy, as described by the
second law of thermodynamics.
In all such
energy transformation processes, the total energy remains the same, and a transfer of energy from one system to another, results in a loss to compensate for any gain. 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.
[4]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 (and higher total energy) relative to the
Earth.
History
The word
energy derives from the Greek
ἐνέργεια energeia, which possibly appears for the first time in the work of
Aristotle in the 4th century BC.
The concept of energy emerged out of the idea of
vis viva (living force), which
Gottfried 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 theorized that thermal energy 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 possibly the first to use the term "energy" instead of
vis viva, in its modern sense.
[5] 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.
William Thomson (
Lord Kelvin) amalgamated all of these laws into the laws of
thermodynamics, which aided in 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.
During a 1961 lecture
[6] 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 all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the
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.
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 instants of time (see
Noether's theorem).
Energy in various contexts
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, as 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 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 invariably accompanied by an increase or decrease of energy of the substances involved. Some energy is transferred between the surroundings and the reactants of the reaction in the form of heat or light; thus the products of a reaction may have more or less energy than the reactants. A reaction is said to be exergonic if the final state is lower on the energy scale than the initial state; in the case of endergonic reactions the situation is the reverse. Chemical reactions are invariably not possible unless the reactants surmount an energy barrier known as the activation energy. The speed of a chemical reaction (at given temperature T) is related to the activation energy E, by the Boltzmann's population factor e−E/kT – that is the probability of molecule to have energy greater than or equal to E at the 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 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 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), lipids, and proteins, which release energy when reacted with oxygen in respiration. 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, assuming an average human energy expenditure of 12,500kJ per day and 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 is about the maximum.[7] 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[8]
- 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.,[9] 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 atmosphere of 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 that 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 that was originally "stored" in heavy isotopes (such as
uranium and
thorium), by
nucleosynthesis, a process ultimately using 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,
radioactive decay of these 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 a kind of gravitational potential energy storage of the thermal energy, which may be later released to active kinetic energy in 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 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 created these atoms.
In another similar chain of transformations beginning at the dawn of the universe,
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 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. 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 turbines or 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 in
photosynthesis, when carbon dioxide and water (two low-energy compounds) are converted into the high-energy compounds carbohydrates, lipids, and proteins. Plants also release oxygen during photosynthesis, which is utilized by living organisms as an
electron acceptor, to release the energy of carbohydrates, lipids, and proteins. Release of 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 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.
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)
[10] 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.
[6][11] Conservation of energy is the mathematical consequence of
translational symmetry of
time (that is, the indistinguishability of time intervals taken at different time)
[12] - 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.
This 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
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.
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.
[13]- 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, while energy is always conserved (in the sense that the total energy does not change despite energy transformations), energy can be converted into a form, e.g., thermal energy, that cannot be utilized to perform work. When one talks about "conserving energy by driving less," one talks about conserving fossil fuels and preventing useful energy from being lost as heat. This usage of "conserve" differs from that of the law of conservation of energy.[11]
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).
[14] 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 the 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 the following equation:
if there are no other energy-transfer processes involved. Here
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:
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).
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 (
Ep) to kinetic energy (
Ek) and then back to potential energy constantly. This is referred to as conservation of energy. In this closed 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
Ep = mgh (mass times acceleration due to gravity times the height) and
(half mass times velocity squared). Then the total amount of energy can be found by adding
Ep + Ek = Etotal.
Energy and the laws of motion
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 a core concept.
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.
[15]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. 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 is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction).
Energy and thermodynamics
Internal energy
Internal energy is the sum of all microscopic forms of energy of a system. It is the energy needed to create the system. It is related to the potential energy, e.g., molecular structure, crystal structure, and other geometric aspects, as well as the motion of the particles, in form of kinetic energy. Thermodynamics is chiefly concerned with changes in internal energy and not its absolute value, which is impossible to determine with thermodynamics alone.
[16]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,
[17] 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 as the following equation:
,
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 d
S 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, d
V, is negative when work is done on the system). Although this equation is the standard textbook 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 the following equation:
,
which is unsatisfactory
[11] 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 (i.e., given new available
energy states that 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 whether the electric energy is 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
m2c4 = E2 − p2c2, the contribution
mc2 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
0.5p2 / m at speeds much smaller than
c, as can be proved by writing
E = mc2 √
(1 + p2m − 2c − 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, a form of energy, is force times distance.
This says that the work (
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 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. In results 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 waves in a vacuum, the resulting energy states are related to the frequency by the
Planck equation
E = hν (where
h is the
Planck's constant and
ν 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 an 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 = mc2,
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 this system the
matter (electrons and positrons) is destroyed and changed to non-matter energy (the photons). However, the total system mass and energy do not change during this interaction.
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.
[14]It is not uncommon to hear that energy is "equivalent" to mass. It would be more accurate to state that every energy has an inertia and gravity equivalent, and because mass is a form of energy, then mass too has inertia and gravity associated with it.
Energy and life
Main article:
BioenergeticsAny 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 (C
6H
12O
6) and
stearin (C
57H
110O
6) 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:
[18]- 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. 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").
[19] 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,
[20] i.e. reconverted into carbon dioxide and heat.
Measurement
Because energy is defined as the ability to do work on objects, there is no absolute measure of energy. Only the transition of a system from one state into another can be defined and thus energy is measured in relative terms. The choice of a baseline or zero point is often arbitrary and can be made in whatever way is most convenient for a problem.
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. In addition to the joule, other units of energy include the
kilowatt hour (kWh) and the
British thermal unit (Btu). These are both larger units of energy. One kWh is equivalent to exactly 3.6 million joules, and one Btu is equivalent to about 1055 joules.
[21]Energy density
Main article:
Energy density is a term used for the amount of useful energy stored in a given system or region of space per unit volume.
For
fuels, the energy per unit volume is sometimes a useful parameter. In a few applications, comparing, for example, the effectiveness of
hydrogen fuel to
gasoline it turns out that hydrogen has a higher
specific energy than does gasoline, but, even in liquid form, a much lower energy
density.
Forms of energy
In the context of
physical sciences, several forms of energy have been defined. These include:
These energies may be divided into two main groups;
kinetic energy and
potential energy. Other familiar types of energy are a varying mix of both potential and kinetic energy. Energy may be
transformed between these forms.
The above list of the known possible forms of energy is not necessarily complete. Whenever physical scientists discover that a certain phenomenon appears to violate the
law of energy conservation, new forms may be added, as is the case with
dark energy, a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe.
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:
These notions of potential and kinetic energy depend on a notion of length scale. For example, one can speak of macroscopic potential and kinetic energy, which do not include thermal potential and kinetic energy. Also what is called chemical potential energy (below) is a macroscopic notion, and closer examination shows that it is really the sum of the potential and kinetic energy on the atomic and subatomic scale. Similar remarks apply to nuclear "potential" energy and most other forms of energy. This dependence on length scale is non-problematic if the various length scales are decoupled, as is often the case ... but confusion can arise when different length scales are coupled, for instance when friction converts macroscopic work into microscopic thermal energy.
Mechanical energy
Mechanical energy manifest in many forms,but can be broadly classified into elastic potential energy and kinetic energy. The term potential energy is a very general term, because it exists in all force fields, such as gravitation, electrostatic and magnetic fields. Potential energy refers to the energy any object gets due to its position in a force field.
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,
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.
Elastic potential energy
As a ball falls freely under the influence of
gravity, it accelerates downward, its initial
potential energy converting into
kinetic energy. On impact with a hard surface the ball deforms, converting the kinetic energy into
elastic potential energy. As the ball springs back, the energy converts back firstly to kinetic energy and then as the ball re-gains height into potential energy. Energy conversion to heat due to
inelastic deformation and
air resistance cause each successive bounce to be lower than the last.
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
only when
k is constant. 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
Main article:
Kinetic energyKinetic 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:
At speeds approaching the
speed of light,
c, this work must be calculated using
Lorentz transformations, which results in the following:
Here the two terms on the right hand side are identified with the total energy and the rest energy of the object, respectively. This equation reduces to the one above it, at small (compared to
c) speed. The kinetic energy is zero at v=0, so that at rest, the total energy is the rest energy. Thus, a resting mass has the amount of total energy equal to:
- Erest = mc2
This energy is thus called
rest mass energy.
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).
Sound energy
Sound is a form of mechanical vibration which propagates through any mechanical medium. It is closely related to the ability of the human ear to perceive sound. The wide outer area of the ear is maximized to collect sound vibrations. It is amplified and passed through the outer ear, striking the eardrum, which transmits sounds into the inner ear. Auditory nerves fire according to the particular vibrations of the sound waves in the inner ear, which designate such things as the pitch and volume of the sound. The ear is set up in an optimal way to interpret sound energy in the form of vibrations.
Gravitational 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
- Ep,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
,
where
r is the separation between the two bodies and
G is the
gravitational constant, 6.6742(10)×10
−11 m
3kg
−1s
−2.
[22] In this case, the reference point is the infinite separation of the two bodies.
Thermal energy
Main article:
Thermal energyThermal 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 molecules in the 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
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
Electrostatic energy
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:
where ε
0 is the
electric constant of a vacuum, 10
7/4π
c0² or 8.854188…×10
−12 F/m.
[22] 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
Electric energy
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:
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:
while the energy stored in an
inductor (of
inductance L) when current
I is passing via it is
.
This second expression forms the basis for
superconducting magnetic energy storage.
Electromagnetic Energy
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:
and
,
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
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 = hν
where
h is the
Planck constant, 6.6260693(11)×10
−34 Js,
[22] 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
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 mutual positions of electric charges, electrons and protons, in the process of aggregation. So, basically it is electrostatic potential energy of electric charges. 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 kilojoules per mole.
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. 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 = ΔH − TΔS
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) = 29.3076 GJ = 8,141 kilowatt hour
- 1 tonne of oil equivalent (TOE) = 41.868 GJ = 11,630 kilowatt hour
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 storage of chemical energy are batteries and food. When food is digested and metabolized (often with oxygen), chemical energy is released, which can in turn be transformed into heat, or by muscles into kinetic energy.
Nuclear energy
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 the more-optimal size of the nucleus allows the
nuclear force (which is opposed by the
electromagnetic force) to bind nuclear particles more tightly together than before the reaction.
The
Weak nuclear force (different from the strong force) provides 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 or 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, although protons change into neutrons, the number of total protons-plus-neutrons does not change. In this system, the radiated light itself (as a system) retains the "missing" mass, 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 are destroyed in the process of turning the Sun's nuclear potential energy into light. Instead, the four nucleons in a helium nucleus in the Sun have an average mass that is less than the protons which formed them, and this mass difference (4 million tons/second) is the mass that moves off as sunlight.
[citation needed]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 gives rise to weight when it 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 =
mc², derived by
Albert Einstein (1905) quantifies the relationship between rest-mass and rest-energy within 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).
Matter may be destroyed and converted to energy (and vice versa), but mass cannot ever be destroyed; rather, mass remains a constant for both the matter and the energy, during any process when they are converted into each other. However, since
c2 is extremely large relative to ordinary human scales, the conversion of ordinary amount of
matter (for example, 1 kg) to other forms of energy (such as heat, light, and other radiation) can liberate tremendous amounts of energy (~
9x1016 joules = 21 megatons of TNT), 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 (loss of mass) from most systems is difficult to measure by weight, unless the energy loss is very large. Examples of energy transformation into matter (i.e., kinetic energy into particles with rest mass) are found in high-energy
nuclear physics.
Transformation of energy into useful work is a core topic of thermodynamics. 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, 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.
See also
Notes and references
- ^ Harper, Douglas. "Energy". Online Etymology Dictionary. Retrieved May 1, 2007.
- ^ "Retrieved on 2010-Dec-05". Faculty.clintoncc.suny.edu. Retrieved 2010-12-12.
- ^ "Retrieved on 2010-Dec-05" (PDF). Retrieved 2010-12-12.
- ^ Lofts, G; O'Keeffe D; et al. (2004). "11 — Mechanical Interactions". Jacaranda Physics 1 (2 ed.). Milton, Queensland, Australia: John Willey & Sons Australia Ltd.. p. 286. ISBN 0701637773.
- ^ Smith, Crosbie (1998). The Science of Energy - a Cultural History of Energy Physics in Victorian Britain. The University of Chicago Press. ISBN 0-226-76420-6.
- ^ a b Feynman, Richard (1964). The Feynman Lectures on Physics; Volume 1. U.S.A: Addison Wesley. ISBN 0-201-02115-3.
- ^ "Retrieved on May-29-09". Uic.edu. Retrieved 2010-12-12.
- ^ Bicycle calculator - speed, weight, wattage etc. [1].
- ^ "Earth's Energy Budget". Okfirst.ocs.ou.edu. Retrieved 2010-12-12.
- ^ "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws". Physics.ucla.edu. 1918-07-16. Retrieved 2010-12-12.
- ^ a b c The Laws of Thermodynamics including careful definitions of energy, free energy, et cetera.
- ^ "Time Invariance". Ptolemy.eecs.berkeley.edu. Retrieved 2010-12-12.
- ^ Berkeley Physics Course Volume 1. Charles Kittel, Walter D Knight and Malvin A Ruderman
- ^ a b Misner, Thorne, Wheeler (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0716703440.
- ^ The Hamiltonian MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007
- ^ I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39
- ^ Kittel and Kroemer (1980). Thermal Physics. New York: W. H. Freeman. ISBN 0-7167-1088-9.
- ^ These examples are solely for illustration, as it is not the energy available for work which limits the performance of the athlete but the power output of the sprinter and the force of the weightlifter. A worker stacking shelves in a supermarket does more work (in the physical sense) than either of the athletes, but does it more slowly.
- ^ Crystals are another example of highly ordered systems that exist in nature: in this case too, the order is associated with the transfer of a large amount of heat (known as the lattice energy) to the surroundings.
- ^ Ito, Akihito; Oikawa, Takehisa (2004). "Global Mapping of Terrestrial Primary Productivity and Light-Use Efficiency with a Process-Based Model." in Shiyomi, M. et al. (Eds.) Global Environmental Change in the Ocean and on Land. pp. 343–58.
- ^ Ristinen, Robert A., and Kraushaar, Jack J. Energy and the Environment. New York: John Wiley & Sons, Inc., 2006.
- ^ a b c Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80: 633–730. doi:10.1103/RevModPhys.80.633.
Further reading
- Alekseev, G. N. (1986). Energy and Entropy. Moscow: Mir Publishers.
- Crowell, Benjamin (2007) [2003]. Conservation Laws. Light and matter series of introductory physics textbooks. 2.2. Fullerton, CA: Light and Matter. ISBN 0970467028.
- Ross, John S. (23 April 2002). "Work, Power, Kinetic Energy". Project PHYSNET. Michigan State University.
- Smil, Vaclav (2008). Energy in nature and society: general energetics of complex systems. Cambridge, USA: MIT Press. ISBN 0-262-19565-8.
- Walding, Richard, Rapkins, Greg, Rossiter, Glenn (1999-11-01). New Century Senior Physics. Melbourne, Australia: Oxford University Press. ISBN 0-19-551084-4.
External links
