An electric potential (also called the electric
field potential, potential drop or the electrostatic
potential) is the amount of work needed to
move a unit of positive charge from a reference
point to a specific point inside the field
without producing an acceleration. Typically,
the reference point is the Earth or a point
at infinity, although any point beyond the
influence of the electric field charge can
be used.
In classical electrostatics, electric potential
is a scalar quantity denoted by V or occasionally
φ, equal to the electric potential energy
of any charged particle at any location (measured
in joules) divided by the charge of that particle
(measured in coulombs). By dividing out the
charge on the particle a quotient is obtained
that is a property of the electric field itself.
This value can be calculated in either a static
(time-invariant) or a dynamic (varying with
time) electric field at a specific time in
units of joules per coulomb (J C−1), or
volts (V). The electric potential at infinity
is assumed to be zero.
In electrodynamics, when time-varying fields
are present, the electric field cannot be
expressed only in terms of a scalar potential.
Instead, the electric field can be expressed
in terms of both the scalar electric potential
and the magnetic vector 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.
== Introduction ==
Classical mechanics explores concepts such
as force, energy, potential etc. Force and
potential energy are directly related. A net
force acting on any object will cause it to
accelerate. As an object moves in the direction
in which 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 it rolls downhill its potential energy
decreases, 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 the
gravitational field and an electric field
(in the absence of time-varying magnetic fields).
Such fields must affect objects due to the
intrinsic properties of the object (e.g.,
mass or charge) and the position of the object.
Objects may possess a property known as electric
charge and an electric field exerts a force
on charged objects. If the charged object
has a positive charge the force will be in
the direction of the electric field vector
at that point while if the charge is negative
the force will be in the opposite direction.
The magnitude of the force is given by the
quantity of the charge multiplied by the magnitude
of the electric field vector.
== Electrostatics ==
The electric potential at a point r in a static
electric field E is given by the line integral
where C is an arbitrary path connecting the
point with zero potential to r. When the curl
∇ × E is zero, the line integral above
does not depend on the specific path C chosen
but only on its endpoints. In this case, the
electric field is conservative and determined
by the gradient of the potential:
Then, by Gauss's law, the potential satisfies
Poisson's equation:
∇
⋅
E
=
∇
⋅
(
−
∇
V
E
)
=
−
∇
2
V
E
=
ρ
/
ε
0
,
{\displaystyle \mathbf {\nabla } \cdot \mathbf
{E} =\mathbf {\nabla } \cdot \left(-\mathbf
{\nabla } V_{\mathbf {E} }\right)=-\nabla
^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0},\,}
where ρ is the total charge density (including
bound charge) and ∇· denotes the divergence.
The concept of electric potential is closely
linked with potential energy. A test charge
q has an electric potential energy UE given
by
U
E
=
q
V
.
{\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 the curl
∇ × E ≠ 0, i.e., in the case of a non-conservative
electric field (caused by a changing magnetic
field; see Maxwell's equations). The generalization
of electric potential to this case is described
below.
=== Electric potential due to a point charge
===
The electric potential arising from a point
charge Q, at a distance r from the charge
is observed to be
V
E
=
1
4
π
ε
0
Q
r
,
{\displaystyle V_{\mathbf {E} }={\frac {1}{4\pi
\varepsilon _{0}}}{\frac {Q}{r}},\,}
where ε0 is the permittivity of vacuum.
V
E
{\displaystyle V_{\mathbf {E} }}
is known as the Coulomb potential.
The electric potential for a system of point
charges is equal to the sum of the point charges'
individual potentials. This fact simplifies
calculations significantly, because addition
of potential (scalar) fields is much easier
than addition of the electric (vector) fields.
The equation given above for the electric
potential (and all the equations used here)
are in the forms required by SI units. In
some other (less common) systems of units,
such as CGS-Gaussian, many of these equations
would be altered.
== Generalization to electrodynamics ==
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
in terms of a scalar potential V because the
electric field is no longer conservative:
∫
C
E
⋅
d
ℓ
{\displaystyle \textstyle \int _{C}\mathbf
{E} \cdot \mathrm {d} {\boldsymbol {\ell }}}
is path-dependent because
∇
×
E
≠
0
{\displaystyle \mathbf {\nabla } \times \mathbf
{E} \neq \mathbf {0} }
(Faraday's law of induction).
Instead, one can still define a scalar potential
by also including the magnetic vector potential
A. In particular, A is defined to satisfy:
B
=
∇
×
A
,
{\displaystyle \mathbf {B} =\mathbf {\nabla
} \times \mathbf {A} ,\,}
where B is the magnetic field. Because the
divergence of the magnetic field is always
zero due to the absence of magnetic monopoles,
such an A can always be found. Given this,
the quantity
F
=
E
+
∂
A
∂
t
{\displaystyle \mathbf {F} =\mathbf {E} +{\frac
{\partial \mathbf {A} }{\partial t}}}
is a conservative field by Faraday's law and
one can therefore write
E
=
−
∇
V
−
∂
A
∂
t
,
{\displaystyle \mathbf {E} =-\mathbf {\nabla
} V-{\frac {\partial \mathbf {A} }{\partial
t}},\,}
where V is the scalar potential defined by
the conservative field F.
The electrostatic potential is simply the
special case of this definition where A is
time-invariant. On the other hand, for time-varying
fields,
−
∫
a
b
E
⋅
d
ℓ
≠
V
(
b
)
−
V
(
a
)
,
{\displaystyle -\int _{a}^{b}\mathbf {E} \cdot
\mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)},\,}
unlike electrostatics.
== Units ==
The SI derived unit of electric potential
is the volt (in honor of Alessandro Volta),
which is why a difference in electric potential
between two points is known as voltage. Older
units are rarely used today. Variants of the
centimeter gram second system of units included
a number of different units for electric potential,
including the abvolt and the statvolt.
== Galvani potential versus electrochemical
potential ==
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 a voltmeter is connected between
two different types of metal, it measures
not the electric potential difference, but
instead the potential difference corrected
for the different atomic environments. The
quantity measured by a voltmeter is called
electrochemical potential or fermi level,
while the pure unadjusted electric potential
V is sometimes called Galvani potential
ϕ
{\displaystyle \phi }
. The terms "voltage" and "electric potential"
are a bit ambiguous in that, in practice,
they can refer to either of these in different
contexts.
== See also ==
Absolute electrode potential
Electrochemical potential
Electrode potential
