Magnetron

Modeling the Magnetron Toy
Chris DuBois May 9, 2005

Abstract
Many real-world phenomena are often too complex for us to capture entirely using mathematical tools, but a careful approach enables an exploration of that complexity and may provide insight into the phenomena. The author found a simple desk toy, the “Magnetron,” to be a perfect target for exploration: while simply designed andconstructed, one quickly realizes the system has very complex dynamics. The toy features two rotors, spaced slightly apart, with magnets mounted at the end of each arm; manually spinning one rotor causes the other rotor to spin due to the interactions between the magnetic fields. In this paper we present a mathematical model that successfully reproduces several of the toy’s behaviors. In the future wehope to explore the transitions between various equilibria of the system.

The Magnetron
The Magnetron toy has two plastic rotors mounted on a wooden block by metal spindles that serve as axles. Each rotor has three evenly spaced arms. At the end of each arm, there is a small bar magnet mounted inside a plastic housing with its North pole oriented away from the rotor center. The two rotors arespaced such that they nearly touch when they are closest to each other. Since dipoles are subject to forces due to external magnetic fields, the fields produced by the magnets in one rotor affect the rotation of the opposite rotor and vice versa.

Introduction
There are several interesting aspects of this physical system that become apparent after a week-long obsession with this toy. First, thesystem has a stable equilibrium when two arms are pointing towards the middle but not directly at each other. This can be explained by the magnets wanting to align themselves in the other magnet’s field. The system has two unstable equilibria: when the rotor arms are as far away from each other as possible and when two arms are positioned directly towards each other. Additionally, there is a steadystate when one rotor is moving fast and the other is such that one arm points directly away from the system center. Aside from equilibria, the toy shows some cool behavior such as “momentum transfer” - where one rotor begins with high angular velocity but suddenly stops while the second rotor begins fast rotation from a standstill. It is quite fascinating to watch how the two transfer energy toeach other so efficiently without physically touching. Another neat effect is “velocity matching,” where the two rotors are spinning in opposite directions such that they keep a nearly constant velocity. Also, there is a bouncing effect that is important for understanding the system since it often happens when the system is winding down to equilibrium. To the author’s knowledge, there is only one otherattempt at modeling this particular system [1]. Whereas this paper also models the magnets as point dipoles, I use the system’s potential energy to find the equations of motion instead of calculating the individual forces present. This model improves on previous models by incorporating a frictional force (and by successfully modeling the dynamics of the toy!).

1

Assumptions of The Model
2Dimensional
Because of the way the two rotors are mounted, rotational motion is restricted a single plane parallel to the base plate. Also, the magnetic fields in the vertical direction are symmetric with respect to this plane, so they do not play an important role in the dynamics of the two rotors. Both of these considerations make a 2D model reasonable.

Energy
We consider the total energy ofthe system to be the sum of the kinetic and potential energy. Using fundamental Newtonian principles, we calculate the rotational kinetic energy from the inertia and angular velocity of each rotor. Using equations from electromagnetic theory, we can calculate the potential energy present from the interacting magnetic fields of the dipoles. Approaching the state of the system in terms of total...