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.

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.

Inductance (Индуктивность)

Inductance is a coefficient between the current and the flux linkage (потокосцепление).

Units: Henry [Hn]

The magnetic energy stored in a coil is derived as:

Mutual inductance (Взаимная индуктивность)

The flux which is induced in second coil is proportional to the current which induces magnetic field.

A coefficient between the current in one coil and a flux (flux coupling) in another coil is called a ‘mutual inductance’

Reciprocity principle (принцип взаимности):

It means that the value of the mutual inductance does not depend on the order of induces.