A PROPOSED EXPLANATION OF THE PROPELLING FORCE BEHIND (AND AT THE FRONT OF) THE MAGNET-BATTERY-MAGNET IN A COIL TRAIN.

A video on YouTube  titled “World’s Simplest Electric Train” demonstrates that a typical cylindrical battery sandwiched between a pair of neodymium magnets can push itself through a copper coil. The driving force for this motion is provided by a magnetic force due to the magnetic field of the coil acting on the intrinsic current of the magnets. A qualitative discussion of the matter is given below with only passing consideration of the order of magnitude of some of the basic physical quantities of the system. A discussion of some of the related complications is also given, sufficient to demonstrate that a rigorous quantitative analysis would not be trivial.

Diagram 1 illustrates the major elements of the system oriented as if it will travel to the right. The battery and magnets are outlined in light black lines and their polarities are noted. The heavy black line marks the coil and the heavy red line marks the current flowing through the coil due to the electromotive of the battery. The green lines show the net intrinsic current flowing on the surface of the magnets. The orange lines represent the magnetic field resulting from the current flowing through the coil.

FIGURE 1

Conduction of the current from the terminals of the battery to the coil is made possible by the magnets that contact their respective ends of the battery and the coil. The current is limited by the resistance at the contact points between the magnets and the battery and between the magnets and the coil along with resistance in the coil and the battery’s internal resistance. The current in the coil generates the magnetic field like an ordinary solenoid.

The magnetic field is perpendicular to the sides of the magnets and perpendicular to the intrinsic current due to the net spin of the electrons of the niobium atoms. This intersection of magnetic field and current defines the cause of magnetic force according to the right-hand rule. This is illustrated in Diagram 2 which is a close-up of the top edge of the leading magnet. The orange and green vectors represent the same elements as before. All the magnetic field vectors in Diagram 2 are set in a vertical plane bisecting the center of the system. The blue vectors point in the direction of the magnetic force over the surface of the magnet. The right hand rule states that when the fingers of the right hand are pointed in the direction of the magnetic field (downward at the magnet’s top surface), and the thumb coincides with the direction of the current (out of the page at the top of the magnet) then the palm of the right hand faces the same direction as the magnetic force.

FIGURE 2

This interaction occurs around the entire circumference of both magnets giving rise to a “sheath” of force on both magnets. The direction of force at all points on the sides (not ends) of the magnets is forward while the direction of the current and intersecting magnetic field rotate as one travels around the circumference of the magnets.

The energy for this motion against friction between the coil and the “train” is clearly supplied by the battery. The rate of mechanical energy consumption would be about 0.5 Watts given about 5 Newtons of  friction force and a velocity of 10 cm/s. This would correspond to less than 0.5 amps of current from a 1.5V battery. The current may be more to account for energy emitted by electromagnetic radiation and heating of the coil by resistive heating.

One serious problem with this explanation is that it requires that the two magnets are aligned with opposing polarities which would cause the magnets to repel each other. The casing of the battery must have enough ferromagnetic metal to hold the two batteries to its ends. This is apparent at the start of the video as the magnets attach themselves to the battery. This attractive force must also be strong enough for the battery to hold the battery on to the leading magnet as the magnet is propelled forward by the magnetic force explained above.

Other magnetic phenomena are at play in this system. The presence of the magnetic field of the magnets within the coil has been ignored up to this point. Rare earth elements like Neodymium are known to have magnetic fields on the order of 1 Tesla, the current in the coil from the battery would have to be 1,000 to 10,000 amps to match this. A few amps is a generous guess for the coil current, considering the fraction of a Watt necessary to push the train through the coil. This means the actual magnetic field as shown in Diagram 1 by no means represents the net field as a superposition of the coil’s field and that from the magnets. But, the field at the surface of the magnets is as described earlier since the field of the magnets is zero at that surface. So the proposed forward force remains possible.

So this begs the question, what about the force between the magnetic field from the magnets on the current in the coil? A proper knee-jerk reaction to this question is an appeal to Newton’s First Law: Every action has an equal and opposite reaction. The field from the coil pushes the magnets forward, so the fields from the magnets must push the coil backwards. It’s left as an exercise for the reader to verify that this is the case with respect to direction of force. The intrinsic current of the magnets must be 1000’s of times greater than the current of the coil for the magnitudes of the first law force pairs to be equal in magnitude since the coil’s magnetic field is 1000’s of times weaker than the magnetic field of the magnets.

Magnetic induction is yet another complication to this picture of the system. Application of Faraday’s Law shows that as the train approaches, a current in the coil immediately ahead of the leading magnet is induced in the same direction as the current from the battery. A current in the opposite direction is induced behind the leading magnet (enhanced by induction from the magnetic field from the coil ahead of the battery). The net effect would be a buildup of (positive) charge in the coil in the neighborhood of the middle of the leading magnet and that buildup would travel down the coil like a wave with the train. This buildup of charge would oppose the effect of induction, and the buildup of charge would grow to a point where the resulting electromotive force reaches an equilibrium with induction. A similar but reversed effect would occur with the trailing magnet. Instead of a buildup of charge in the coil in the neighborhood of the middle of the magnet, there would be a scarcity of positive charge.