2019-03-17

Dynamic vector

Of Faraday's law and the Lorentz force involves the appearance of EMF in the conductor, if he has a relative transverse movement of the magnetic induction vector. The key word here is "movement". But how to "get to work" permanent and stationary magnetic field? In the first part of this work we conducted a small analysis of the values of EMF when moving a static vector. Here we will discuss about dynamic vector, characterized in that its magnetic lines of force can pereodevatsya and thereby capturing the energy of a stationary magnetic field.

Magnetic reconnection is the process of changing the topology of the magnetic field lines. As the result of reconnection of magnetic field tends to move to a state with less energy. Given that the magnetic lines must not overlap between themselves and be open, you can submit a picture of their distribution without resorting to the formulas.

From the theory and observations the reconnection of magnetic lines is known [1,2] that this process can occur with the release of energy, and without it. We are interested in the latter option, though, given our assumption of full utilization of energy generated, in the presence of the appropriate device it is possible to consider and the first.

To understand the principle, here, we'll look at a simplified one-dimensional model of reconnection of magnetic lines. To do this, imagine that we have a constant magnetic field with direction given by the vector \(\vec B_1\) (Fig.

*2a*). Perpendicular to it there is another magnetic field vector \(\vec B_2\) and begins to occur the process of partial switching of the magnetic lines (Fig.*2b*). The second field is variable and constantly increasing. At the same time, the number of magnetic lines increases, and the distance between them is reduced (Fig.*2c-2d*). If you follow the motion of one magnetic line, the vector will move and at the same time be rotated relative to the static vector \(\vec B_1\). Fig.2. The switching process of the magnetic lines in perpendicular magnetic fields |

This turn we take as a basis for further reasoning. More details it consists of two electro-magnets

**M**(Fig.*3a*), which form a growing magnetic field with an induction of \(B_2\). Between them is a constant magnetic field with induction of \(B_1\). Also, the electromagnets are located between the two nonmagnetic conductor**w1**and**w2**, the centers are slightly shifted relative to the common center of the design \(h/2\). We believe that this offset and the diameters of the conductors are extremely small compared to \(h\), and their length equal to the length of the whole structure — \(l\). When the field is \(B_2\) starts to increase, the process of reconnection of magnetic lines and their passage through conduits with an average speed of \(\vartheta\), which is always perpendicular to the resulting vector \(B\) (Fig.*3C*). Fig.3. The design of the device, geometry of the resulting vectors and their movement-rotation |

Next, we will consider only one of the conductors, because the second will be similar but opposite in sign processes. Geometric diagram of the process is shown in figure

*3b*, where the center is the conductor, through which passes the resulting vector \(B\). In fact, these vectors are parallel to each other, there is plenty, we consider a total. It is obvious that: \[B = \sqrt{B_1^2 + B_2^2} \qquad (2.1)\] the Segment \(a\), the same direction and proportional to the average speed is thus: \[a = \frac{h}{2} \sin\alpha, \quad \sin\alpha = {B_2 \over B} \qquad (2.2)\] Then the rate will are derived from this value. Given that \[\quad B_1 = const, \quad B_2 = B_2(t), \quad \dot B_2 = {\Bbb{d}B_2 \over \Bbb{d}t} \qquad (2.3)\] we find this velocity: \[\vartheta = {\Bbb{d}a \over \Bbb{d}t} = \frac{h}{2} {\dot B_2\,B_1^2 \over \left(B_1^2 + B_2^2\right)^{3/2}} \qquad (2.4)\] Substituting it into formula (1.6) and considering that in the end we put the EMF of the two bond wires**w1**and**w2**, obtain the desired EMF: \[\mathcal{E}(t) = 2\,l\,\vartheta\,B = l\ h {\dot B_2\,B_1^2 \over B_1^2 + B_2^2} \qquad (2.5)\] As you can see, in the creation of the EMF involved a constant magnetic field \(B_1\). If it is zero, then EMF is zero. But this field will remain inoperative, unless the rate of rise of dynamic fields \(\dot B_2\). Only together they can create the desired effect! It is obvious that the EMF will also depend on the size of the whole structure (\(l, h\)).Output power

To obtain the output power required two factor: the voltage and current. Voltage we already know, and talk — find out conditions of transient processes in the circuit of

There is a circuit consisting of a voltage source \(\mathcal{E}(t)\) is our EMF, the inductance of the two conductors \(L\) and resistive loads \(R\), which we included losses in the conductors. Then the expression for our circuit will be: \[\mathcal{E}(t) = L {\Bbb{d}I \over \Bbb{d}t} + R\,I \qquad (2.6)\] From the theory of transient processes [3], without tiring the readers detailed calculations, just write the current value of: \[I(t) = {e^{-t/\tau} \over L} \int \limits_0^{t} e^{t/\tau} \big(\mathcal{E}(t) - \mathcal{E}(0) \big)\, \Bbb{d}t \qquad (2.7)\] the Instantaneous output power will be, as the product of voltage and current: \[P(t) = {\mathcal{E}(t) \over L} e^{-t/\tau} \int \limits_0^{t} e^{t/\tau} \big(\mathcal{E}(t) - \mathcal{E}(0) \big)\, \Bbb{d}t \qquad (2.8)\] Here: \(\tau = L/R\) — time constant of an RL-circuit, \(\mathcal{E}(0)\) is the value of the EMF at the beginning of the pulse. A graph of the output power on time will have first a sharp rise and then a smooth decrease, approximately as shown in

*Fig. 4*. Fig.4. Scheme for transient study in RL-circuit | Fig.5. An exemplary graph of output power from time to time |

*Fig. 5*. For finding the efficiency of such system it is necessary to know the average power output to \(P_2\), which can then be compared with the cost of \(P_1\): \[P_2 = f_G \int \limits_0^{1/f_G} P(t)\, \Bbb{d}t \qquad (2.9)\] where: \(f_G\) is the frequency of pulses or oscillations of the field \(B_2\). Then efficiency will be as \[\eta = {P_2 \over P_1} \qquad (2.10)\]

Insights

The use of permanent magnetic fields as energy source offers tremendous opportunities in the energy sector. In this work we have deliberately considered only a simplified version of the construction of the device that demonstrated the principle. Using such a mechanism, and a modern element base, you can build better systems.

From the formulas it is clearly seen that for capturing the energy constant field needs another dynamic, with steep rise front of the pulse. This front will be crucial for manifestation of the effect. Box \(B_1\) can be variable, with a small frequency such that a relatively dynamic field it might be considered as stationary. In this case, the first field can be generated with the aid of reactive power than can greatly reduce costs. And if to generate a dynamic magnetic field (\(B_2\)) to use the pulse technology on the bias currents, the cost of creating it possible to further reduce, and thereby increase the efficiency of the entire scheme.

Enhance the effect can be if the conductors

**w1**and**w2**placed in the environment with high magnetic permeability. In this case, the value of induction \(B_1\) increases to \(\mu\) times, and hence will increase in proportion and other parameters.__The materials used__

- Priest E., Forst T. Magnetic reconnection. M.: Fizmatlit, 2005.
- Wikipedia. The magnetic reconnection.
- Lecture 7. Transients in circuits of the first order.