Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. It is a scalar quantity.
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.
3 Mathematical introduction
4 Generalization to electrodynamics
5 Special cases and computational devices
6 Applications in electronics
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, it
produces an electric field, which will create a force on charged
Mathematically, it is the potential φ (a scalar field)
associated with the conservative electric field () that occurs when
the magnetic field is time invariant (so that from Faraday's law of
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 φ without affecting (gauge invariance).
The electric potential φ is therefore measured in units of energy per unit of electric charge. In SI units, this is:
joules/coulombs = volts.
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).
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
Classical mechanics explores the concepts such as force, energy, potential etc. in more detail.
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.
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.
potential and the magnetic vector potential together form a four vector,
so that the two kinds of potential are mixed under Lorentz
The concept of electric potential (denoted by: φ, φE or V) is closely linked with potential energy, thus:
UE = qφ
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.
The proper definition of the electric potential uses the electric field :
E is equal to the electric field, ds is an unknown, and 'C' is an
arbitrary path connecting the point with zero potential to the point
under consideration. When , the line integral above does not depend on
the specific path C chosen but only on its endpoints. Equivalently, the
electric potential determines the electric field via its gradient:
and therefore, by Gauss's law, the potential satisfies Poisson's equation:
where ρ is the total charge density (including bound charge).
these equations cannot be used if , i.e., in the case of a
nonconservative electric field (caused by a changing magnetic field; see
Maxwell's equations). The generalization of electric potential to this
case is described below.
Generalization to electrodynamics
time-varying magnetic fields are present (which is true whenever there
are time-varying electric fields and vice versa), one cannot describe
the electric field simply in terms of a scalar potential φ because the
electric field is no longer conservative: is path-dependent because .
one can still define a scalar potential by also including the magnetic
vector potential . In particular, is defined by:
the magnetic flux density. One can always find such an because (the
absence of magnetic monopoles). Given this, the quantity is a
conservative field by Faraday's law and one can therefore write:
where φ is the scalar potential defined by the conservative field .
electrostatic potential is simply the special case of this definition
where is time-invariant. On the other hand, for time-varying fields,
note that , unlike electrostatics.
Note that this definition of φ
depends on the gauge choice for the vector potential (the gradient of
any scalar field can be added to without changing ). One choice is the
Coulomb gauge, in which we choose . In this case, we obtain , where ρ is
the charge density, just as for electrostatics. Another common choice
is the Lorenz gauge, in which we choose to satisfy .
Special cases and computational devices
The electric potential at a point due to a constant electric field can be shown to be:
The electric potential created by a point charge q, at a distance r from the charge, can be shown to be, in SI units:
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:
where q is the total charge, is obtained by solving the Poisson's equation (in cgs units):
The solution is given by:
erf(x) is the error function. This solution can be checked explicitly
by a careful manual evaluation of . Note that, for r much greater than
σ, erf(x) approaches unity and the potential approaches the point
charge potential seen above, as expected.
Applications in electronics
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.
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
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