The cut in the space (Разрез в пространстве)

How to resolve the problem which does not allow to consider the magnetic field as the potential.

The yellow ring is wire with the current. Current induces magnetic field around itself. So if we shall go around the wire we immediately will have a ratio but it is forbidden. If so that we can’t use a scalar magnetic potential at all. In such case we should make the cut in the space.

Black line – cut line. Starts at the infinity, goes around the wire and then go to the infinity.

We shall consider that our problem domain is the space around this line.

Now any closed loop will not contain a current inside. In such case we can use a definition of the scalar magnetic potential.

T he cut in the space is necessary to do if we want to consider the systems with the current.

Laplace equation for the scalar magnetic potential (Уравнение Лапласа для скалярного магнитного потенциала)

Basic equations:

In general case:

For the medium with the constant magnetic permeability μ:

– Laplace equation for the scalar magnetic potential

This equation has solution only if the correct boundary conditions will be applied to the problem.

10. Vector magnetic potential. Inductance (Векторный магнитный потенциал. Индуктивность)

Vector magnetic potential (Векторный магнитный потенциал)

Vector magnetic potential is universal. It exists both in conducting media with currents, in insulating area where are no currents at all, may be used in static magnetic field and may be used in electrodynamics, where magnetic field depends on time.

Main equations:

Consider a vector satisfying a relation:

Why it is possible. Let’s apply an operation div to both parts of this relation. We shall get:

and as we know

Magnetic flux (Магнитный поток)

Definition of the flux:

Stokes theorem:

Differential equation for the vector magnetic potential (Дифференциальное уравнение для векторного магнитного потенциала)

Ampere’s law:

Magnetic field intensity and flux density are related by:

Taking into account , we get:

If μ is constant:

Identical transformation:

Gauging of the vector magnetic potential (Калибровка векторного магнитного потенциала)

The vector potential defined by the relation is not unique (it can’t give us unique definition of the vector A).

Adding a term of to the value of the vector potential does not change the flux density, because

We can invent many different functions which will satisfy to this relation. How to get rid from all of this big number of solution and keep only one of them? We can amply additional property at vector A. And this additional property is called gauging.

The most often ‘Coulomb gauging’ is used:

Калибровка вектора магнитного потенциала необходима, чтобы обеспечить единственность вектора в пространстве. Существует несколько калибровок, позволяющих прийти к этой цели. Зачастую используется Кулоновская калибровка.

Integral presentation of the vector magnetic potential (Интегральное представление векторного потенциала)

For the Coulomb gauging of the vector potential we get:

In Cartesian coordinate system this vector equation results in 3 scalar ones:

Comparing to electric field:

Each scalar equation has an integral solution of:

Comparing to electric field:

We can unite these expressions into one vector formula:

This expression gives us unique value of the . Also, for physical systems we can say that the vector magnetic potential tends to 0 in infinitely remote point as same as potential.