Electrical potential

Electric potential is the potential energy per unit charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts.

Explanation
Electric potential may be conceived of as "electric pressure". Where this "pressure" is uniform, nothing happens, just as we do not feel the tremendous atmospheric pressure at sea level. However, where the pressure varies, it produces a force that can push charged objects to different locations.

Mathematically, it is the potential &phi; (a scalar field) associated with the conservative electric field E (E = &minus;&nabla;&phi;) that occurs when the magnetic field is time invariant (so that &nabla; &times; E = 0 from Faraday's law of induction).

Like any potential function, only the potential difference (voltage) between two points is physically meaningful (neglecting quantum Aharonov-Bohm effects), since any constant can be added to &phi; without affecting E.

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


 * joules/coulombs = volts.

The electric potential can also be generalized to handle situations with time-varying magnetic 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. 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 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 is mixed under Lorentz transformations.

Mathematical introduction
The concept of electric potential (denoted by: &phi;, $$\phi_\mathrm{E}$$ or V) is closely linked with potential energy, thus:



U_\mathrm{E} = q\phi $$

where $$U_\mathrm{E}$$ 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.

The proper definition of the electric potential uses the electric field E:



\phi_\mathrm{E} = - \int_C \mathbf{E} \cdot \mathrm{d}\mathbf{s} $$

where s is an arbitrary path connecting the point with zero potential to the point under consideration. When $$\nabla\times\mathbf E = 0$$, the line integral above does not depend on the specific path C chosen but only on its endpoints. Note: this equation cannot be used and the electric potential is not defined if $$\nabla\times\mathbf E \ne 0$$, i.e., in the case of a nonconservative electric field (caused by a changing magnetic field; see Maxwell's equations).

Special cases and computational devices
The electric potential at a point $$\mathbf{s}$$ due to a constant electric field $$\mathbf{E}$$ can be shown to be:


 * $$\phi_\mathrm{E} = - \mathbf{E} \cdot \mathbf{ds}$$

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 $$ \rho(r) $$ 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)

$$

where erf(x) is the error function. This solution can be checked explicitly by a careful manual evaluation of $$\nabla^2 \phi_\mathbf{E}$$. Note that, for r much greater than &sigma;, erf(x) approaches unity and the potential $$\phi_\mathbf{E}$$ approaches the point charge potential $$\frac{q}{r}$$ seen above, as expected.

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.