In
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
, a vector potential is a
vector field whose
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
is a given vector field. This is analogous to a ''
scalar potential
In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
'', which is a scalar field whose
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
is a given vector field.
Formally, given a vector field v, a ''vector potential'' is a
vector field A such that
Consequence
If a vector field v admits a vector potential A, then from the equality
(
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
of the
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was fi ...
is zero) one obtains
which implies that v must be a
solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
.
Theorem
Let
be a
solenoidal vector field
In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field:
\nabla \cdot \mathbf ...
which is twice
continuously differentiable. Assume that decreases at least as fast as
for
.
Define
Then, A is a vector potential for , that is,
Here,
is curl for variable y.
Substituting curl
'' for the current density j of the
retarded potential
In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so th ...
, you will get this formula. In other words, v corresponds to the
H-field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
.
You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v;
A generalization of this theorem is the
Helmholtz decomposition
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
which states that any vector field can be decomposed as a sum of a solenoidal vector field and an
irrotational vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
.
By
analogy
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ( ...
with
Biot-Savart's law, the following
is also qualify as a vector potential for v.
:
Substitute j (
current density
In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ar ...
) for v and H (
H-field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
)for A, we will find the Biot-Savart law.
Let
and let the Ω be a
star domain
In geometry, a Set (mathematics), set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lie ...
centered on the ''p'' then,
translating
Poincaré's lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
for
differential forms
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
into vector fields world, the followng
is also a vector potential for the
Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If is a vector potential for , then so is
where
is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires
choosing a gauge.
See also
*
Fundamental theorem of vector calculus
In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth function, smooth, rapidly decaying vector field in three dimensions can b ...
*
Magnetic vector potential
In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
*
Solenoid
upright=1.20, An illustration of a solenoid
upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines
A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
*
Closed and Exact Differential Forms In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
References
* ''Fundamentals of Engineering Electromagnetics'' by David K. Cheng, Addison-Wesley, 1993.
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Concepts in physics
Potentials
Vector calculus
Vector physical quantities