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

General solution of the Laplace’s equation in a cylindrical coordinate system (Общее решение уравнения Лапласа в цилиндрической системе координат)

Combining the solutions, we will get:

This function is known as circular (angular) harmonic of order k.

General solution:

c_k, d_k, g_k, h_k – these coefficients correspond to different numbers (different harmonics) of the order of k.

15. The Faraday’s law (Закон электромагнитной индукции).

The Faraday's Law links together these two sides of one electromagnetic field: magnetic field, from one side and electric field, from the another side. The Faraday’s Law is based on Maxwell equations system. The main idea of this law is: the magnetic field which depends in time, induces some electric field.

The origin of the induced voltage: - time varying magnetic fields;

- moving of the coil in stationary magnetic field.

This interaction between electric field and magnetic field finally induces, for example – electromagnetic wave. In this process (electromagnetic wave) it is important to consider together the Faraday’s Law and very special form of the Ampere’s Law. Ampere’s Law at some stage also tells us that, the time dependent electric field may induce magnetic field. But today the main part of the electromagnetic field theory is the Faraday’s Law.

The Faraday’s Law describes several processes, which takes place in electromagnetic systems. For example: if we have static system, which configuration is stable, doesn’t dependent on time, in principle in such electromagnetic system, the electric voltage may be induced or electric field may be induced, if the magnetic field changes in time. On the other hand, the opposite situation is possible: the magnetic field doesn’t dependent on time, nevertheless the electric field and the voltage, which is induced in some contour will be induced. That is possible if the contour, where we consider the voltage or electric field has unstable shape, shape may depend on time. Or this contour has a stable shape, but the contour itself moves in the external magnetic fields. These two sides of the same law, the Law which calls Faraday’s Law. They’re also very important parts of the Faraday’s Law, so called Lenz’s Law.