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Electric potential

From Wikipedia, the free encyclopedia
Line integral of the electric field
Not to be confused withVoltage.
Electric potential
Electric potential around two oppositely charged conducting spheres. Purple represents the highest potential, yellow zero, and cyan the lowest potential. The electricfield lines are shown leaving perpendicularly to the surface of each sphere.
Common symbols
V,φ
SI unitvolt
Other units
statvolt
InSI base unitsV = kg⋅m2⋅s−3⋅A−1
Extensive?yes
DimensionML2T−3I−1
Articles about
Electromagnetism
Solenoid

Electric potential (also called theelectric field potential, potential drop, theelectrostatic potential) is defined as the amount ofwork/energy needed per unit ofelectric charge to move the charge from a reference point to a specific point in an electric field. More precisely, the electric potential is the energy per unit charge for atest charge that is so small that the disturbance to the field (due to the test charge's own field) under consideration is negligible. 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 isearth or a point atinfinity, although any point can be used.

In classicalelectrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is ascalar quantity denoted byV or occasionallyφ,[1] equal to theelectric potential energy of anycharged particle at any location (measured injoules) divided by thecharge of that particle (measured incoulombs). 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 theelectric 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) orvolt (V). The electric potential at infinity is assumed to be zero.

Inelectrodynamics, when time-varying fields are present, the electric field cannot be expressed only as ascalar potential. Instead, the electric field can be expressed as both the scalar electric potential and themagnetic vector potential.[2] The electric potential and the magnetic vector potential together form afour-vector, so that the two kinds of potential are mixed underLorentz transformations.

Practically, the electric potential is acontinuous function in all space, because a spatial derivative of a discontinuous electric potential yields an electric field of impossibly infinite magnitude. Notably, the electric potential due to an idealizedpoint charge (proportional to1 ⁄r, withr the distance from the point charge) is continuous in all space except at the location of the point charge. Though electric field is not continuous across an idealizedsurface charge, it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. Additionally, an idealized line of charge has electric potential (proportional toln(r), withr the radial distance from the line of charge) is continuous everywhere except on the line of charge.

Introduction

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Classical mechanics explores concepts such asforce,energy, andpotential.[3] Force and potential energy are directly related. A net force acting on any object will cause it toaccelerate. As an object moves in the direction of a force acting on it, its potential energy decreases. For example, thegravitational 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 agravitational field and an electric field (in the absence of time-varying magnetic fields). Such fields affect objects because of the intrinsic properties (e.g.,mass or charge) and positions of the objects.

An object may possess a property known aselectric charge. Since anelectric field exerts force on a charged object, if the object has a positive charge, the force will be in the direction of theelectric field vector at the location of the charge; if the charge is negative, the force will be in the opposite direction.

The magnitude of force is given by the quantity of the charge multiplied by the magnitude of the electric field vector,

|F|=q|E|.{\displaystyle |\mathbf {F} |=q|\mathbf {E} |.}

Electrostatics

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Main article: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.

An electric potential at a pointr in a staticelectric fieldE is given by theline integral

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

whereC is an arbitrary path from some fixed reference point tor; it is uniquely determined up to a constant that is added or subtracted from the integral. In electrostatics, theMaxwell-Faraday equation reveals that thecurl×E{\textstyle \nabla \times \mathbf {E} } is zero, making the electric fieldconservative. Thus, the line integral above does not depend on the specific pathC chosen but only on its endpoints, makingVE{\textstyle V_{\mathbf {E} }} well-defined everywhere. Thegradient theorem then allows us to write:

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

This states that the electric field points "downhill" towards lower voltages. ByGauss's law, the potential can also be found to satisfyPoisson's equation:E=(VE)=2VE=ρ/ε0{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} =\mathbf {\nabla } \cdot \left(-\mathbf {\nabla } V_{\mathbf {E} }\right)=-\nabla ^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0}}

whereρ is the totalcharge density and{\textstyle \mathbf {\nabla } \cdot } denotes thedivergence.

The concept of electric potential is closely linked withpotential energy. Atest charge,q, has anelectric potential energy,UE, given by

UE=qV.{\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×E0{\textstyle \nabla \times \mathbf {E} \neq \mathbf {0} }, i.e., in the case of anon-conservative electric field (caused by a changingmagnetic field; seeMaxwell'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

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See also:Coulomb's law
The electric potential created by a charge,Q, isV =Q/(4πε0r). Different values ofQ yield 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 location ofQ is observed to beVE=14πε0Qr,{\displaystyle V_{\mathbf {E} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}},}whereε0 is thepermittivity of vacuum[4],VE is known as theCoulomb potential. Note that, in contrast to the magnitude of anelectric field due to a point charge, the electric potential scales respective to the reciprocal of the radius, rather than the radius squared.

The electric potential at any location,r, in a system of point charges is equal to the sum of the individual electric potentials due to every point charge in the system. 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 chargesqi at pointsri becomes

VE(r)=14πε0i=1nqi|rri|{\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}}{|\mathbf {r} -\mathbf {r} _{i}|}}\,}

where

  • r is a point at which the potential is evaluated;
  • ri is a point at which there is a nonzero charge; and
  • qi is the charge at the pointri.

And the potential of a continuous charge distributionρ(r) becomes

VE(r)=14πε0Rρ(r)|rr|d3r,{\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{R}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}r'\,,}

where

  • r is a point at which the potential is evaluated;
  • R is a region containing all the points at which the charge density is nonzero;
  • r' is a point insideR; and
  • ρ(r') is the charge density at the pointr'.

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

Generalization to electrodynamics

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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 as a scalar potentialV because the electric field is no longerconservative:CEd{\displaystyle \textstyle \int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}} is path-dependent because×E0{\displaystyle \mathbf {\nabla } \times \mathbf {E} \neq \mathbf {0} } (due to theMaxwell-Faraday equation).

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

B=×A{\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} }

whereB is themagnetic field. By thefundamental theorem of vector calculus, such anA can always be found, sincethe divergence of the magnetic field is always zero due to the absence ofmagnetic monopoles. Now, the quantityF=E+At{\displaystyle \mathbf {F} =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}}is a conservative field, since the curl ofE{\displaystyle \mathbf {E} } is canceled by the curl ofAt{\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}} according to theMaxwell–Faraday equation. One can therefore writeE=VAt,{\displaystyle \mathbf {E} =-\mathbf {\nabla } V-{\frac {\partial \mathbf {A} }{\partial t}},}

whereV is the scalar potential defined by the conservative fieldF.

The electrostatic potential is simply the special case of this definition whereA is time-invariant. On the other hand, for time-varying fields,abEdV(b)V(a){\displaystyle -\int _{a}^{b}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)}}unlike electrostatics.

Gauge freedom

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Main article:Gauge fixing

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,𝜓, we can perform the followinggauge transformation to find a new set of potentials that produce exactly the same electric and magnetic fields:[5]

V=VψtA=A+ψ{\displaystyle {\begin{aligned}V^{\prime }&=V-{\frac {\partial \psi }{\partial t}}\\\mathbf {A} ^{\prime }&=\mathbf {A} +\nabla \psi \end{aligned}}}

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

2V=ρε0{\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}

just like in electrostatics. However, in theLorenz gauge, the electric potential is aretarded potential that propagates at the speed of light and is the solution to aninhomogeneous wave equation:

2V1c22Vt2=ρε0{\displaystyle \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}

Units

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TheSI derived unit of electric potential is thevolt (in honor ofAlessandro Volta), denoted as V, which is why the electric potential difference between two points in space is known as avoltage. Older units are rarely used today. Variants of thecentimetre–gram–second system of units included a number of different units for electric potential, including theabvolt and thestatvolt.

Galvani potential versus electrochemical potential

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Main articles:Galvani potential,Electrochemical potential, andFermi level

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 avoltmeter is connected between two different types of metal, it measures thepotential difference corrected for the different atomic environments.[6] The quantity measured by a voltmeter is calledelectrochemical potential orfermi level, while the pure unadjusted electric potential,V, is sometimes called theGalvani potential,ϕ. The terms "voltage" and "electric potential" are a bit ambiguous but one may refer toeither of these in different contexts.

Common formulas

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Charge configurationFigureElectric potential
Infinite wireV=λ2πε0lnx,{\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}

whereλ{\displaystyle \lambda } is uniform linear charge density.

Infinitely large surfaceV=σx2ε0,{\displaystyle V=-{\frac {\sigma x}{2\varepsilon _{0}}},}

whereσ{\displaystyle \sigma } is uniform surface charge density.

Infinitely long cylindrical volumeV=λ2πε0lnx,{\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}

whereλ{\displaystyle \lambda } is uniform linear charge density.

Spherical volumeV=Q4πε0x,{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},}

outside the sphere, whereQ{\displaystyle Q} is the total charge uniformly distributed in the volume.

V=Q(3R2r2)8πε0R3,{\displaystyle V={\frac {Q(3R^{2}-r^{2})}{8\pi \varepsilon _{0}R^{3}}},}

inside the sphere, whereQ{\displaystyle Q} is the total charge uniformly distributed in the volume.

Spherical surfaceV=Q4πε0x,{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},}

outside the sphere, whereQ{\displaystyle Q} is the total charge uniformly distributed on the surface.

V=Q4πε0R,{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}R}},}

inside the sphere for uniform charge distribution.

Charged RingV=Q4πε0R2+x2,{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}{\sqrt {R^{2}+x^{2}}}}},}

on the axis, whereQ{\displaystyle Q} is the total charge uniformly distributed on the ring.

Charged DiscV=σ2ε0[x2+R2x],{\displaystyle V={\frac {\sigma }{2\varepsilon _{0}}}\left[{\sqrt {x^{2}+R^{2}}}-x\right],}

on the axis, whereσ{\displaystyle \sigma } is the uniform surface charge density.

Electric DipoleV=0,{\displaystyle V=0,}

on the equatorial plane.

V=p4πε0x2,{\displaystyle V={\frac {p}{4\pi \varepsilon _{0}x^{2}}},}

on the axis (given thatxd{\displaystyle x\gg d}), wherex{\displaystyle x} can also be negative to indicate position at the opposite direction on the axis, andp{\displaystyle p} is the magnitude ofelectric dipole moment.

See also

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References

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  1. ^Goldstein, Herbert (June 1959).Classical Mechanics. United States: Addison-Wesley. p. 383.ISBN 0201025108.
  2. ^Griffiths, David J. (1999).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. ^"2022 CODATA Value: vacuum electric permittivity".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
  5. ^Griffiths, David J. (1999).Introduction to Electrodynamics (3rd ed.). Prentice Hall. p. 420.ISBN 013805326X.
  6. ^Bagotskii VS (2006).Fundamentals of electrochemistry. John Wiley & Sons. p. 22.ISBN 978-0-471-70058-6.

Further reading

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Wikimedia Commons has media related toElectric potential.
  • 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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