# 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 defined as 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. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.

## Contents

In classical electrostatics, the electrostatic field is a vector quantity that 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, an electric potential is the electric potential energy per unit charge.

This value can be calculated in either a static (time-invariant) or a dynamic (time-varying) electric field at a specific time with the unit joules per coulomb (J⋅C−1) or volt (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, the 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 of the force that is acting, 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 and is 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. Since 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; 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:

${\displaystyle F=qE.}$

## 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

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

where C is an arbitrary path from some fixed reference point to r. In electrostatics, the Maxwell-Faraday equation reveals that the curl ${\textstyle \nabla \times \mathbf {E} }$ is zero, making the electric field conservative. Thus, the line integral above does not depend on the specific path C chosen but only on its endpoints, making ${\textstyle V_{\mathbf {E} }}$ well-defined everywhere. The gradient theorem then allows us to write:

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

This states that the electric field points "downhill" towards lower voltages. By Gauss's law, the potential can also be found to satisfy 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 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 ${\textstyle \nabla \times \mathbf {E} \neq \mathbf {0} }$, 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 in the section § Generalization to electrodynamics.

### 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 ${\displaystyle \mathbf {\nabla } \times \mathbf {E} \neq \mathbf {0} }$ (due to the Maxwell-Faraday equation).

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.

### Gauge freedom

The electrostatic potential could have any constant added to it without affecting the electric field. In electrodynamics, the electric potential has infinitely many degrees of freedom. For any (possibly time-varying or space-varying) scalar field ${\displaystyle \psi }$, we can perform the following gauge transformation to find a new set of potentials that produce exactly the same electric and magnetic fields: [5]

${\displaystyle V^{\prime }=V-{\frac {\partial \psi }{\partial t}}}$
${\displaystyle \mathbf {A} ^{\prime }=\mathbf {A} +\nabla \psi }$

Given different choices of gauge, the electric potential could have quite different properties. In the Coulomb gauge, the electric potential is given by Poisson's equation

${\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}$

just like in electrostatics. However, in the Lorenz gauge, the electric potential is a retarded potential that propagates at the speed of light, and is the solution to an inhomogeneous wave equation:

${\displaystyle \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

## 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 the potential difference corrected for the different atomic environments. [6] 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, however 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, or Maxwell-Heaviside 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. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

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 physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.

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.

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 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.

"A Dynamical Theory of the Electromagnetic Field" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

In classical electromagnetism, magnetic vector potential is the vector quantity 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 the four-gradient and 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.

Electric potential energy is a potential energy 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.

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.

In electromagnetism, Jefimenko's equations give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay of the fields due to the finite speed of light and relativistic effects. Therefore they can be used for moving charges and currents. They are the particular solutions to Maxwell's equations for any arbitrary distribution of charges and currents.

In electromagnetism and applications, an inhomogeneous electromagnetic wave equation, or nonhomogeneous electromagnetic wave equation, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations inhomogeneous, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

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 electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light c, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution to another point in space, see figure below.

Coulomb's inverse-square law, or simply Coulomb's 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. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point, as it made it possible to discuss the quantity of electric charge in a meaningful way.

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