Potentials in the problem domain (Потенциалы в проблемной области)

Inside the cylinder. General case.
Case of the line current source: (potential on the x-axes = 0)
Outside the cylinder. General case. (reduced potential)
Case of the line current source:
Current potential on the surface of the cylinder.

Magnetic field intensity in the problem domain (Напряжённость магнитного поля в проблемной области)

Inside the cylinder.
Outside the cylinder.

Magnetic field intensity induced by the wire with the current (Напряжённость магнитного поля, индуцируемого проводом с током)

Radial component of the field intensity induced by the current line at the cylinder surface
Angular component of the field intensity

The field intensity induced by the current line may be calculated directly:

Definition of coefficients (Определение коэффициентов)

The above general relations are used for definition of coefficients.

At the surface of the cylinder:

Solution of the problem (Решение задачи)

After substituting this solution into the expression of potential, we will have a scalar potential and reduced potential:

Scalar potential:
Reduced potential:

Magnetic field on the system axis (φ = 0):

For total magnetic field inside the cylinder:

For reduced magnetic field outside the cylinder:

Magnetic field directions (Направления магнитного поля)

Let’s assume that there is a current which comes to us. Then the direction of will be downward and this at the axes is an angular component of the magnetic field which is induced by the wire. The total magnetic field inside the cylinder should have the same direction, but much smaller than . is the reduced magnetic field, it also has only one angular component, the magnitude of this vector approximately the same as . The difference between them is very small in the vicinity (окрестность) of this surface.

Inductance of the two-wire transmission line per unit length (Индуктивность двухпроводной линии передачи на единицу длины)

Total inductance: L  L_2wire  L, L – additional inductance, it is induced by this cylinder.

L_2wire is the inductance of 2-wire line without a cylinder

The flux induced by the magnetized cylinder (Поток, индуцируемый намагниченным цилиндром)

The flux induced by magnetized cylinder:

After integrating and substitution:

Additional inductance for the current line:
Additional inductance for the two-wire current line:

Лекция 6. Time dependent electromagnetic fields (Зависящие от времени электрические поля)

The basement of the theory of time dependent electromagnetic fields – the Faraday’s Law.

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.