Corrections and extensions to A superconductor electromechanical oscillator. Departamento de Física, Universidade Federal de Santa Catarina, Campus,

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1 1 Corrections and extensions to A superconductor electromechanical oscillator and its potential application in energy storage. Osvaldo F Schilling Departamento de Física, Universidade Federal de Santa Catarina, Campus, Trindade, , Florianópolis, SC. Brazil. We have remade the calculations in [1] by obtaining the magnetic force between coil and magnet from the derivatives of the expression for the electromagnetic energy of the system[2,3]. Such calculations have confirmed all the main results of [1], i.e., undamped oscillations of the superconducting system with frequency Ω while transporting a rectified current for both FC and ZFC initial conditions; oscillations persisting in a resistive system provided that the ratio R/L is smaller than Ω; and the continuous charging process of a resistanceless capacitor connected to the coil, at a rate of (F eff /(B 0 a))(ω/ω C ) 2 Coulombs/sec. A very important new result obtained from the present calculations is that the trapped flux inside the coil area also contributes a static force to the system, in addition to the external F already accounted for. The magnitude of such term can be made much greater than usual values of F so that the stored energy deliverable per cycle can be much greater than previously estimated.

2 2 The purpose of this note is to discuss the corrections to be applied in (1) of [1]( the notation used here is the same as in [1]; numbers between brackets designate equations in [1]) in order that such motion equation becomes consistent with the precise Lagrangian formulation of the problem. We apply these corrections to the three cases treated in [1]: resistanceless coil, resistanceless coil linked to a resistor, and resistanceless coil linked to a capacitor. First of all we call the attention of the reader to Figure 1. Such Figure should replace the original figure in [1]. Note that a now stands for the horizontal dimension of the central upper and lower magnets. The coil horizontal sides must be greater than a. With this arrangement of magnets it is possible to make the coil oscillate without flux density variations inside the coil wire, and without the need for the paramagnetic coating suggested in [1]. Therefore, with this setup hysteresis losses might be greatly decreased or even eliminated. All fields in the figure should be greater than B c1. The expression for the magnetic force between a permanent magnet and a coil transporting a current i is not, in general, given simply by F m = i dl B, where the integration is carried out around the perimeter of the coil. Such result is strictly valid only if the current is constant, which is not the case in

3 3 [1]. The magnetic force must, in general, be deduced from the gradient of the expression for the total electromagnetic potential energy by simultaneously imposing Faraday s Law as a constraint[2,3]. We have made the detailed calculations, which confirm that (1) (combined with (2)) describes the motion of a superconducting coil which was cooled through T c under zero field (ZFC) conditions (see (E2) below and the comments that follow it). For these conditions the inner area of the coil should be perfectly shielded from the magnetic flux Φ m0 imposed by the magnet at t=0+, so that the total flux Φ t is initially zero and should be conserved null thereafter. This requires that a sufficiently large initial current i 0 be induced in the coil by the application of the external magnetic field. The value of this current is given by the equation Φ t = 0 = Li 0 + Φ m0. The combination of (1) and (2) results in (3), which remains correct. However, the solution (4) for the ZFC conditions would need corrections due to the finite initial current i 0, which is assumed null in [1]. We will not discuss this particular correction here, but will go on straight to Equation (5) for the current. Such equation is correct but, again, the initial condition should be i(0)=i 0, rather than 0. This implies that the ZFC solution for the current is: i(t) = F/ (B 0 a) ( F/ (B 0 a) + Φ m0 /L ) cos(ωt) (E1)

4 4 rather than simply (6). Note that the solution (E1) can be substantially different from (6) provided F/(B 0 a) << Φ m0 /L. This is a very important conclusion. We shall continue by analyzing the field-cooled(fc) case, in which the coil becomes superconducting with the magnetic field already applied. In this case the total flux to be conserved is Φ t = Φ m0 and the initial current for a high-κ type-ii superconductor above B c1 is relatively small and may be neglected. Equation (1) in the paper should be rewritten in the form[2,3] mv dv/dt = Fv + (i Φ t /L) dφ m /dt (E2) with dφ m /dt = B 0 av as in [1]. We note that if Φ t = 0 ( the ZFC case) we immediately recover (1), since i dφ m /dt = ib 0 av. Otherwise, if Φ t 0 and constant ( FC case), the important alteration is that instead of simply being subjected to an external force F the coil feels an effective force F eff = F + B 0 aφ t /L. That is, the total flux trapped inside the coil area imposes upon the system a force equivalent to an extra weight. Equations (4) and (6) for v(t) and i(t), respectively, remain valid with F eff replacing F. In summary, the recalculations have shown that carrying out the experiment starting from either ZFC or FC conditions will affect the amplitude of the speed, current and oscillations obtained, but not the natural frequency, which remains with the value Ω as deduced in [1]. The fact that the trapped flux contributes an extra

5 5 weight is a very important finding as far as applications are concerned. Firstly, the term B 0 aφ t /L can indeed be made much greater than F. This means that the magnetic energy stored in the coil may become ¾Φ 2 t /L, which is a function of the trapped flux, a term that can be made quite large by adopting strongfield magnets and large coil cross-sectional areas. Secondly, the existence of this term opens the field of application to situations in which gravity is absent, like in outer space missions. Since any external system to be connected to the oscillator will have some electrical resistance we considered in [1] the effects of adding a resistor R in series with the oscillator coil. Equation (1) needs correction in this case, to become[2,3] mv dv/dt = Fv ( dφ m /dt + ir) Φ m /L Ri 2 (E3) which reduces to (E2) if R=0, since Φ t = Φ m + Li would be fixed. The set of equations (1) and (7) was linear and could easily be solved analytically, something not possible with the additional nonlinear terms included. We have solved equations (E3) and (7) numerically and the results are shown in Figure 2. The results are representative of the motion and currents in the loop for underdamped conditions, R/L < Ω. Figure 2a shows the resulting currents when F=0, m= 10-2 Kg, B 0 a= 10-3 Tm, Φ m0 = 2x10-4 Wb, L=10-7 H and R/L = 1 s -1 and 5 s -1 ( for Ω= 32 s -1 ). For these parameters and R=0 the rectified current

6 6 would have a period of 0.2 seconds ( =2π/Ω) and peak value of 4000 A, and the speed would have a sinusoidal behavior of same frequency and amplitude 6 m/s. The main conclusion is that, if the approximate condition R/L < Ω is satisfied the system oscillates and the maximum current is obtained from the outset[1]. For greater values of R/L the oscillations disappear. The interpretation of this result, not predicted by [1] is as follows. Take for instance a type-i superconducting sphere. If such sphere is placed in a region of space where there is an uniform magnetic field, surface currents will be induced in the sphere to cancel the total magnetic flux density inside it, to keep London s fluxoid null inside the superconductor( Meissner Effect). One might say that such effect, explained by the BCS Theory of Superconductivity by means of a properly chosen wave-function for the electron pairs, imposes that the energy contained in the field be transferred to the surface currents in the sphere. That is, such effect, of quantum origin, allows the sphere to spontaneously capture energy from the field. In the present case the superconducting loop also has the ability of capturing energy and for low values of the ratio R/L the oscillations displayed in Figure 2 are obtained. Another situation considered in [1] was the addition of a resistanceless capacitor in series with the oscillator. This time the corrected equation that replaces (1) has an extra term proportional to the charge q of the capacitor C:

7 7 mv dv/dt = Fv ( dφ m /dt + q/c) Φ m /L (E4) The first term on the right reinforced by the flux weight usually dominates the other terms. Therefore, even with the additional terms the numerical solution of (E4) together with (8) has confirmed the main result previously obtained from the analytical solution of (1) and (8). That is, the capacitor connected to the oscillator will have its charge increased at a time-averaged rate (in Coulombs/sec.) of (F eff /(B 0 a))(ω/ω C ) 2 [1]. This solution to the superconductor coil plus capacitor problem displays mechanical oscillations only for irrealistically large values of the product LC. The component of v(t) which is linear in t dominates the solution, so that the coil just drops without oscillating as energy is transferred from its magnetic field to the capacitor. In conclusion, we have remade the calculations in [1] by obtaining the magnetic force between coil and magnet from the derivatives of the expression for the electromagnetic energy of the system[2,3]. Such calculations have confirmed all the main results of [1], i.e., undamped oscillations of the superconducting system with frequency Ω while transporting a rectified current for both FC and ZFC initial conditions; oscillations persisting in a resistive system provided that the ratio R/L is smaller than Ω; and the continuous charging process of a resistanceless capacitor connected to the coil, at a rate of (F eff /(B 0 a))(ω/ω C ) 2 Coulombs/sec.

8 8 The existence of an extra force due to the flux trapped inside the coil was revealed by these mew calculations. The result is that much greater amounts of potential energy may be stored for future use, and also that the system does not need to rely upon the existence of gravity for its operation.

9 9 References [1] Schilling O F 2004 Supercond.Sci.Technol. 17 L17. [2] Panofsky W K H and Phillips M 1962 Classical Electricity and Magnetism( Reading: Addison-Wesley) 2 nd edition, chapter 10. [3] Wilson M N 1983 Superconducting Magnets ( Oxford: Oxford University Press)p. 47.

10 10 Figure Captions: Figure 1: A model system to replace the one displayed in [1]. Four magnets produce uniform fields in such way that no part of the coil is subjected to time-variable magnetic fields. In this case magnetic hysteresis related to the time variation of the external fields might be much diminished or even eliminated ( see discussion in [1]). Figure 2: Simulated behavior of the speed of the coil (a) and currents (b) for two values of the ratio R/L < Ω. See text for details.

11 Figure 1 11

12 12 v(m/s) 6 4 R/L= 1 s -1 R/L= 5 s -1 (b) t(sec.) Figure 2 a

13 13 i(a) R/L= 1 s -1 R/L= 5 s -1 (a) t(sec.) Figure 2 b