# Electric potential

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Electric potential
Electric potential around two oppositely charged conducting spheres. Purple represents the highest potential, yellow zero, and cyan the lowest potential. The electric field lines are shown leaving perpendicularly to the surface of each sphere.
Common symbols
V, φ
SI unit volt
Other units
statvolt
In SI base units V = kg⋅m2⋅s−3⋅A−1
Extensive?yes
Dimension ML2T−3I−1

The electric potential (also called the electric field potential, potential drop, the electrostatic potential) is the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field with negligible acceleration of the test charge to avoid producing kinetic energy or radiation by test charge. Typically, the reference point is the Earth or a point at infinity, although any point can be used. More precisely it is the energy per unit charge for a small test charge that does not disturb significantly the field and the charge distribution producing the field under consideration.

## Contents

In classical electrostatics, the electrostatic field is a vector quantity which is expressed as the gradient of the electrostatic potential, which is a scalar quantity denoted by V or occasionally φ, [1] equal to the electric potential energy of any charged particle at any location (measured in joules) divided by the charge of that particle (measured in coulombs). By dividing out the charge on the particle a quotient is obtained that is a property of the electric field itself. In short, electric potential is the electric potential energy per unit charge.

This value can be calculated in either a static (time-invariant) or a dynamic (varying with time) electric field at a specific time in units of joules per coulomb (J⋅C−1), or volts (V). The electric potential at infinity is assumed to be zero.

In electrodynamics, when time-varying fields are present, the electric field cannot be expressed only in terms of a scalar potential. Instead, the electric field can be expressed in terms of both the scalar electric potential and the magnetic vector potential. [2] The electric potential and the magnetic vector potential together form a four vector, so that the two kinds of potential are mixed under Lorentz transformations.

Practically, electric potential is always a continuous function in space; Otherwise, the spatial derivative of it will yield a field with infinite magnitude, which is practically impossible. Even an idealized point charge has 1 ⁄ r potential, which is continuous everywhere except the origin. The electric field is not continuous across an idealized surface charge, but it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. An idealized linear charge has ln(r) potential, which is continuous everywhere except on the linear charge.

## Introduction

Classical mechanics explores concepts such as force, energy, and potential. [3] Force and potential energy are directly related. A net force acting on any object will cause it to accelerate. As an object moves in the direction in which the force accelerates it, its potential energy decreases. For example, the gravitational potential energy of a cannonball at the top of a hill is greater than at the base of the hill. As it rolls downhill its potential energy decreases, being translated to motion, kinetic energy.

It is possible to define the potential of certain force fields so that the potential energy of an object in that field depends only on the position of the object with respect to the field. Two such force fields are the gravitational field and an electric field (in the absence of time-varying magnetic fields). Such fields must affect objects due to the intrinsic properties of the object (e.g., mass or charge) and the position of the object.

Objects may possess a property known as electric charge and an electric field exerts a force on charged objects. If the charged object has a positive charge the force will be in the direction of the electric field vector at that point while if the charge is negative the force will be in the opposite direction. The magnitude of the force is given by the quantity of the charge multiplied by the magnitude of the electric field vector.

## Electrostatics

Electric potential of separate positive and negative point charges shown as color range from magenta (+), through yellow (0), to cyan (−). Circular contours are equipotential lines. Electric field lines leave the positive charge and enter the negative charge.
Electric potential in the vicinity of two opposite point charges.

The electric potential at a point r in a static electric field E is given by the line integral

${\displaystyle V_{\mathbf {E} }=-\int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\,}$

where C is an arbitrary path connecting the point with zero potential to r. When the curl × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. In this case, the electric field is conservative and determined by the gradient of the potential:

${\displaystyle \mathbf {E} =-\mathbf {\nabla } V_{\mathbf {E} }.\,}$

Then, by Gauss's law, the potential satisfies Poisson's equation:

${\displaystyle \mathbf {\nabla } \cdot \mathbf {E} =\mathbf {\nabla } \cdot \left(-\mathbf {\nabla } V_{\mathbf {E} }\right)=-\nabla ^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0},\,}$

where ρ is the total charge density (including bound charge) and · denotes the divergence.

The concept of electric potential is closely linked with potential energy. A test charge q has an electric potential energy UE given by

${\displaystyle U_{\mathbf {E} }=q\,V.\,}$

The potential energy and hence also the electric potential is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero.

These equations cannot be used if the curl × E ≠ 0, i.e., in the case of a non-conservative electric field (caused by a changing magnetic field; see Maxwell's equations). The generalization of electric potential to this case is described below.

### Electric potential due to a point charge

The electric potential arising from a point charge Q, at a distance r from the charge is observed to be

${\displaystyle V_{\mathbf {E} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}},}$

where ε0 is the permittivity of vacuum. [4] VE is known as the Coulomb potential.

The electric potential for a system of point charges is equal to the sum of the point charges' individual potentials. This fact simplifies calculations significantly, because addition of potential (scalar) fields is much easier than addition of the electric (vector) fields. Specifically, the potential of a set of discrete point charges qi at points ri becomes

${\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i}{\frac {q_{i}}{|\mathbf {r} -\mathbf {r} _{i}|}},}$

Where

${\displaystyle \mathbf {r} }$ is a point at which the potential is evaluated.
${\displaystyle \mathbf {r} _{i}}$ is a point at which there is a nonzero charge.
${\displaystyle q_{i}}$ is the charge at the point ${\displaystyle \mathbf {r} _{i}}$.

and the potential of a continuous charge distribution ρ(r) becomes

${\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{R}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}d^{3}r'.}$

Where

${\displaystyle \mathbf {r} }$ is a point at which the potential is evaluated.
${\displaystyle R}$ is a region containing all the points at which the charge density is nonzero.
${\displaystyle \mathbf {r} '}$ is a point inside ${\displaystyle R}$.
${\displaystyle \rho (\mathbf {r} ')}$ is the charge density at the point ${\displaystyle \mathbf {r} '}$.

The equations given above for the electric potential (and all the equations used here) are in the forms required by SI units. In some other (less common) systems of units, such as CGS-Gaussian, many of these equations would be altered.

## Generalization to electrodynamics

When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential V because the electric field is no longer conservative: ${\displaystyle \textstyle \int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}$ is path-dependent because ${\displaystyle \mathbf {\nabla } \times \mathbf {E} \neq \mathbf {0} }$ (Faraday's law of induction).

Instead, one can still define a scalar potential by also including the magnetic vector potential A. In particular, A is defined to satisfy:

${\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} ,\,}$

where B is the magnetic field. By the fundamental theorem of vector calculus, such an A can always be found, since the divergence of the magnetic field is always zero due to the absence of magnetic monopoles. Now, the quantity

${\displaystyle \mathbf {F} =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}}$

is a conservative field, since the curl of ${\displaystyle \mathbf {E} }$ is canceled by the curl of ${\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}$ according to the Maxwell–Faraday equation. One can therefore write

${\displaystyle \mathbf {E} =-\mathbf {\nabla } V-{\frac {\partial \mathbf {A} }{\partial t}},\,}$

where V is the scalar potential defined by the conservative field F.

The electrostatic potential is simply the special case of this definition where A is time-invariant. On the other hand, for time-varying fields,

${\displaystyle -\int _{a}^{b}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)},\,}$

unlike electrostatics.

## Units

The SI derived unit of electric potential is the volt (in honor of Alessandro Volta), which is why a difference in electric potential between two points is known as voltage. Older units are rarely used today. Variants of the centimetre–gram–second system of units included a number of different units for electric potential, including the abvolt and the statvolt.

## Galvani potential versus electrochemical potential

Inside metals (and other solids and liquids), the energy of an electron is affected not only by the electric potential, but also by the specific atomic environment that it is in. When a voltmeter is connected between two different types of metal, it measures not the electric potential difference, but instead the potential difference corrected for the different atomic environments. [5] The quantity measured by a voltmeter is called electrochemical potential or fermi level, while the pure unadjusted electric potential V is sometimes called Galvani potential ${\displaystyle \phi }$. The terms "voltage" and "electric potential" are a bit ambiguous in that, in practice, they can refer to either of these in different contexts.

## Related Research Articles

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

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

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.

An electric field is the physical field that surrounds electrically-charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges, or from time-varying magnetic fields. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p).

Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician 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.

Scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

In electromagnetism, displacement current density is the quantity D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

Magnetic vector potential, A, is the vector quantity in classical electromagnetism defined so that its curl is equal to the magnetic field: . Together with the electric potential φ, the magnetic vector potential can be used to specify the electric field E as well. Therefore, many equations of electromagnetism can be written either in terms of the fields E and B, or equivalently in terms of the potentials φ and A. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

In physics, the electric displacement field or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in the related concept of displacement current in dielectrics. In free space, the electric displacement field is equivalent to flux density, a concept that lends understanding of Gauss's law. In the International System of Units (SI), it is expressed in units of coulomb per meter square (C⋅m−2).

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.

In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring The name is frequently confused with Hendrik Lorentz, who has given his name to many concepts in this field. The condition is Lorentz invariant. The condition does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function. The Lorenz condition is used to eliminate the redundant spin-0 component in the (1/2, 1/2) representation theory of the Lorentz group. It is equally used for massive spin-1 fields where the concept of gauge transformations does not apply at all.

In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Diametric. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field. Paramagnetic materials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed. Ferromagnetic and ferrimagnetic materials have strong magnetization in a magnetic field, and can be magnetized to have magnetization in the absence of an external field, becoming a permanent magnet. Magnetization is not necessarily uniform within a material, but may vary between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector M.

The Maxwell stress tensor is a symmetric second-order tensor used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand.

Lorentz–Heaviside units constitute a system of units within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant ε0 and magnetic constant µ0 do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Lorentz–Heaviside units may be regarded as normalizing ε0 = 1 and µ0 = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.

There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In physics, a field is a physical quantity, represented by a number or another tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each point on the map; the temperature can be considered at a certain point in time or over some interval of time, to study the dynamics of temperature change. A surface wind map, assigning an arrow to each point on a map that describes the wind speed and direction at that point, is an example of a vector field, i.e. a 1-dimensional (rank-1) tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank-1 tensor field, while electrodynamics can be formulated in terms of two interacting vector fields at each point in spacetime, or as a single-rank 2-tensor field.

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, a commonly used unit in atomic physics and chemistry is the debye (D).

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