Electric Potential · aiwiki.page

Electric potential is a scalar quantity that describes the electrical conditions at a point in space. In electrostatics, it equals the potential energy of a test charge divided by its electric charge, relative to a chosen reference. Usually denoted $V$ or $\phi$ , it provides an alternative to describing electrical interactions directly through the electric field. A difference in potential between two points determines the energy change associated with moving a charge between them. (openstax.org)

Definition and units

For a test charge $q$ with electrostatic potential energy $U$ ,

V=\frac{U}{q}.

The test charge is assumed sufficiently small not to appreciably alter the source-charge distribution. Potential therefore characterizes the source field, rather than the particular test charge used to probe it. The corresponding relationship between changes is

\Delta U=q\Delta V.

Potential difference, commonly called voltage, is distinct from energy: the same potential difference produces different energy changes for different charges. (openstax.org)

The SI unit is the volt, defined by

1\ \mathrm{V}=1\ \mathrm{J/C},

where J denotes the joule and C the coulomb. The unit is named after Alessandro Volta. The electronvolt, by contrast, is an energy unit: it is the energy gained by a charge of elementary-charge magnitude when accelerated through a potential difference of one volt. (openstax.org)

Work, motion, and reference level

The electrostatic force is a conservative force, so its work depends only on the initial and final positions. For motion from $A$ to $B$ ,

W_{\mathrm{electric}} =q[V(A)-V(B)] =-\Delta U.

If no other force does work, the change in kinetic energy is $\Delta K=-q\Delta V$ . A positive charge released from rest accelerates toward lower potential; a negative charge, such as an electron, accelerates toward higher potential. Both motions decrease electrostatic potential energy. These statements describe acceleration, not necessarily the direction of an already moving particle’s velocity. (openstax.org)

The zero of potential is arbitrary. Adding the same constant to every potential leaves potential differences and the electric field unchanged. For localized charge distributions whose potential approaches a finite limit far away, it is customary to set $V=0$ at infinity. Other problems use a conducting surface or a designated circuit node as the reference. Zero potential does not imply zero electric field: the potential’s spatial variation, rather than its value alone, determines the field. (openstax.org)

Relationship to the electric field

In electrostatics, the potential difference is the negative line integral of the electric field:

V(B)-V(A) =-\int_A^B\mathbf E\cdot d\boldsymbol{\ell}.

The integral is independent of the path. Locally, the same relationship is expressed using the gradient:

\mathbf E=-\nabla V.

Thus the field points in the direction of the steepest decrease of potential. In Cartesian coordinates, $E_x=-\partial V/\partial x$ , with analogous expressions for the other components. For a uniform field and displacement $\mathbf d$ , $\Delta V=-\mathbf E\cdot\mathbf d$ ; a displacement perpendicular to the field produces no potential change. (openstax.org)

Potentials produced by charges

For a stationary point charge $Q$ in vacuum, with zero potential at infinity,

V(r)=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r},

where $r$ is the distance from the charge and $\varepsilon_0$ is the vacuum permittivity. This follows by integrating the field given by Coulomb’s law. Potential is positive around a positive isolated charge and negative around a negative one under this reference convention. (farside.ph.utexas.edu)

The superposition principle allows potentials to be added algebraically. For discrete charges,

V(\mathbf r)=\frac{1}{4\pi\varepsilon_0} \sum_i\frac{Q_i}{|\mathbf r-\mathbf r_i|}.

For a continuous charge density $\rho$ , the sum becomes an integral:

V(\mathbf r)=\frac{1}{4\pi\varepsilon_0} \int\frac{\rho(\mathbf r')}{|\mathbf r-\mathbf r'|}\,d^3r'.

Unlike electric fields, individual potential contributions require no vector addition. Opposite contributions can cancel at a point even when the electric field there remains nonzero. (farside.ph.utexas.edu)

Equipotentials and boundary-value problems

An equipotential surface consists of points sharing the same potential. Moving a charge along it requires no work against the electrostatic force. Where the field is nonzero, it is perpendicular to the surface. A connected electrical conductor in electrostatic equilibrium has constant potential throughout its interior and surface, because its internal electric field vanishes. Its potential need not be zero. (openstax.org)

Combining the field–potential relationship with Gauss’s law gives Poisson’s equation in vacuum:

\nabla^2V=-\frac{\rho}{\varepsilon_0}.

In a charge-free region this reduces to Laplace’s equation, $\nabla^2V=0$ . Determining the potential is consequently a boundary-value problem: the charge distribution must be supplemented by suitable information at the boundaries, such as specified potentials or normal field components. (farside.ph.utexas.edu)

Time-dependent electromagnetic fields

In general electromagnetism, a changing magnetic field can produce an electric field that is not conservative. A scalar potential alone then cannot describe the entire electric field. Introducing the magnetic vector potential $\mathbf A$ gives

\mathbf E=-\nabla\phi-\frac{\partial\mathbf A}{\partial t}.

The scalar and vector potentials are not unique: coordinated gauge transformations leave the physical fields unchanged. Accordingly, outside electrostatics, the electric-field integral between two points need not be path independent or equal simply to their scalar-potential difference. (farside.ph.utexas.edu)