A two-dimensional gas is a collection of objects constrained to move in a planar or other two-dimensional space in a gaseous state. The objects can be: ideal gas elements such as rigid disks undergoing elastic collisions; the concept of a two-dimensional gas is used either because: the issue being studied takes place in two dimensions. While physicists have studied simple two body interactions on a plane for centuries, the attention given to the two-dimensional gas is a 20th-century pursuit. Applications have led to better understanding of superconductivity, gas thermodynamics, certain solid state problems and several questions in quantum mechanics. Research at Princeton University in the early 1960s posed the question of whether the Maxwell–Boltzmann statistics and other thermodynamic laws could be derived from Newtonian laws applied to multi-body systems rather than through the conventional methods of statistical mechanics. While this question appears intractable from a three-dimensional closed form solution, the problem behaves differently in two-dimensional space.
In particular an ideal two-dimensional gas was examined from the standpoint of relaxation time to equilibrium velocity distribution given several arbitrary initial conditions of the ideal gas. Relaxation times were shown to be fast: on the order of mean free time. In 1996 a computational approach was taken to the classical mechanics non-equilibrium problem of heat flow within a two-dimensional gas; this simulation work showed. While the principle of the cyclotron to create a two-dimensional array of electrons has existed since 1934, the tool was not used to analyze interactions among the electrons. An early research investigation explored cyclotron resonance behavior and the de Haas–van Alphen effect in a two-dimensional electron gas; the investigator was able to demonstrate that for a two-dimensional gas, the de Haas–van Alphen oscillation period is independent of the short-range electron interactions. In 1991 a theoretical proof was made. In the same work an experimental recommendation was made.
In general, 2D molecular gases are experimentally observed on weakly interacting surfaces such as metals, graphene etc. at a non-cryogenic temperature and a low surface coverage. As a direct observation of individual molecules is not possible due to fast diffusion of molecules on a surface, experiments are either indirect or integral. An example of the indirect observation of a 2D gas is the study of Stranick et al. who used a scanning tunnelling microscope in ultrahigh vacuum to image an interaction of a two-dimensional benzene gas layer in contact with a planar solid interface at 77 kelvins. The experimenters were able to observe mobile benzene molecules on the surface of Cu, to which a planar monomolecular film of solid benzene adhered, thus the scientists could witness the equilibrium of the gas in contact with its solid state. Integral methods that are able to characterize a 2D gas fall into a category of diffraction; the exception is the work of Matvija et al. who used a scanning tunneling microscope to directly visualize a local time-averaged density of molecules on a surface.
This method is of special importance as it provides an opportunity to probe local properties of 2D gases. If the surface coverage of adsorbates is increased, a 2D liquid is formed, followed by a 2D solid, it was shown that the transition from a 2D gas to a 2D solid state can be controlled by a scanning tunneling microscope which can affect the local density of molecules via an electric field. A multiplicity of theoretical physics research directions exist for study via a two-dimensional gas. Examples of these are Complex quantum mechanics phenomena, whose solutions may be more appropriate in a two-dimensional environment. Bose gas Fermi gas Melting point Optical lattice Three-body problem Riemann problems for a two-dimensional gas Two-dimensional gas of disks
Degrees of freedom (physics and chemistry)
In physics, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, degrees of freedom of the system, are the dimensions of the phase space; the location of a particle in three-dimensional space requires. The direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic, where the state at one instant uniquely determines its past and future position and velocity as a function of time, such a system has six degrees of freedom. If the motion of the particle is constrained to a lower number of dimensions, for example, the particle must move along a wire or on a fixed surface the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.
In classical mechanics, the state of a point particle at any given time is described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism. In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system; the specification of all microstates of a system is a point in the system's phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer, it is useful to specify quadratic degrees of freedom. These are degrees of freedom. In three-dimensional space, three degrees of freedom are associated with the movement of a particle. A diatomic gas molecule has 6 degrees of freedom; this set may be decomposed in terms of translations and vibrations of the molecule. The center of mass motion of the entire molecule accounts for 3 degrees of freedom. In addition, the molecule has two rotational degrees of one vibrational mode.
The rotations occur around the two axes perpendicular to the line between the two atoms. The rotation around the atom–atom bond is not a physical rotation; this yields, for a diatomic molecule, a decomposition of: N = 6 = 3 + 2 + 1. For a general, non-linear molecule, all 3 rotational degrees of freedom are considered, resulting in the decomposition: 3 N = 3 + 3 + which means that an N-atom molecule has 3N − 6 vibrational degrees of freedom for N > 2. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one; as defined above one can count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows: For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space, thus its degree of freedom in a 3-D space is 3. For a body consisting of 2 particles in a 3-D space with constant distance between them we can show its degrees of freedom to be 5.
Let's say the other has coordinate with z2 unknown. Application of the formula for distance between two coordinates d = 2 + 2 + 2 results in one equation with one unknown, in which we can solve for z2. One of x1, x2, y1, y2, z1, or z2 can be unknown. Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules makes negligible contributions to the heat capacity; this is because these degrees of freedom are frozen because the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures. In the following table such degrees of freedom are disregarded because of their low effect on total energy. Only the translational and rotational degrees of freedom contribute, in equal amounts, to the heat capacity ratio; this is why γ = 7/5 for diatomic gases at room temperature. However, at high temperatures, on the order of the vibrational temperature, vibrational motion cannot be neglected. Vibrational temperatures are between 103 K and 104 K.
The set of degrees of freedom X1, ... , XN of a system is independent if the energy associated with the set can be written in the following form: E = ∑ i = 1 N E i, where Ei is a function of the sole variable Xi. example: if X1 and X2 are two degrees of freedom, E is the associated energy: If E = X 1 4 + X 2 4 the two degrees of freedom are independent. If E = X 1 4 + X 1
The neutron is a subatomic particle, symbol n or n0, with no net electric charge and a mass larger than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behave within the nucleus, each has a mass of one atomic mass unit, they are both referred to as nucleons, their properties and interactions are described by nuclear physics. The chemical and nuclear properties of the nucleus are determined by the number of protons, called the atomic number, the number of neutrons, called the neutron number; the atomic mass number is the total number of nucleons. For example, carbon has atomic number 6, its abundant carbon-12 isotope has 6 neutrons, whereas its rare carbon-13 isotope has 7 neutrons; some elements occur in nature with only one stable isotope, such as fluorine. Other elements occur with many stable isotopes, such as tin with ten stable isotopes. Within the nucleus and neutrons are bound together through the nuclear force. Neutrons are required for the stability of nuclei, with the exception of the single-proton hydrogen atom.
Neutrons are produced copiously in nuclear fusion. They are a primary contributor to the nucleosynthesis of chemical elements within stars through fission and neutron capture processes; the neutron is essential to the production of nuclear power. In the decade after the neutron was discovered by James Chadwick in 1932, neutrons were used to induce many different types of nuclear transmutations. With the discovery of nuclear fission in 1938, it was realized that, if a fission event produced neutrons, each of these neutrons might cause further fission events, etc. in a cascade known as a nuclear chain reaction. These events and findings led to the first self-sustaining nuclear reactor and the first nuclear weapon. Free neutrons, while not directly ionizing atoms, cause ionizing radiation; as such they can be a biological hazard, depending upon dose. A small natural "neutron background" flux of free neutrons exists on Earth, caused by cosmic ray showers, by the natural radioactivity of spontaneously fissionable elements in the Earth's crust.
Dedicated neutron sources like neutron generators, research reactors and spallation sources produce free neutrons for use in irradiation and in neutron scattering experiments. An atomic nucleus is formed by a number of protons, Z, a number of neutrons, N, bound together by the nuclear force; the atomic number defines the chemical properties of the atom, the neutron number determines the isotope or nuclide. The terms isotope and nuclide are used synonymously, but they refer to chemical and nuclear properties, respectively. Speaking, isotopes are two or more nuclides with the same number of protons; the atomic mass number, symbol A, equals Z+N. Nuclides with the same atomic mass number are called isobars; the nucleus of the most common isotope of the hydrogen atom is a lone proton. The nuclei of the heavy hydrogen isotopes deuterium and tritium contain one proton bound to one and two neutrons, respectively. All other types of atomic nuclei are composed of two or more protons and various numbers of neutrons.
The most common nuclide of the common chemical element lead, 208Pb, has 82 protons and 126 neutrons, for example. The table of nuclides comprises all the known nuclides. Though it is not a chemical element, the neutron is included in this table; the free neutron has 1.674927471 × 10 − 27 kg, or 1.00866491588 u. The neutron has a mean square radius of about 0.8×10−15 m, or 0.8 fm, it is a spin-½ fermion. The neutron has no measurable electric charge. With its positive electric charge, the proton is directly influenced by electric fields, whereas the neutron is unaffected by electric fields; the neutron has a magnetic moment, however. The neutron's magnetic moment has a negative value, because its orientation is opposite to the neutron's spin. A free neutron is unstable, decaying to a proton and antineutrino with a mean lifetime of just under 15 minutes; this radioactive decay, known as beta decay, is possible because the mass of the neutron is greater than the proton. The free proton is stable. Neutrons or protons bound in a nucleus can be stable or unstable, depending on the nuclide.
Beta decay, in which neutrons decay to protons, or vice versa, is governed by the weak force, it requires the emission or absorption of electrons and neutrinos, or their antiparticles. Protons and neutrons behave identically under the influence of the nuclear force within the nucleus; the concept of isospin, in which the proton and neutron are viewed as two quantum states of the same particle, is used to model the interactions of nucleons by the nuclear or weak forces. Because of the strength of the nuclear force at short distances, the binding energy of nucleons is more than seven orders of magnitude larger than the electromagnetic energy binding electrons in atoms. Nuclear reactions therefore have an energy density, more than ten million times that of chemical reactions; because of the mass–energy equivalence, nuclear binding energies reduce the mass of nuclei. The ability of the nuclear force to store energy arising from the electromagnetic repulsion of nuclear components is the basis for most of the energy that makes nuclear reactors or bombs possible.
In nuclear fission, the absorption of a neutron by a heavy nuclide causes the nuclide to become unstable and break into light nuclides and additional neu
Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, astronomical objects, such as spacecraft, planets and galaxies. If the present state of an object is known it is possible to predict by the laws of classical mechanics how it will move in the future and how it has moved in the past; the earliest development of classical mechanics is referred to as Newtonian mechanics. It consists of the physical concepts employed by and the mathematical methods invented by Isaac Newton and Gottfried Wilhelm Leibniz and others in the 17th century to describe the motion of bodies under the influence of a system of forces. More abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics; these advances, made predominantly in the 18th and 19th centuries, extend beyond Newton's work through their use of analytical mechanics. They are, with some modification used in all areas of modern physics.
Classical mechanics provides accurate results when studying large objects that are not massive and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In case that objects become massive, general relativity becomes applicable. However, a number of modern sources do include relativistic mechanics into classical physics, which in their view represents classical mechanics in its most developed and accurate form; the following introduces the basic concepts of classical mechanics. For simplicity, it models real-world objects as point particles; the motion of a point particle is characterized by a small number of parameters: its position and the forces applied to it. Each of these parameters is discussed in turn. In reality, the kind of objects that classical mechanics can describe always have a non-zero size.
Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom, e.g. a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles; the center of mass of a composite object behaves like a point particle. Classical mechanics uses common-sense notions of how matter and forces interact, it assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics assumes that forces act instantaneously; the position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O.
In cases where P is moving relative to O, r is defined as a function of time. In pre-Einstein relativity, time is considered an absolute, i.e. the time interval, observed to elapse between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space; the velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time: v = d r d t. In classical mechanics, velocities are directly subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at 60 − 50 = 10 km/h. However, from the perspective of the faster car, the slower car is moving 10 km/h to the west denoted as -10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities. Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve, where u is the speed of the first object, v is the speed of the second object, d and e are unit vectors in the directions of motion of each object then the velocity of the first object as seen by the second object is u ′ = u − v. Similarly, the first object sees the velocity of the second object as v ′ = v − u.
When both objects are moving in the same direction, this equation can be simplified to u ′ = d. Or, by ignoring direction, the difference can be given in terms of speed only: u ′ = u − v; the acceleration, or rate of change of velocity, is th
A molecule is an electrically neutral group of two or more atoms held together by chemical bonds. Molecules are distinguished from ions by their lack of electrical charge. However, in quantum physics, organic chemistry, biochemistry, the term molecule is used less also being applied to polyatomic ions. In the kinetic theory of gases, the term molecule is used for any gaseous particle regardless of its composition. According to this definition, noble gas atoms are considered molecules as they are monatomic molecules. A molecule may be homonuclear, that is, it consists of atoms of one chemical element, as with oxygen. Atoms and complexes connected by non-covalent interactions, such as hydrogen bonds or ionic bonds, are not considered single molecules. Molecules as components of matter are common in organic substances, they make up most of the oceans and atmosphere. However, the majority of familiar solid substances on Earth, including most of the minerals that make up the crust and core of the Earth, contain many chemical bonds, but are not made of identifiable molecules.
No typical molecule can be defined for ionic crystals and covalent crystals, although these are composed of repeating unit cells that extend either in a plane or three-dimensionally. The theme of repeated unit-cellular-structure holds for most condensed phases with metallic bonding, which means that solid metals are not made of molecules. In glasses, atoms may be held together by chemical bonds with no presence of any definable molecule, nor any of the regularity of repeating units that characterizes crystals; the science of molecules is called molecular chemistry or molecular physics, depending on whether the focus is on chemistry or physics. Molecular chemistry deals with the laws governing the interaction between molecules that results in the formation and breakage of chemical bonds, while molecular physics deals with the laws governing their structure and properties. In practice, this distinction is vague. In molecular sciences, a molecule consists of a stable system composed of two or more atoms.
Polyatomic ions may sometimes be usefully thought of as electrically charged molecules. The term unstable molecule is used for reactive species, i.e. short-lived assemblies of electrons and nuclei, such as radicals, molecular ions, Rydberg molecules, transition states, van der Waals complexes, or systems of colliding atoms as in Bose–Einstein condensate. According to Merriam-Webster and the Online Etymology Dictionary, the word "molecule" derives from the Latin "moles" or small unit of mass. Molecule – "extremely minute particle", from French molécule, from New Latin molecula, diminutive of Latin moles "mass, barrier". A vague meaning at first; the definition of the molecule has evolved. Earlier definitions were less precise, defining molecules as the smallest particles of pure chemical substances that still retain their composition and chemical properties; this definition breaks down since many substances in ordinary experience, such as rocks and metals, are composed of large crystalline networks of chemically bonded atoms or ions, but are not made of discrete molecules.
Molecules are held together by ionic bonding. Several types of non-metal elements exist only as molecules in the environment. For example, hydrogen only exists as hydrogen molecule. A molecule of a compound is made out of two or more elements. A covalent bond is a chemical bond; these electron pairs are termed shared pairs or bonding pairs, the stable balance of attractive and repulsive forces between atoms, when they share electrons, is termed covalent bonding. Ionic bonding is a type of chemical bond that involves the electrostatic attraction between oppositely charged ions, is the primary interaction occurring in ionic compounds; the ions are atoms that have lost one or more electrons and atoms that have gained one or more electrons. This transfer of electrons is termed electrovalence in contrast to covalence. In the simplest case, the cation is a metal atom and the anion is a nonmetal atom, but these ions can be of a more complicated nature, e.g. molecular ions like NH4+ or SO42−. An ionic bond is the transfer of electrons from a metal to a non-metal for both atoms to obtain a full valence shell.
Most molecules are far too small to be seen with the naked eye. DNA, a macromolecule, can reach macroscopic sizes, as can molecules of many polymers. Molecules used as building blocks for organic synthesis have a dimension of a few angstroms to several dozen Å, or around one billionth of a meter. Single molecules cannot be observed by light, but small molecules and the outlines of individual atoms may be traced in some circumstances by use of an atomic force microscope; some of the largest molecules are supermolecules. The smallest molecule is the diatomic hydrogen, with a bond length of 0.74 Å. Effective molecular radius is the size; the table of permselectivity for different substances contains examples. The chemical formula for a molecule uses one line of chemical element symbols and sometimes al
Coulomb's law, or Coulomb's inverse-square law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force or Coulomb force; the quantity of electrostatic force between stationary charges is always described by Coulomb’s law. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb, was essential to the development of the theory of electromagnetism, maybe its starting point, because it was now possible to discuss quantity of electric charge in a meaningful way. In its scalar form, the law is: F = k e q 1 q 2 r 2, where ke is Coulomb's constant, q1 and q2 are the signed magnitudes of the charges, the scalar r is the distance between the charges; the force of the interaction between the charges is attractive if the charges have opposite signs and repulsive if like-signed. Being an inverse-square law, the law is analogous to Isaac Newton's inverse-square law of universal gravitation, but gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive.
Coulomb's law can be used to derive Gauss's law, vice versa. The two laws are equivalent; the law has been tested extensively, observations have upheld the law on a scale from 10−16 m to 108 m. Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers. Thales of Miletus made a series of observations on static electricity around 600 BC, from which he believed that friction rendered amber magnetic, in contrast to minerals such as magnetite, which needed no rubbing. Thales was incorrect in believing the attraction was due to a magnetic effect, but science would prove a link between magnetism and electricity. Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber, he coined the New Latin word electricus to refer to the property of attracting small objects after being rubbed.
This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646. Early investigators of the 18th century who suspected that the electrical force diminished with distance as the force of gravity did included Daniel Bernoulli and Alessandro Volta, both of whom measured the force between plates of a capacitor, Franz Aepinus who supposed the inverse-square law in 1758. Based on experiments with electrically charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not elaborate on this. In 1767, he conjectured. In 1769, Scottish physicist John Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as x−2.06. In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had been discovered, but not published, by Henry Cavendish of England.
In 1785, the French physicist Charles-Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism, he used a torsion balance to study the repulsion and attraction forces of charged particles, determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The torsion balance consists of a bar suspended from its middle by a thin fiber; the fiber acts as a weak torsion spring. In Coulomb's experiment, the torsion balance was an insulating rod with a metal-coated ball attached to one end, suspended by a silk thread; the ball was charged with a known charge of static electricity, a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the instrument.
By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls and derive his inverse-square proportionality law. Coulomb's law states that: The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them; the force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive. Coulomb's law can be stated as a simple mathematical expression; the scalar and vector forms of the mathematical equation are | F | = k e | q 1 q
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential energy of an extended spring, the electric potential energy of an electric charge in an electric field; the unit for energy in the International System of Units is the joule, which has the symbol J. The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. Potential energy is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space; these forces, that are called conservative forces, can be represented at every point in space by vectors expressed as gradients of a certain scalar function called potential.
Since the work of potential forces acting on a body that moves from a start to an end position is determined only by these two positions, does not depend on the trajectory of the body, there is a function known as potential that can be evaluated at the two positions to determine this work. There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy has two components: the kinetic energy of random motions of particles and the potential energy of their mutual positions. Forces derivable from a potential are called conservative forces; the work done by a conservative force is W = − Δ U where Δ U is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, while work done by the force field decreases potential energy.
Common notations for potential energy are PE, U, V, Ep. Potential energy is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity; the action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. This work is stored in the force field, said to be stored as potential energy. If the external force is removed the force field acts on the body to perform the work as it moves the body back to the initial position, reducing the stretch of the spring or causing a body to fall. Consider a ball whose mass is m and whose height is h; the acceleration g of free fall is constant, so the weight force of the ball mg is constant. Force × displacement gives the work done, equal to the gravitational potential energy, thus U g = m g h The more formal definition is that potential energy is the energy difference between the energy of an object in a given position and its energy at a reference position.
Potential energy is linked with forces. If the work done by a force on a body that moves from A to B does not depend on the path between these points the work of this force measured from A assigns a scalar value to every other point in space and defines a scalar potential field. In this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for an applied force is independent of the path the work done by the force is evaluated at the start and end of the trajectory of the point of application; this means that there is a function U, called a "potential," that can be evaluated at the two points xA and xB to obtain the work over any trajectory between these two points. It is tradition to define this function with a negative sign so that positive work is a reduction in the potential, W = ∫ C F ⋅ d x = U − U where C is the trajectory taken from A to B; because the work done is independent of the path taken this expression is true for any trajectory, C, from A to B.
The function U is called the potential energy associated with the applied force. Examples of forces that have potential energies are spring forces. In this section the relationship between work and potential energy is presented in more detail; the line integral that defines work along curve C takes a special form if the force F is related to a scalar field φ so that F = ∇ φ = ( ∂ φ ∂ x, ∂