Forces. The second line. (Силы. 2ая линия)

We can consider the force which is applied to the left wire with the coordinates of x_A (A), may be calculated as a sum of three components: , where - the force from the sight of a second wire; – the force from the sight of first image; - the force from the sight of second image.

, it has positive sign because force is attractive

, it has positive sign because force is attractive

, it has minus because force is repulsive

Finally:

Лекция 5. Solution of Laplace’s equation by separation of variables. (Решение уравнения Лапласа методом разделения переменных) Application of Laplace’s equation (Применение уравнения Лапласа).

Electrostatic field.

This equation for the potential is valid in the absence (отсутствие) of the distributed (распределённого) electric charges.

Magnetostatic field.

U here is magnetic scalar potential. The equation is valid if there are no currents in the problem domain.

Laplace’s equation has a unique solution when the boundary conditions are defined at the border of the problem domain:

- 1st type boundary conditions. (defenition of the potential along the border(part of the border))

- 2nd type boundary conditions. (normal derivative of the potential along the border (part of the border) of the boundary of the problem domain)

Choice of a coordinate system (Выбор системы координат)

Reasonable choice is a cylindrical system with the center in O (inside the ferromagnetic object)

Variable separation in cylindrical coordinates (Разделение переменных в цилиндрических координатах)

The variable separation method starts with assumption (предположение): we can present the potential or unknown function as a product of two special functions – function that depends on radius and function that depends on angle.

Cylindrical coordinate system; variables:
Laplace’s equation in cylindrical system:
Presentation of the potential:
Laplace’s equation: (просто подставляем)
or: (разделили обе части на RS/r2)

- function that depends on radius

- function that depends on angle

There is only solution: both sides are the constants, that’s only function which satisfy this strange condition.

Angular function (Угловая функция)

Equation for the angular function: (k may be positive, but then the form of the solution will be different)
or: (умножили обе части на S)
Solution:	, g, h = const
	Evidently k is integer number. (потому что при других k один оборот не будет происходить за 360 градусов)

Radial function (Радиальная функция)

Equation for the radial function:

or:

Let us try to find a solution of this equation by substituting: , α – some unknown power.

The solution: , c, d = const