20. Diffusion of magnetic fields into conductors (Распространение электромагнитного поля в проводнике)

When we shall talk about this topic, we shall assume, that’s we have conducting media, usually, it’s metal. And the electromagnetic field satisfies the Faraday’s Law and the Ampere’s Law. The specific feature of this fields is there are no displacement currents. That’s usual situation in conductors, sometimes even in conductors the displacement currents are taken into account. That is a case, which we called bad conductors. But today we shall neglect them. Inside in conducting medium there are no displacement currents and there are no free charges. In such a case the electromagnetic fields may be described by several equations:

The Faraday’s Law:
In this case this is conductivity current, which may be described by the Ohm’s Law:
Also, the electric and the magnetic fields satisfies the Gauss’s Law (for the magnetic field it’s a universal law, so it takes place in any case. In the case of electric field, it’s some special case: divE=0 only in a free charge medium (это следствие из divD=0)):
The First Kirchhoff’s Law:

Let’s make some transformations with these equations:

Differential equations for the electric field intensity:

M athematical transformation:

Diffusion equations for the electromagnetic field characteristics:

Such equations (applied to scalar variables) describe processes of the particle diffusion, thermal processes.

One-dimensional equations: (here we assume that only x-component of the E and y-component of H exist)

21. Periodic electromagnetic fields in conductors. (Периодическое электромагнитное поле в проводниках)

These approaches are very important form of existence of electromagnetic field – electromagnetic wave. That’s the case when the electric field intensity or magnetic field intensity depends on time according to sinusoidal function:

Quasi-stationary approaches:

The electromagnetic field is often called a planner electromagnetic wave. To describe it let’s use the complex method (using to describe currents and voltage in electromagnetic circuits):

O n the picture: in the left part γ is equal to 0, so electromagnetic wave comes from somewhere to this planner surface and then it starts to penetrate the conducting material (metal):

Equation:
Solution for the complex field intensity:
Parameter α:
Using designation:
Solution for the field intensity:

(A₂=0, because: decreases, when we are travelling into the conducting medium and field intensity is growing to infinity)

Penetration of the electromagnetic field into a conductor. (Проникновение электромагнитного поля в проводник)

E verywhere inside the conductor the magnetic field intensity will be oscillating function, but the amplitude of this oscillation will be less and less:

The penetration length:

Amplitude of the electromagnetic wave dumps according to exponential dependence: