Electric potential energy | |
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Common symbols | U_{E} |
SI unit | joule (J) |
Derivations from other quantities | U_{E} = C · V ^{2} / 2 |
Electric potential energy, or electrostatic potential energy, is a potential energy (measured in joules) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged objects.
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.
The joule is a derived unit of energy in the International System of Units. It is equal to the energy transferred to an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre. It is also the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second. It is named after the English physicist James Prescott Joule (1818–1889).
A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path. Equivalently, if a particle travels in a closed loop, the net work done by a conservative force is zero.
The term "electric potential energy" is used to describe the potential energy in systems with time-variant electric fields, while the term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields.
A time-variant system is a system that is not time invariant (TIV). Roughly speaking, its output characteristics depend explicitly upon time. In other words, a system in which certain quantities governing the system's behavior change with time, so that the system will respond differently to the same input at different times.
An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.
A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain, then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".
The electric potential energy of a system of point charges is defined as the work required assembling this system of charges by bringing them close together, as in the system from an infinite distance.
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The electrostatic potential energy can also be defined from the electric potential as follows:
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The SI unit of electric potential energy is the joule (named after the English physicist James Prescott Joule). In the CGS system the erg is the unit of energy, being equal to 10^{−7} J. Also electronvolts may be used, 1 eV = 1.602×10^{−19} J.
James Prescott Joule was an English physicist, mathematician and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work. This led to the law of conservation of energy, which in turn led to the development of the first law of thermodynamics. The SI derived unit of energy, the joule, is named after him.
The erg is a unit of energy and work equal to 10^{−7} joules. It originated in the centimetre–gram–second (CGS) system of units. It has the symbol erg. The erg is not an SI unit. Its name is derived from ergon (ἔργον), a Greek word meaning work or task.
The electrostatic potential energy, U_{E}, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is:
,
where is Coulomb's constant, r is the distance between the point charges q & Q, and q & Q are the charges (not the absolute values of the charges—i.e., an electron would have a negative value of charge when placed in the formula). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb's law to this formula.
The electron is a subatomic particle, symbol ^{}e^{−}
_{} or ^{}β^{−}
_{}, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.
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, and was essential to the development of the theory of electromagnetism, maybe even its starting point, because it was now possible to discuss quantity of electric charge in a meaningful way.
Outline of proof |
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The electrostatic force F acting on a charge q can be written in terms of the electric field E as
By definition, the change in electrostatic potential energy, U_{E}, of a point charge q that has moved from the reference position r_{ref} to position r in the presence of an electric field E is the negative of the work done by the electrostatic force to bring it from the reference position r_{ref} to that position r.
where:
In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors. A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. Usually U_{E} is set to zero when r_{ref} is infinity: so When the curl ∇×E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. This happens in time-invariant electric fields. When talking about electrostatic potential energy, time-invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb's law can be used. Using Coulomb's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. By the definition of the position vector r and the displacement vector s, it follows that r and s are also radially directed from Q. So, E and ds must be parallel: Using Coulomb's law, the electric field is given by and the integral can be easily evaluated: |
The electrostatic potential energy, U_{E}, of one point charge q in the presence of n point charges Q_{i}, taking an infinite separation between the charges as the reference position, is:
,
where is Coulomb's constant, r_{i} is the distance between the point charges q & Q_{i}, and q & Q_{i} are the signed values of the charges.
The electrostatic potential energy U_{E} stored in a system of N charges q_{1}, q_{2}, ..., q_{N} at positions r_{1}, r_{2}, ..., r_{N} respectively, is:
, |
| (1) |
where, for each i value, Φ(r_{i}) is the electrostatic potential due to all point charges except the one at r_{i},^{ [note 3] } and is equal to:
,
where r_{ij} is the distance between q_{j} and q_{i}.
Outline of proof |
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The electrostatic potential energy U_{E} stored in a system of two charges is equal to the electrostatic potential energy of a charge in the electrostatic potential generated by the other. That is to say, if charge q_{1} generates an electrostatic potential Φ_{1}, which is a function of position r, then Doing the same calculation with respect to the other charge, we obtain The electrostatic potential energy is mutually shared by and , so the total stored energy is This can be generalized to say that the electrostatic potential energy U_{E} stored in a system of N charges q_{1}, q_{2}, ..., q_{N} at positions r_{1}, r_{2}, ..., r_{N} respectively, is: . |
The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic potential against which an external agent must do work in moving the point charge from infinity to its final location.
A common question arises concerning the interaction of a point charge with its own electrostatic potential. Since this interaction doesn't act to move the point charge itself, it doesn't contribute to the stored energy of the system.
Consider bringing a point charge, q, into its final position near a point charge, Q_{1}. The electrostatic potential Φ(r) due to Q_{1} is
Hence we obtain, the electric potential energy of q in the potential of Q_{1} as
where r_{1} is the separation between the two point charges.
The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q_{1} due to two charges Q_{2} and Q_{3}, because the latter doesn't include the electrostatic potential energy of the system of the two charges Q_{2} and Q_{3}.
The electrostatic potential energy stored in the system of three charges is:
Outline of proof |
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Using the formula given in ( 1 ), the electrostatic potential energy of the system of the three charges will then be: Where is the electric potential in r_{1} created by charges Q_{2} and Q_{3}, is the electric potential in r_{2} created by charges Q_{1} and Q_{3}, and is the electric potential in r_{3} created by charges Q_{1} and Q_{2}. The potentials are: Where r_{ab} is the distance between charge Q_{a} and Q_{b}. If we add everything: Finally, we get that the electrostatic potential energy stored in the system of three charges: |
The energy density, or energy per unit volume, , of the electrostatic field of a continuous charge distribution is:
Outline of proof |
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One may take the equation for the electrostatic potential energy of a continuous charge distribution and put it in terms of the electrostatic field. Since Gauss's law for electrostatic field in differential form states where
then, so, now using the following divergence vector identity we have using the divergence theorem and taking the area to be at infinity where So, the energy density, or energy per unit volume of the electrostatic field is: |
Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat. This is known as the Joule effect. A capacitor stores it in its electric field. The total electric potential energy stored in a capacitor is given by
where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor.
Outline of proof |
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One may assemble charges to a capacitor in infinitesimal increments, , such that the amount of work done to assemble each increment to its final location may be expressed as The total work done to fully charge the capacitor in this way is then where is the total charge on the capacitor. This work is stored as electrostatic potential energy, hence, Notably, this expression is only valid if , which holds for many-charge systems such as large capacitors having metallic electrodes. For few-charge systems the discrete nature of charge is important. The total energy stored in a few-charge capacitor is which is obtained by a method of charge assembly utilizing the smallest physical charge increment where is the elementary unit of charge and where is the total number of charges in the capacitor. |
The total electrostatic potential energy may also be expressed in terms of the electric field in the form
where is the displacement of the electric field within a dielectric material and integration is over the entire volume of the dielectric.
The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge, ,
where integration is over the entire volume of the dielectric.
These latter two expressions are valid only for cases when the smallest increment of charge is zero () such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges.
In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , also or . Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.
In physics the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with a velocity v in an electric field E and a magnetic field B experiences a force of
In electromagnetism, absolute permittivity, often simply called permittivity, usually denoted by the Greek letter ε (epsilon), is the measure of capacitance that is encountered when forming an electric field in a particular medium. More specifically, permittivity describes the amount of charge needed to generate one unit of electric flux in a particular medium. Accordingly, a charge will yield more electric flux in a medium with low permittivity than in a medium with high permittivity. Permittivity is the measure of a material's ability to store an electric field in the polarization of the medium.
An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point beyond the influence of the electric field charge can be used.
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.
In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.
Electrostatics is a branch of physics that studies electric charges at rest.
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.
In plasmas and electrolytes, the Debye length, named after Peter Debye, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. A Debye sphere is a volume whose radius is the Debye length. With each Debye length, charges are increasingly electrically screened. Every Debye‐length , the electric potential will decrease in magnitude by 1/e. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density , charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas-Fermi length and the Thomas-Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.
The chemists Peter Debye and Erich Hückel noticed that solutions that contain ionic solutes do not behave ideally even at very low concentrations. So, while the concentration of the solutes is fundamental to the calculation of the dynamics of a solution, they theorized that an extra factor that they termed gamma is necessary to the calculation of the activity coefficients of the solution. Hence they developed the Debye–Hückel equation and Debye–Hückel limiting law. The activity is only proportional to the concentration and is altered by a factor known as the activity coefficient . This factor takes into account the interaction energy of ions in solution.
A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field. It is an arbitrary closed surface S = ∂V used in conjunction with Gauss's law for the corresponding field by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.
The method of image charges is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.
Spherical multipole moments are the coefficients in a series expansion of a potential that varies inversely with the distance R to a source, i.e., as 1/R. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to calculate accurately the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
In the natural sciences, an intermolecular force is an attraction between two molecules or atoms. They occur from either momentary interactions between molecules or permanent electrostatic attractions between dipoles. They can be explained using a simple phenomenological approach, or using a quantum mechanical approach.
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).