Electric potential

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At a point in space, the electric potential is the potential energy per unit of charge that is associated with a static (time-invariant) electric field. It is typically measured in volts, and is a Lorentz scalar quantity. The difference in electrical potential between two points is known as voltage.

There is also a generalized electric scalar potential that is used in electrodynamics when time-varying electromagnetic fields are present. This generalized electric potential cannot be simply interpreted as a potential energy, however.

Contents

Explanation

Electric potential may be conceived of as "electric pressure". Where this "pressure" is uniform, no current flows and nothing happens. This is similar to why people do not feel normal atmospheric air pressure: there is no difference between the pressure inside the body and outside, so nothing is felt. However, where this electrical pressure varies, an electric field exists, which will create a force on charged particles.

\mathbf{E}\mathbf{E}

The electric potential φ is therefore measured in units of energy per unit of electric charge. In SI units, this is:

joule/coulomb = volt.

The electric potential can also be generalized to handle situations with time-varying potential fields, in which case the electric field is not conservative and a potential function cannot be defined everywhere in space. There, an effective potential drop is included, associated with the inductance of the circuit. This generalized potential difference is also called the electromotive force (emf).

Introduction

Objects may possess a property known as electric charge. An electric field exerts a force on charged objects, accelerating them in the direction of the force, in either the same or the opposite direction of the electric field. If the charged object has a positive charge, the force and acceleration will be in the direction of the field. This force has the same direction as the electric field vector, and its magnitude is given by the size of the charge multiplied with the magnitude of the electric field.

Classical mechanics explores the concepts such as force, energy, potential etc. in more detail.

Force and potential energy are directly related. As an object moves in the direction that 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 the object falls, that potential energy decreases and is translated to motion, or inertial (kinetic) energy.

For certain forces, it is possible to define the "potential" of a field such that the potential energy of an object due to a field is dependent only on the position of the object with respect to the field. Those forces must affect objects depending only on the intrinsic properties of the object and the position of the object, and obey certain other mathematical rules.

Two such forces are the gravitational force (gravity) and the electric force in the absence of time-varying magnetic fields. The potential of an electric field is called the electric potential.

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.

Mathematical introduction

The concept of electric potential (denoted by: φ, φE or V) is closely linked with potential energy, thus:

UE = qφ

where UE is the electric potential energy of a test charge q due to the electric field. Note that 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 is zero.

\mathbf{E}

\phi_ \mathrm{E} = - \int_C \mathbf{E} \cdot \mathrm{d} \mathbf{\ell}
\mathbf{\nabla} \times \mathbf{E} = 0

\mathbf{E} = - \mathbf{\nabla} \phi_\mathrm{E}

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


\mathbf{\nabla} \cdot \mathbf{E} = \mathbf{\nabla} \cdot \left (- \mathbf{\nabla} \phi_\mathrm{E} \right ) = -\nabla^2 \phi_\mathrm{E} = \rho / \varepsilon_0

where ρ is the total charge density (including bound charge).

\mathbf{\nabla}\times\mathbf{E} \ne 0

Generalization to electrodynamics

\int \mathbf{E}\cdot \mathrm{d}\mathbf{S}\mathbf{A}
\mathbf{B} = \mathbf{\nabla} \times \mathbf{A}
\mathbf{B}
\mathbf{E} = -\mathbf{\nabla}\phi - \frac{\partial\mathbf{A}}{\partial t}
\mathbf{F}\mathbf{A}\mathbf{A}

Special cases and computational devices

\mathbf{l}
\phi_\mathrm{E} = - \int \mathbf{E} \cdot \mathrm{d}\mathbf{l}.

The electric potential created by a point charge q, at a distance r from the charge, can be shown to be, in SI units:

\phi_\mathbf{E} = \frac{q} {4 \pi \epsilon_o r}.

The electric potential due to a system of point charges is equal to the sum of the point charges' individual potentials. This fact simplifies calculations significantly, since addition of potential (scalar) fields is much easier than addition of the electric (vector) fields.

The electric potential created by a tridimensional spherically symmetric gaussian charge density ρ(r) given by:

 \rho(r) = \frac{q}{\sigma^3\sqrt{2\pi}^3}\,e^{-\frac{r^2}{2\sigma^2}},

where q is the total charge, is obtained by solving the Poisson's equation (in cgs units):

\nabla^2 \phi_\mathbf{E} = - 4 \pi \rho.

The solution is given by:

 \phi_\mathbf{E}(r) = \frac{q}{r}\,\mbox{erf}\left(\frac{r}{\sqrt{2}\sigma}\right)
\nabla^2 \phi_\mathbf{E}

Applications in electronics

This electric potential, typically measured in volts, provides a simple way to analyze electric circuits without requiring detailed knowledge of the circuit shape or the fields within it.

The electric potential provides a simple way to analyze electrical networks with the help of Kirchhoff's voltage law, without solving the detailed Maxwell's equations for the fields of the circuit.

Units

The SI unit of electric potential is the volt (in honour of Alessandro Volta), which is so widely used that the terms voltage and electric potential are almost synonymous. Older units are rarely used nowadays. Variants of the centimeter gram second system of units included a number of different units for electric potential, including the abvolt and the statvolt.