S17_ThomsonJumpingRing

Examining the Electrodynamics of the Thomson Jumping Ring Experiment

Alec Twaites and Taylor Chadwick

University of Minnesota

Abstract

The purpose of our experiment was to construct a model for the motion of an aluminum ring as it undergoes an accelerating force due to a magnetic field from a solenoid coil. We achieved this by measuring a series of height dependent auxiliary functions, and then using second order Euler approximations and Newton's second law of motion to generate a model of the expected motion. We found that our model described the motion of the ring to within 0.42 standard deviations in the regime where the magnetic force is driving the motion.

Introduction

Faraday's law states that the closed path integral of the electric field is proportional to the change in the magnetic flux over some time. For the purposes of this experiment the closed metal ring acts as the path on which the electromagnetic field induces a current. Lenz’s law defines the direction that the electromagnetic field induced by a magnetic field acts. More specifically, the induced current in the ring generates a magnetic field opposing the magnetic field from the solenoid, thus creating a repulsive force, which propels the ring in the axial direction (which we have defined as positive z). While Faraday’s and Lenz’s laws govern a large part of the physics in the Thomson jumping ring experiment, there are a plethora of complex electrodynamic interactions happening between the induced magnetic fields and electric currents that the electrostatic theory fails to account for [1,2,5].

If one were only to consider Faraday’s and Lenz's laws, then the attractive and repulsive forces on the ring average to zero. When an AC current is applied to the solenoid the electromotive force described by the law is positive in the first quadrant of the AC current. This reasoning is called the ‘first–quadrant explanation’ of this phenomenon, because it only utilizes the first quarter-cycle of the applied current and ignores the remainder of the cycle [1]. As Amiri [1] explains, if one only considers the first quarter of the applied current, then for the ring to jump over a 10 cm core in a 4 ms time frame, the average acceleration would have to be over 1200g which, given the parameters of the experiment, is impossible. Thus, the ring must require multiple cycles to travel past the end of the extended core [1,2]. However, the net force is still zero! To explain why there is a net positive - or upward - force on the ring, we must consider a phase difference between the magnetic field in the solenoid and the induced magnetic field in the ring. This phase differenceb results from the fact that circuits with inductive loads experience current lag, i.e. the induced current in the ring flows slightly behind the current in the solenoid [1-3,6].

Theory

Essentially, this project used the expansion of the well known force relation

F=IL×B

as derived by Ladera and Donoso [4] and modified for the geometry of our set up, which is

Using the 10-turn axial magnetic field probe we mapped the profile of the axial component, b_z, of the generated magnetic field produced by the solenoid-core. Using a hall probe and a Gauss meter we mapped the profile of the radial component, b_r. The profile of the phase difference was mapped by holding the ring at fixed heights above the solenoid and measuring the peak-to-peak difference between the induced current in a "split ring" probe held at the top of the solenoid and the induced current in long pick-up coils. The height dependent mutual inductance of the ring-coil system was calculated by Ladera and Donoso [4] and verified experimentally. The height dependent self-inductance was calculated based on the relationship tan(φ)=ω*l(z)*L0/R.

Left: the profile of the instantaneous force exerted on the ring as a function of the ring's position by the solenoid-core. This curve was generated using spline data in MATLAB.

Right: Predicted motion (solid lines) of the ring for 4,6, and 7 cycles at 13.03 applied amps (80 V) compared to video data (dots). The large dots represent when power was turned off; to the left of the dots represent driven motion and to the right represents ballistic motion. Predictions for the ballistic motion were calculated based on predicted position and instantaneous velocity.

Video data was collected to compare our predicted motion against real motion. Above are some samples of the jumping motion of the ring when 18.8 A is applied through the solenoid coil for 2, 4, and 6 power cycles respectively. The videos were then analyzed using Tracker, a free video analysis software package. Click on the images to view the motion.

Further Investigation

One major complication we encountered was our inability to sync our pulses with the power cycles coming from the AC main. Future groups could program a FPGA to connect the coil to the power main at exactly the beginning of a cycle. We were also limited to about 7 power cycles, which a FPGA could extend to many more cycles.

The weight of our aluminum rings were heavy compared to other referenced literature; future groups could use light rings, or rings of different materials.

Finally, one of the big unknowns in our project was the relative magnetic permeability of the steel core. Future groups could use the apparatus to precisely measure this quantity.

Acknowledgements

We would like to thank Kurt Wick and Kevin Booth for their expertise, wisdom, and labor they have given us throughout the semester. We would also like to thank Bill Gilbert for building, and then rebuilding our apparatus. Thanks to Jochen Mueller and Clem Pryke, and the rest of the MXP2 class for their valuable feedback.

References

1. R. N. Jeffery and F. Amiri, “The Phase Shift in the Jumping Ring,” Phys. Teach.46 350-357 (2008).

2. J.Hall, “Forces on the Jumping Ring,” Phys. Teach. 35, 80-82 (1997).

3. E. J. Churchill and J. D. Noble, “A Demonstration of Lenz’ Law?,” Am. J. Phys. 39 285-287 (1971).

4. C. L. Ladera and G. Donoso, “Unveiling the Physics of the Thompson Jumping Ring,” Am. J. Phys. 83 341-347 (2015).

5. Paul J. H. Tjossem and Victor Cornejo, “Measurements and mechanisms of Thomson’s jumping ring,” Am. J. Phys. 68, 238–244 (2000).

6. P. Tanner, J. Loebach, J. Cook, and H. D. Hallen, “A pulsed jumping ring apparatus for demonstration of Lenz’s law,” Am. J. Phys. 69, 911–916 (2001).