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

Let's consider the reflection of the magnetic field in flat boundary.

Boundary conditions:

For the chosen, coordinate system:

Equivalent magnetic charge density (Эквивалентная плотность магнитного заряда)

Assuming the same magnetic permeability in the whole space:

This wire with the current infinitely long, infinitely thin wire with the current, induces magnetic charges at the interface. The results may be different because the current wire induces magnetic field, which circulate around the wire. The electric field has another direction. Nevertheless, the basic ideas may be similar but results may be difference.

Charge density

The field in the presence of a cylindrical magnetic object (Поле в присутствии цилиндрического магнитного объекта)

If we shall place current in this point (A), then again, we can find reflections or mirror images of this currents. These images will be placed in the same positions as before.

The field sources for the magnetic field intensity in the external domain (Источники поля для напряжённости магнитного поля во внешней области)

The current which passes in the point A, then when we calculate magnetic field outside the cylinder, we use reflections. ‘B’ the current which is placed in inverse point . The same story with the current, which placed in the center. For two wireline (two lines with identical but opposite current). The final current, which is placed in the point C, will vanish. It will be equal to 0.

The field sources for the magnetic field intensity in the internal domain (Источники поля для напряжённости магнитного поля во внутренней области)

И почему здесь везде μ₀ я тоже не выкупил.

If we want to calculate the magnetic field inside the cylinder, we simply replace this current by another one, which has this magnitude .

Images of a two-wire transmission line (external domain) (Изображения двухпроводной линии электропередачи (внешняя область))

Positions of two wire lines are A and A’. The central image vanishes because both wires induce the same current in the center of cylinder. The same current but with opposite signs. B is the point, which is inverse with respect to point A. This point is closer to the surface if we shall compare the inverse point to the A’ current. The current line in point B corresponds the positive current . B’ is also image current with the current of negative ‘i’: . The magnetic field outside the cylinder, which is induced by two-wire line and two cylinders, is really a some of the magnetic field, which is induced by two wire lines or by form unique lonely separate current lines.

Dependence of the field intensity on the coordinate (Зависимость напряжённости поля от координаты)

The tangential component of the field intensity is continues, so this curve should be smooth. The current flows in only one line then field which is induced by the line changes strongly by several orders when we go out from the wire, when we approach the cylinder. That is why the curve comes down to the 0 very quickly. That is important to keep the same values in the left and right side.