Vector potential

The vector potential is, historically, a mathematical tool that in classical electrodynamics, a subfield of physics, has been introduced to deal with the magnetic induction or flux density (clearly spoken to the " magnetic field" ) to simplify.

Mathematically, the vector potential ( in contrast to the scalar potential ) is a vector field whose rotation according to the following formula

A second vector field supplies.

Vector potentials can inter alia be used to decouple the so-called Maxwell's equations used to describe the electromagnetic field and thereby make it easier to solve. So turns out that the vector potential by a convolution of a given position-dependent current density ( ie, an array of current-carrying conductors in space, such examples of a coil ) shows, so you can calculate the vector potential at a given current density, and from this the measurable magnetic induction or flux density that is generated by this arrangement.

The vector potential has the unit.

Definition

The vector potential is defined as

Applies. Here, the rotation of the vector potential. This approach is the divergence of zero because for every two times continuously differentiable vector fields. This is required by the Maxwell equations.

In electrodynamics, the above formula is unchanged, whereas for the electric field

Applies. Here, the scalar potential.

Both of these approaches, together with the Lorenz gauge, may be used to decouple the Maxwell equations. In magnetostatics, the Coulomb gauge is used usually representing the static limiting case of the Lorenz gauge.

Scalar potential and vector potential are in the theory of relativity and quantum electrodynamics to the four-potential

Summarized.

Properties of the vector potential

( 1) The vector potential is determined up to a gradient, because the rotation of a gradient disappears forever. So For any scalar function

( 2) The vector potential is not as conservative vector field. Otherwise, it would be represented by the gradient of a scalar field and it would apply:

( 3) In the magnetostatic vector potential of the Coulomb gauge can be made free source, which means

( 4) In the electrodynamics, that is, in non- static conditions, one usually used, however, the so-called Lorenz gauge, namely the following relationship, which is useful for the calculation of electromagnetic wave fields:

( 5) In the magnetostatic vector potential satisfies the Poisson equation, for which it holds ( with the Vakuumpermittivität and the vacuum permeability ):

And the terms are sometimes referred to as respectively.

(6 ) In the electrodynamics of the Poisson equation for the ( inhomogeneous ) wave equation extended for the vector potential

The ( inhomogeneous ) solution of this equation is the so-called retarded vector potential

The homogeneous solution is determined by the initial conditions.

( 7) The three components, and the vector potential and the scalar potential can be combined into a so-called four-vector in electrodynamics, which in the Lorentz transformations of special relativity by Albert Einstein as the quadruple (x, y, z, ct ) transformed. c is the vacuum speed of light.

Electric vector potential

In the calculation of fields in charge- and circuit- current-free areas, such as in waveguides one encounters the electric vector potential.

Based on the true source of freedom of the considered fields

To a functional relationship between and subtracted to obtain the equations, and from each other, and receives:

The eddy field is called electric vector potential. It describes only time-varying electric fields.

Relations between vector and scalar potential

According to the Helmholtz theorem between can ( almost) every vector field can be regarded as a superposition of two components, the first of the gradient of a scalar potential, the second, however, the rotation of a vector potential:

If a conservative force field, in which the force of the principle of least constraint to the direction of maximum increase of the potential is always in the opposite direction following, alternatively The spelling