Equivalent charge density (эквивалентная плотность заряда)
Assuming the same dielectric constant in whole
space:
Substituting the expressions for the derived linear charge densities, we shall get:
Charge density
There are no free charges at the interface!
The last expression is universal in any case for any shape of this surface. Primary source and field may describe the field, which is induced in the case of non-homogenous medium (неоднородная среда), which is induced by the surface charges. And surface charges may be calculated by this expression.
Method of images for cylindrical boundaries between dielectrics (метод изображений цилиндрических границ между диэлектриками) Problem formulation (постановка задачи)
There is a primary field source is shown here with this yellow dot. It is infinitely long wire and not very far from this cylinder wire. There is a cylinder with the dielectric permittivity, which differs from the permittivity of surrounding space. The external space is the vacuum so dielectric permittivity of this space is ε0. ε is always bigger than ε0. Our goal is again satisfied the boundary conditions.
The inverse point (обратная точка)
Normal component of the field intensity (нормальная составляющая напряжённости поля)
The normal component of the external field
intensity:
A radial line that passes through this P point and the normal direction in this point corresponds to the radial direction, they are the same. The only difference is between real normal component and direction. In mathematics, positive normal component should be directed from the center. The direction of two vectors will be the same. The magnitude may be different.
The field that is induced both by initial charge line and the polarized cylinder may be expressed by means of this surface charge, which is placed and distributed somehow over this cylindrical surface.
Surface charges:
Field displacement induced by surface charges:
This total field intensity is the superposition of two sources. First of them is normal component, which is induced by the external source, and another is the field intensity, which is induced by the surface charges:
Normal component of the field intensity outside the cylinders:
To find a proper solution of the problem outside the cylinder it is enough to ensure the right values of the normal component of the field intensity along the interface!
Geometrical relations (геометрические соотношения)
Wire with the linear charge density of τ is located in the point A. P is an arbitrary (произвольный) point at the circle. B is the inverse point. That is inverse point B shall connect with the arbitrary point of the surface P and we shall have triangle. One of them is ACP. Another one is CPB. There are similar (подобны)!
Consider triangles APC and PCB:
But we need to extract only one affect from this similarity. Angle of β is the same in two triangles and it is conclusion, which is necessary for the further transformations.