Axial Magnetic Field of Superconducting Loops

Transcription

1 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p.1 Axial Magnetic Field of Superconducting Loops Inbum Lee, Tianhao Yang, and M. J. Madsen Department of Physics, Wabash College, Crawfordsville, IN (Dated: December 16, 2015) In this experiment, we attempted to create a closed superconducting loop. Two designs of the loop were tested. One was created by splitting the middle part of a strip of HTS thin film conductor and opening up the hole. The other one was by taping the two ends of another HTS strip. We tested them by measuring the axial magnetic field produced the two loops. We expected to obtain a magnetic field close to that of a magnetic dipole, and the data show that the magnetic field of our split loop agrees with the expectation. Hence we produced a closed loop.

2 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p.2 I. INTRODUCTION A current inside metal produces a magnetic field[1]. When a superconductor reaches its critical temperature, the resistance inside the superconductor becomes zero[5]. Then, if we use a change in magnetic flux to induce a current inside a superconducting loop according Lenz s law[1], it will become a cryo-permanent magnet until it loses its superconductivity. Due to the discovery of High Temperature Superconductor(HTS), superconductivity became more easily reachable using liquid nitrogen[2]. In this experiment, we used HTS thin films[3] and designed closed superconducting loops that would act as a cryo-permanent dipole magnet with a persistent current. II. MODEL By inspection, a perfectly closed superconducting loop would produce the same magnetic field as a closed loop or a single-turn coil. Therefore, we model the magnetic field of our superconducting loops we designed with the axial magnetic field of a closed circular loop (see Fig.1) whose equation is B = µ 0 2AI, (1) 4π ((z + z 0 ) 2 + R 2 ) 3/2 where µ 0 = 4π 10 7 T m/a is the vacuum magnetic permeability, A = πr 2 is the area of the loop, I is the current flowing inside, R is the radius of an effective circle that has the same area as our loop and z 0 is the z-position offset[4]. If our loops are truly closed loops without disconnection, the data points of the magnetic field of the superconducting loops should be fitted well with the formula. The two left figures in Fig. 1 show the expected current flow, induced magnetic field, and the graph of the axial magnetic field versus z- position of a closed loop. The magnetic field should peak when z = 0. The direction of the magnetic field, represented by the sign of the magnetic field, will remain constant above and below the loop. On the other hand, the figures on the right (Fig. 1) show those of a broken loop, in which two counter-propagating currents flow with an offset by a small distance z 0 from the center of the loop. It can modelled with a magnetic quadrupole whose equation is,

3 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p.3 B z = µ 0 4π 2AI ((z z 0 ) 2 + R 2 ) 3/2 µ 0 4π and all the variables are same as mentioned above[4]. 2AI, (2) ((z + z 0 ) 2 + R 2 ) 3/2 Current Area Radius Current FIG. 1. The figures above show the possible outcomes and the corresponding axial magnetic field graphs from the superconductors made and used in this experiment. The left two figures are the diagram and the axial magnetic field graph of a closed superconducting loop. The figures on the right are those of a disconnected loop. The current inside the broken loop cannot flow in a circle so that its magnetic field looks like that of a magnetic quadrupole. Our expectation is to obtain a graph that looks like the one in the lower left corner. III. SETUP We began our experiment by designing superconducting loops in two ways using strips of HTS thin film. The first one was created by cutting the middle of a strip while leaving the edges connected. Then we opened it up to make a hole in the middle. Its physical shape looks like the upper left figure in Fig. 1. We also created another loop by taping the the end of a superconducting strip. We predicted that the current inside it would flow like the current in the upper right figure in Fig. 1. The two superconducting loops were placed inside an insulating container one at a time and were submerged in liquid nitrogen to cool their temperature below the critical temper-

4 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p.4 ature. Before they reach below the critical temperature, we put a 0.25T permanent magnet on top of them in order to induce the current inside the loops by removing the magnet after the loops turn to superconductors. The change in magnetic field strength induces a voltage inside the loop and hence a current according to Faraday s law [1]. Although literatures suggest the hysteresis effect in superconductors[6], and we could put a magnet after the loop gains superconductivity, using the effect, it would be easier to induce the current by removing a magnetic flux existing beforehand. The induced current then produces magnetic field according to Lenz s law[1]. We measured the magnetic field along the z-axis with a HMC5883 magneto-resistive sensor, and we used a stepper motor to change the position of the sensor at 3mm increments. In total, the axial magnetic field at 56 locations, above and below the loops, were measured. HMC 5883 Magneto-resistive Sensor FIG. 2. This is a diagram of our setup. A stepper motor was used to move the z-position of the HMC5883 magneto-resistive sensor at 3mm increments. The magnetized superconducting loops were placed inside an insulating container with liquid nitrogen In this experiment, we measured the on-axis magnetic field of the superconducting loops at 56 locations, above and below, with fixed x- and y-position. IV. DATA AND ANALYSIS We removed the influence of the earth s magnetic field by subtracting the background magnetic field from the data obtained in the presence of the superconducting loops. In agreement with our expectation, the circular, taped loop produced the magnetic field of a

5 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p.5 quadrupole. Its magnetic field versus z-position graph is shown at the top of Fig. 3. On the other hand, the magnetic field data of the split loop appeared to match the magnetic dipole model. Since our interest was more in the split loop, we only conducted the analysis of the split loop data. First, we fitted the data with the magnetic dipole model whose equation is Eq. 1. The parameters obtained then were R = ± 0.29mm (95 % CI, Gaussian pdf.) and z 0 = ± 0.15mm (95 % CI, Gaussian pdf.). From the fit analysis, we also obtained IA value, and in order to calculate the current from it, we approximated the actual area of the loop. We began approximating by graphing its upper and lower outlines with a tracker software. Then, we integrated the area under the two graphs and calculated the enclosed area using their difference. We assumed the uncertainty in this analysis by calculating the area difference coming from 1 mm of error in tracking outlines. As a result, the measured area was A = (3.59±0.14) 10 3 mm 2 (95% CI) the calculated current inside the superconducting loop was I = ± 0.83 A (95% CI). This current is unexpectedly high, but with the certainty in our calculation, it is rather exciting to have such strong current. V. CONCLUSION Among the two designed and created superconducting loops, the split loop met our goal. Its magnetic field graph that resembles the magnetic field of a magnetic dipole shows that it is a genuine closed loop in which the current flows in circle. Due to its superconductivity, the current is persistent, and thus the loop acts as a cryo-permanent magnet. The calculated current inside was well above our expectation, but we are certain in our calculation. The next possible research area can be testing the hysteresis effect or measuring the critical current of the loop. [1] David J. Griffiths, Introduction to Electrodynamics, 3rd edition (Prentice Hall, Upper Saddle River, NJ, 1999). [2] Allen, Philip B., et al. High temperature superconductivity. Ed. Jeffrey W. Lynn. Springer Science and Business Media, [3] Correra, L., ed. High Tc Superconductor Thin Films. Elsevier, 2012.

6 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p B-magnetic field(mg) Magnetic Field (mg) z-position(mm) z-position(mm) FIG. 3. The graphs above are the magnetic fields of the two loops along the z-axis. The top graph shows the magnetic field produced by the circular, taped loop whereas the bottom one shows the field produced by the split loop. The graphs were shifted in order to match the expected graphs in Fig 1. [4] John D. Jackson, Classical Electrodynamics, 3rd edition (Wiley, Hoboken, NJ, 1998) [5] Onnes, H. Kamerlingh. Disappearance of the electrical resistance of mercury at helium temperatures. Akad. Van Wetenschappen (1911). [6] Silcox, J., and R. W. Rollins. Hysteresis in hard superconductors. Applied Physics Letters 2.12 (1963):

7 WJP, PHY381 (2015) Wabash Journal of Physics v2.3, p.7 Magnetic Field(mG) z-position(mm) FIG. 4. The magnetic field of the split loop is fitted with the magnetic dipole model, Eq.1. The graph was shifted in order to match the expected graphs in the model section, Fig 1. From the fit parameters, we calculated that the radius of the effective circle is R = ± 0.29mm (95 % CI, Gaussian pdf.), z-position offset is z 0 = 62.48±0.15mm (95 % CI, Gaussian pdf.). Also, with measured area of the loop A = (3.59 ± 0.14) 10 3 mm 2 (95% CI), we calculated the current inside I = ± 0.83 A (95% CI).