Vector potential

From a historical perspective, the vector potential is a mathematical aid that was introduced in classical electrodynamics to simplify the handling of magnetic induction or flux density (clearly speaking, the “magnetic field”). ${\displaystyle \mathbf {A} (\mathbf {r} )}$ ${\displaystyle \mathbf {B} (\mathbf {r} )}$

Mathematically, the vector potential (in contrast to the scalar potential ) is a vector field whose rotation is according to the following formula ${\displaystyle \mathbf {A} (\mathbf {r} )}$

${\displaystyle \mathbf {B} (\mathbf {r} ){\overset {\underset {\mathrm {def} }{}}{=}}\operatorname {rot} \mathbf {A} (\mathbf {r} )=\nabla \times \mathbf {A} (\mathbf {r} )}$

supplies a second vector field . ${\displaystyle \mathbf {B} (\mathbf {r} )}$

Vector potentials can u. a. to decouple the Maxwell equations used to describe the electromagnetic field and thereby make them easier to solve. It turns out that the vector potential arises via a convolution from a given location-dependent current density (i.e. an arrangement of current-carrying conductors in space, such as a coil ), i.e. one can calculate the vector potential for a given current density, and from this the measurable magnetic one Induction or flux density that is generated by this arrangement ( Biot-Savart law ). This vector potential has the unit . ${\displaystyle \mathbf {j} (\mathbf {r} )}$ ${\displaystyle \mathbf {B} (\mathbf {r} )}$ ${\displaystyle [\mathbf {A} ]=\mathrm {\frac {V\,s}{m}} }$

Although it was initially only introduced as a mathematical aid, it is a physical reality in quantum mechanics, as the Aharonov-Bohm experiment showed.

definition

The vector potential is defined so that ${\displaystyle \mathbf {A} (\mathbf {r} )}$

${\displaystyle \mathbf {B} (\mathbf {r} ){\overset {\underset {\mathrm {def} }{}}{=}}\nabla \times \mathbf {A} (\mathbf {r} )}$

applies. Here is the rotation of the vector potential. Through this approach , the divergence of zero because for every two times continuously differentiable vector fields. This is required by the Maxwell equations . ${\displaystyle \nabla \times \mathbf {A} (\mathbf {r} )}$ ${\displaystyle \mathbf {B} }$ ${\displaystyle \operatorname {div} \mathbf {B} =\operatorname {div} \operatorname {rot} \mathbf {A} =\nabla \cdot (\nabla \times \mathbf {A} )=0}$

In electrodynamics , the above formula applies unchanged, whereas for the electric field ${\displaystyle \mathbf {E} (\mathbf {r} ,t)}$

${\displaystyle \mathbf {E} (\mathbf {r} ,t)=-\nabla \Phi (\mathbf {r} ,t)-\partial _{t}\mathbf {A} (\mathbf {r} ,t)}$

applies. Here is the scalar potential . ${\displaystyle \Phi }$

These two approaches, along with the Lorenz gauge , are used to decouple the Maxwell equations . In magnetostatics, the Coulomb calibration is usually used, which represents the static limit case of the Lorenz calibration.

In the theory of relativity and quantum electrodynamics, scalar potential and vector potential become a four-potential

${\displaystyle A^{\mu }=\left(\Phi /c,\mathbf {A} \right)}$

summarized.

properties

• The vector potential is only determined up to a gradient field because the rotation of a gradient field always disappears. The following applies to every scalar function ${\displaystyle \chi (\mathbf {r} ,t)}$

${\displaystyle {\begin{aligned}\mathbf {A} (\mathbf {r} ,t)'&=\mathbf {A} (\mathbf {r} ,t)+\nabla \chi (\mathbf {r} ,t)\\\Rightarrow \;\;\mathbf {B} (\mathbf {r} ,t)'&=\nabla \times \mathbf {A} (\mathbf {r} ,t)'=\nabla \times \mathbf {A} (\mathbf {r} ,t)+\nabla \times \nabla \chi =\nabla \times \mathbf {A} (\mathbf {r} ,t)=\mathbf {B} (\mathbf {r} ,t)\,.\end{aligned}}}$

Different calibrated vector potentials lead to the same magnetic field. This is called gauge invariance .

• The vector potential is not conservative as a vector field . Otherwise it could be represented by the gradient of a scalar field and the following would apply: ${\displaystyle \alpha }$

${\displaystyle \mathbf {B} (\mathbf {r} )=\nabla \times \mathbf {A} (\mathbf {r} )=\nabla \times \nabla \alpha \equiv 0\,\,.}$

• In magnetostatics , the vector potential can be made source-free via the Coulomb calibration , that means

${\displaystyle \nabla \cdot \mathbf {A} (\mathbf {r} )=0}$.

• In electrodynamics , i.e. H. in the case of non-static conditions, on the other hand, the following Lorenz calibration is usually used , which is useful for calculating electromagnetic wave fields:

${\displaystyle \nabla \cdot \mathbf {A} (\mathbf {r} ,t)+{\frac {1}{\ c^{2}}}\partial _{t}\Phi (\mathbf {r} ,t)=0\,.}$Here is the scalar potential (see below) and the speed of light . ${\displaystyle \Phi (\mathbf {r} ,t)}$ ${\displaystyle c}$

• In magnetostatics, the vector potential fulfills the Poisson equation , for which applies (with vacuum permittivity and vacuum permeability ): ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \mu _{0}}$

${\displaystyle \nabla ^{2}\mathbf {A} (\mathbf {r} )=-{\frac {1}{\varepsilon _{0}c^{2}}}\mathbf {j} \equiv -\mu _{0}\mathbf {j} }$.

This gives the following simple representation of the vector potential via a convolution (see Green's function ):

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {j} (\mathbf {r} ')}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} ^{3}r'\,.}$

• In electrodynamics, the Poisson equation extends to the (inhomogeneous) wave equation for the vector potential

${\displaystyle \Box \mathbf {A} (\mathbf {r} )={\frac {1}{c^{2}}}\partial _{t}^{2}\mathbf {A} (\mathbf {r} )-\nabla ^{2}\mathbf {A} (\mathbf {r} )=\mu _{0}\mathbf {j} }$,

where is the D'Alembert operator . ${\displaystyle \Box }$

The inhomogeneous solutions of this equation are the retarded or advanced vector potential

${\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int {\frac {\mathbf {j} (\mathbf {r} ',t')}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} ^{3}r'}$, with . ${\displaystyle t'=t\mp {\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}$

• The three components , and of the vector potential and the scalar potential can be combined in electrodynamics to a four-vector, which transforms like the quadruple in the Lorentz transformations of Albert Einstein's special theory of relativity . is the speed of light. ${\displaystyle A_{x}}$ ${\displaystyle A_{y}}$ ${\displaystyle A_{z}}$ ${\displaystyle \Phi /c}$ ${\displaystyle (ct,x,y,z)}$ ${\displaystyle c}$

Electric vector potential

When calculating fields in areas free of charge and conduction current, e.g. B. in waveguides one encounters the electrical vector potential . ${\displaystyle \mathbf {F} }$

Due to the lack of sources in the fields under consideration, the following applies

${\displaystyle \operatorname {div} \mathbf {D} =0}$       or.

${\displaystyle \operatorname {div} \mathbf {E} =0}$       such as

${\displaystyle \operatorname {div} \operatorname {rot} \mathbf {F} =0}$.

To get a functional relationship between and, one subtracts the equations and from each other and gets: ${\displaystyle \mathbf {D} (r)}$ ${\displaystyle \mathbf {F} (r)}$ ${\displaystyle \operatorname {div} \mathbf {D} =0}$ ${\displaystyle \operatorname {div} \operatorname {rot} \mathbf {F} =0}$

${\displaystyle \operatorname {div} (\mathbf {D} -\operatorname {rot} \mathbf {F} )=0}$

The vortex field is called electrical vector potential. It only describes electric fields that change over time . ${\displaystyle \mathbf {F} }$

Relationships between vector and scalar potential

According to Helmholtz's theorem , (almost) every vector field can be understood as a superposition of two components and , the first of which is the gradient of a scalar potential , the second the rotation of a vector potential : ${\displaystyle \mathbf {K} (\mathbf {r} )}$ ${\displaystyle \mathbf {F} (\mathbf {r} )}$ ${\displaystyle \mathbf {G} (\mathbf {r} )}$ ${\displaystyle \Phi (\mathbf {r} )}$ ${\displaystyle \mathbf {\Gamma } (\mathbf {r} )}$

${\displaystyle \mathbf {K} (\mathbf {r} )=\mathbf {F} (\mathbf {r} )+\mathbf {G} (\mathbf {r} )=\operatorname {grad} \,\Phi (\mathbf {r} )+\operatorname {rot} \,\mathbf {\Gamma } (\mathbf {r} )=\nabla \Phi (\mathbf {r} )+\nabla \times \mathbf {\Gamma } (\mathbf {r} )}$

If there is a conservative force field in which the force, following the principle of least constraint, is always directed in the opposite direction to the direction of the maximum increase in potential , the notation applies as an alternative ${\displaystyle \mathbf {F} (\mathbf {r} )\,}$ ${\displaystyle \mathbf {F} \,}$ ${\displaystyle \Phi \ }$

${\displaystyle \mathbf {K} (\mathbf {r} )=\mathbf {F} (\mathbf {r} )+\mathbf {G} (\mathbf {r} )=-\operatorname {grad} \,\Phi (\mathbf {r} )+\operatorname {rot} \,\mathbf {\Gamma } (\mathbf {r} )=-\nabla \Phi (\mathbf {r} )+\nabla \times \mathbf {\Gamma } (\mathbf {r} ).}$

literature

• Adolf J. Schwab: Conceptual world of field theory. Springer Verlag, 2002. ISBN 3-540-42018-5 .