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Permanent magnetic field as an energy source

Our world is immersed in a huge ocean of energy, we are flying in an endless space with inconceivable speed. All around floats — all energy. Before us is a daunting task — to find ways to extract this energy. Then, removing her from this inexhaustible source, humanity will advance with giant steps.
N. Tesla, 1891

Nikola Tesla tells about all kinds of energy, including potential, including the Earth's magnetic field. But since it is static in time, then use that energy directly, for example using inductors and resistive load, will not work. The purpose of this note is to establish the possibility of using permanent magnetic fields as a source of energy and necessary conditions. Also, we will try to get some formulas for approximate calculations of the devices working on this principle.
Static and dynamic magnetic induction vector
For starters, we split the magnetic field on static and dynamic. Of course, we speak of the magnetic induction vector, which describes well this field. Figure 1a (or 1c — projection) presented a classic case study in school when the lines of force of the magnetic field \(\vec B\) intersect the conductor w of nonmagnetic material, a section of which is depicted in orange. The rate of intersection of \(\vartheta\) is directed across the conductor, along the axis y. Such movement causes the conductor electromotive force (EMF) [1].
Пересечение проводника линиями магнитной индукции
Fig.1. The intersection of the conductor lines of magnetic induction.
From the theory (and practice) it is known that in this case there is no difference regarding what will happen the movement: the lines of force will move relative to the conductor or the conductor relative to the field lines. The result is the same and is called the principle of reversibility. It is used in motors and generators of electric energy.
Figure 1b depicts another possible embodiment in which the field source and conductor fixed relative to each other, but the intensity of the field increases with time (shows the time moments t1 and t2). This means that its lines of force are compacted, and hence move along the conductor. Such as electric transformer. And if you think how to teach it in schools and universities, very surprised, what separates these two very similar phenomena.

The first and second embodiments are described by different formulas [2,3], although in reality reflect a single law, a special case which we will demonstrate below.

Common in all cases is the continuity of the magnetic lines. This field and the vector of the induction we call static. If power lines prisoedinyaetsya [4], such a vector we will call dynamic. Another property of dynamic vector — the absence of the principle of reversibility. But we'll talk about it in the second part of this work, but for now will examine in more detail with the first embodiment.
Static vector
At the intersection of the conductor w of the magnetic lines of the field \(\vec B\), on the moving charges in the conductor the force of Lorentz [2]: \[ {\displaystyle \mathbf {F} = q\,(\mathbf {v} \times \mathbf {B} )} \qquad (1.1)\] where: \(\mathbf {v}\) is the velocity vector, \(\mathbf {B}\) — vector of magnetic induction. The electric field \(\mathbf {E}\) is missing in our problem and not used in the formula.
If we rassmatrivaem the second variant (Fig. 1b), then it may use a different form of the same phenomenon in the form of Faraday's law [3]: \[ |\mathcal{E}| = \left|{\Bbb{d}\Phi \over \Bbb{d}t}\right|, \quad \Phi = \iint \limits _{S}{\mathbf{B}}\, \Bbb{d}\mathbf{S} \qquad (1.2)\] where: \(\mathcal{E}\) is the magnitude of the EMF, \(\Phi\) — magnetic flux, \(\mathbf{S}\) is a square, pierced through the darkness of the flood. Although it is not principle for us, but we will prove that formula (1.1) and (1.2) describe the same phenomenon, and that it is one and the same law, but written in different forms. The proof will hold for we need a special case, because for General — will need some work.
For this, assume that the conductor w is strictly perpendicular to the magnetic induction \(B\) and velocity \(\vartheta\) (Fig. 1c). Then the formula (1.1) will be simplified, and the Lorentz force will be like this: \[ F = q\,\vartheta\,B \qquad (1.3)\] But we know that \(F = E\,q\), where \(E\) is the electric field strength. We also know the length of the conductor is \(l\), where can I find EMF: \(\mathcal{E} = E / l\) hence: \[ \mathcal{E} = l\,\vartheta\,B \qquad (1.4)\]
Now look at formula (1.2). The induction is constant over the entire area, and the area itself is the product of the length of the conductor at a distance \(h\), which overcomes the induction vector for time \(t\). In this design time only change this distance: \(h = h(t)\), and therefore formula (1.2) can now be written as: \[ \mathcal{E} = {\Bbb{d}[B\,\ell\,h] \over \Bbb{d}t} = B\,\ell {\Bbb{d}h\over \Bbb{d}t} \qquad (1.5)\] Believe what EMF is taken modulo. As the ratio of \(\Bbb{d}h \over \Bbb{d}t\) is the speed, we finally get: \[ \mathcal{E} = l\,\vartheta\,B \qquad (1.6)\] As you can see, the formulas (1.4) and (1.6) are equal, and this means that the processes in Fig. 1a and 1b are the same as Faraday's law and Lorentz law for them are written the same formulas, but in different forms. This equivalence will be needed next.
Static vector well researched and of interest to us only as a comparison with the dynamic that open up new possibilities for the use of permanent magnetic fields as energy source. This — the second part of the notes.
The materials used
  1. Wikipedia. Electromotive force.
  2. Wikipedia. The Lorentz Force.
  3. Wikipedia. The law of electromagnetic induction of Faraday.
  4. Wikipedia. The magnetic reconnection.