Hering’s paradox (Парадокс Геринга)

We shall consider a magnet that is c-shape magnet. The magnetic field is induced in the gap of the magnet, it may be induced by the electromagnet, it means there should be coil which induces the magnetic field. What is the experiment? The magnet without leakage. The magnetic flux density lines goуы around the ferromagnetic core, then these lines leave the core just at this surface. Then they go parallel to each other to the opposite side of the ferromagnetic core. The coil and the amperemeter are placed far from this magnet and somebody take this system and start to move to the magnet. In the beginning the amperemeter or voltmeter demonstrates nothing, but then when it cross magnetic field, the EMF is induced in this contour and detector just demonstrates us something happened to the voltage or current are induced in this contour. Outside the magnet, there is a conducting loop with elastic contacts that ensure a closed circuit at all times. The circuit also induces a voltmeter.

T hen we started move the coil down. In such a case our voltmeter shows nothing, because magnetic flux which is linked with this contour is constant.

Movement of the coil causes a voltage impulse:

So, we move contour from this position to that one

T hen, we're taking this contour this magnetic system and started to move outside the magnet. In such a case this spring (пружинка) contacts now nevertheless forms a closed circuit. Now closed circuit starts here (на картинке). It passes through the voltmeter or amperemeter, then comes to the opposite side of the core, this is spring contact with iron core. We moved outside, then at the exit this spring again comes to each other and forms a closed contour.

It's shown here what is going on: in the beginning this spring contact form this contour

Then they contact only the core, nevertheless there is the contour here

Finally, again form a closed circuit

I f we shall consider this situation from the opposite side, from the formulation of the Faraday's Law which deals with the moving conductors, then it may be understandable. When we move this conductor (На первой картинке, где Ф не равно 0) with this spring contact, we see here contact moves, but there is no magnetic field here, that is why equal to zero.

Now, in this system (вторая картинка) there is no field outside the core, that is why this spring contact moves, but they aren't in the magnetic field, that is why the electric field is not induced inside this contacts and inside the wires which connect these contacts with the detector. About this inner part of the core, this part doesn't move, there is a magnetic field and strong magnetic field inside, but the velocity equal to zero, that is why the field intensity not induces inside. So, from this point it's clear, they shouldn't be electric field at all, they shouldn't be induced EMF. And the experiment demonstrate - there is no induced EMF.

This may appear a paradox which represents a contradiction to Faraday's Law. Essential is, however, that the flux change is not related to the motion of the conductor.

I nside the magnet: field exists but there is no movement.

Outside the magnet: conductor moves, but there is no field.

Лекция 7. Time dependent electromagnetic fields. (Зависящие от времени электрические поля)

Diffusion of electromagnetic fields in conducting media. (Распространение электромагнитного поля в проводящей среде)

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:

Mathematical transformation:

Diffusion equations for the electromagnetic field characteristics:

Such equations (applied to scaler 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)