# Electric potential

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.

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 unitvolt
Other units
statvolt
In SI base unitsV = kg⋅m2⋅A−1⋅s−3
Extensive?yes
DimensionM L2 T−3 I−1

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 created by a charge Q is V =Q/(4πε0r). Different values of Q will make different values of electric potential V (shown in the image).

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.

## See also

• Absolute electrode potential
• Electrochemical potential
• Electrode potential

## References

1. Goldstein, Herbert (June 1959). Classical Mechanics. United States: Addison-Wesley. p. 383. ISBN 0201025108.
2. Griffiths, David J. Introduction to Electrodynamics. Pearson Prentice Hall. pp. 416–417. ISBN 978-81-203-1601-0.
3. Young, Hugh A.; Freedman, Roger D. (2012). Sears and Zemansky's University Physics with Modern Physics (13th ed.). Boston: Addison-Wesley. p. 754.
4. "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
5. Bagotskii VS (2006). Fundamentals of electrochemistry. p. 22. ISBN 978-0-471-70058-6.
• Politzer P, Truhlar DG (1981). Chemical Applications of Atomic and Molecular Electrostatic Potentials: Reactivity, Structure, Scattering, and Energetics of Organic, Inorganic, and Biological Systems. Boston, MA: Springer US. ISBN 978-1-4757-9634-6.
• Sen K, Murray JS (1996). Molecular Electrostatic Potentials: Concepts and Applications. Amsterdam: Elsevier. ISBN 978-0-444-82353-3.
• Griffiths DJ (1999). Introduction to Electrodynamics (3rd. ed.). Prentice Hall. ISBN 0-13-805326-X.
• Jackson JD (1999). Classical Electrodynamics (3rd. ed.). USA: John Wiley & Sons, Inc. ISBN 978-0-471-30932-1.
• Wangsness RK (1986). Electromagnetic Fields (2nd., Revised, illustrated ed.). Wiley. ISBN 978-0-471-81186-2.
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