Image method for the flat boundary between magnetic media (Метод изображений для плоской границы между магнитными носителями)

There are 2 half-spaces with different magnetic permeability µ₁ and µ₂ and the point that corresponds to the infinitely long line with current i. This line suspended above this surface at the high h. We need to find the field intensity distribution everywhere in the space in both sub-spaces (above the interface and below the partition boundary.

The magnetostatic problem is described by the Poisson and Laplace equations. This equation has a unique solution in the case when we have defined proper boundary conditions. This may be definition of the potential at the border of the considered space; or normal derivative of the magnetic potential or intensity of the magnetic field because the normal derivative of the potential is really the normal component of the magnetic field intensity or if the medium has the constant magnetic permeability, then it is identical to the case when we shall define a normal component of the flux density.

– the field induced by line source. This expression works only in the case when there are no surfaces. There is only one space with one magnetic permeability is everywhere the same.

Let's suppose that the magnetic field in the upper half-space above the interface may be calculated as the superposition of two magnetic field. The first of them induced by initial wire itself and second of them induced by another wire, which is placed under the surface, and which has a current i₁.

Magnetic constant is the same in both half-spaces µ₁. Let us place the image into the point of . Current of the image is ₁. So we can find field intensity in the boundaries:

Let’s suppose that the magnetic permeability of the whole space is µ₂. Now there should not be any current in the lower space. But above the surface there will be a unknown current . This current, which is placed above the surface at the distance h. Field intensity at the boundaries:

Boundary conditions are:

Boundary conditions for horizontal component of the field intensity:

In the upper half-space:

In the lower half-space:

First relation:

Boundary conditions for vertical component of the field intensity:

In the upper half-space

In the lower half-space

Second relation:

Combining with the first relation:

If the boundary conditions are satisfied, then the solution is unique one. There are no other solutions. That is why such mirror reflection is the only one possible.

8. Static magnetic field. Biot–Savart’s Law. Ampere’s Law (Статическое магнитное поле. Закон Био–Савара. Закон Ампера)

Basic property of static (or stationary) magnetic field that it shall not depend on time.

Variables and units (Переменные и единицы измерения)

Variable	Symbol	Units
Flux density Индукция магнитного поля	B	Tesla	[T]
Field intensity Напряжённость поля	H	Ampere/meter	[A/m]
Magnetic permeability Магнитная проницаемость	μ	Henry/meter	[H/m]
Inductance Индуктивность	L	Henry	[H]
Flux Поток	Ф	Weber	[Wb]
Flux linkage Потокосцепление	ψ	Weber	[Wb]
Scalar magnetic potential Скалярный магнитный потенциал	Um	Ampere	[A]
Vector magnetic potential Векторный магнитный потенциал	A	T·m (Wb/m)	[T·m]
Magnetization Намагниченность	M	Ampere/meter	[A/m]