Magnetic Potential Energy of a Linear Magnetic Track Gun

Introduction

There are a number of youtube video clips which show a type of magnetic ‘gun’ in which a slider permanent magnet is constrained to move along a straight track which has a series of close packed permanent bar magnets placed either side of the track in a fixed ‘v’ arrangement (see “Magnetic rail gun science project” and “magnetic track energy test”) :-

The slider is placed at the breach of the gun (at x=0) and released. It fires along the track and emerges with considerable kinetic energy.

This phenomena has lead to a number of speculative comments such as:

1) If the track were made longer by adding more magnets the slider could be given even more kinetic energy.

2) If a series of such tracks were placed in a line we could accelerate the slider to arbitrarily high speeds.

3) If a large enough circle of these straight segments were made we could get the slider to go around for ever, perhaps accelerating until the machine blows itself apart.

4) That this device maybe used as a source of energy generation giving more energy than is put in (a perpetual motion device or free energy device).

All of these conclusions are in error, but the rather mysterious nature of magnetic interactions makes them seem plausible at first sight. If we could calculate (and also measure) the potential of the slider at each point along the track we could predict how it will move and understand the errors in the speculations and understand the limitations of the device. This calculation is presented below.

Calculation of Potential energy of magnetic slider:

For a magnetic slider oriented as shown in the figure above, the energy at each position on the track, UN(x), will depend on the orientation of the fixed magnets, they are arranged in pairs with mirror symmetry about the track line); the spacing of the magnets, L; the distance from the track

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centre, K; the number of magnet pairs, N; the strength of the individual magnets in the array, m2; and the strength of the slider magnet, m1.

To a first approximation the interaction energy of the slider with the array can be calculated as if the magnetic forces were due to the sum of magnetic dipole interactions between the slider and each of the individual fixed magnets. We ignore the mutual interactions between the fixed magnets in the array of permanent magnets as these do not change as the slider passes. Consider just two dipoles:

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The energy of interaction of the slider dipole with all the other dipoles can now be found:

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Results of calculation:

For typical values of the dimensions of the magnetic tracks we get K=2 and L=1. The potential energy curve calculated for a track with 25 pairs of magnets with the repulsive orientation of the slider is shown below. This situation corresponds to that studied in the video “magnetic energy track test”.

Imagine the slider approaching from the left along a frictionless track. At first it experiences a small attraction as its potential energy decreases (and the kinetic energy will increase). But it rapidly experiences a strong repulsive force at the breach of the gun. The slider will not go into the barrel unless work equal to 0.1c is done to push the slider to x=0. (i.e. energy has to be given to the system by an outside agency). If the slider is released from x=0 it will rapidly accelerate up the barrel with nearly all the work done being converted to kinetic energy by position x=10. It will travel with almost constant velocity from x=10 to x=18 but it will then begin a small deceleration due to the small increase in PE around x=21. The slider then experiences a large accelerating force, which increases the kinetic energy to almost twice the work done loading the gun. The maximum kinetic energy occurs at about x=25 near the muzzle of the gun. The slider then experiences rapid deceleration as its PE increases again. By the time it moves beyond x=35 and so escapes the attraction of the magnets the kinetic energy is just equal to the work done loading the gun i.e 0.1c. If the track is not frictionless, then the kinetic energy of the escaped slider will be less than the energy given to the system loading the gun. If the friction is large enough the slider may even be trapped by the magnets and oscillate back and forth about a position at the muzzle end of the gun loosing energy to friction and coming to rest at the position corresponding to the bottom of the potential well. If we initially place a non magnetic ball with the same mass as the slider at this position in the track, the slider will collide with it, come to a stop as it transfers all its kinetic energy to the ball, and the ball will emerge from the gun with twice the kinetic energy the slider would have had after it escapes the magnetic attraction i.e. it looks like we got more energy out than we put in. However, in order to restore the gun to the initial

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state, with the track clear, we must do work to pull the magnetic slider free. This work is equal to the extra kinetic energy we got out by using the non magnetic ball. We have not gained any “free” energy.

What if we change the orientation of the slider?

The potential energy curve is just that obtained by reflecting the curve we have just examined about the x axis. It then looks like the figure below:

If the slider approaches from the left it experiences a small repulsive force at first and energy equal to the height of the small PE hill at about x=-3 must be supplied for the slider to enter the array. It then experiences a strong attractive force pulling it into the mouth of the gun and increasing its kinetic energy; but as it enters, the attraction rapidly changes to repulsion and the slider decelerates so that by x=10 the kinetic energy of the slider is equal to the small difference PE(-3) – PE(10). It will gradually accelerate as the PE drops to a local minimum at x=21. Beyond that the slider rapidly decelerates and is stopped by the PE barrier. (The slider will not escape from the gun in this orientation unless it was originally given a kinetic energy greater than the height of the barrier (0.1c here) by an outside agent). On a frictionless track the slider would return to the left and emerge with the kinetic energy equal to the energy initially supplied by the external agent if it were less than 0.1c. If there is friction the slider may be trapped at the muzzle end of the gun at a position corresponding to the small potential well there at x=21.

If the slider approaches from the right, in this orientation, then work has to be done to place the slider at the mouth of the array. This work is converted into kinetic energy of the slider emerging from the gun. This is the situation in the video “magnetic rail gun science project”.

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Dependence of the PE on the Number of magnets, their spacing and the track width

The interaction energy UN(x) depends on the parameters K,L,N which define the separation of the fixed magnets either side of the track, the closeness of packing of the magnets along the track and the number of fixed pairs of magnets . The appendix shows graphs of the calculated values of UN(x)/c for various K,L,N .

Since the kinetic energy of the slider fired by the ‘gun’ is all produced by the work done placing it at x=o we would like to know how this work depends on the number of magnets, and the dimensions of the fixed array. This is the same as asking how the barrier height in the UN(x)/c curves depends on these factors. For the track geometry corresponding to K=2 and L=1 we find:

The barrier height does not change by very much on extending the track by adding magnet pairs beyond the first 4 pairs! This is because the magnetic forces between magnets fall off rapidly with distance so adding more magnets has little effect beyond a certain limit. The same result is obtained for other values of K and L provided the ratio K/L < ~4 . For larger values of this ratio we typically see that the barrier does increase as we increase the number of magnets but again it eventually becomes constant for a given K and L (=U max/c). In the examples below (with K/L=8) this insensitivity to number occurs by the time 16 magnet pairs are present. Increasing the length of the track beyond this would not increase the kinetic energy of the projectile.

In order to see a difference in barrier height for all lengths of track up to 25 magnet pairs the ratio K/L has to be 16 or larger (see appendix) which is much larger than encountered in practical set ups where the ratio is typically in the range 2< K/L <6.

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Umax/c is the maximum kinetic energy a magnetic projectile can have from a track characterised by K and L. The actual maximum barrier height (and so the maximum kinetic energy of the projectile) increases as K decreases (as we bring the fixed magnets closer to the centre of the track) and is proportional to 1/K2. As L decreases (as we squeeze the magnets closer together along the track) the maximum barrier height increases in a manner proportional to 1/L:-

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So Maximum kinetic energy of magnetic projectile from gun = 0.41c/L.K2

KE ~ 0.41 2 1/2 o m1m23 = 1.7392x 10-7 m1m2 (Joules) (4)L K2 L K2

For a track with K (cm) and L (cm) and slider mass 0.2Kg the velocity,V, of the projectile would be

V= Sqrt (1.7392 m1m2 /L K2).

Ordinary bar magnets have magnetic moments, m, between 0.5 and 10 (Amp m2) so for K=2 L=1 cm and m1=m2=10 V=6.6 m/s = 24km/hr. For exceptionally strong neo magnets m1=m2 = 50 so velocity of the slider V= 120 km/hr. (i.e a very low velocity gun!)

Discussion of misconceptions

The behaviour of experimental sliders in the magnetic tracks observed in the videos are in accord with the expected motion predicted on the basis of the results of the calculation of the potential energy curves using the dipole approximation to the magnetic fields.

We can give definitive answers to the 4 speculations with which we began:

1) Adding more magnets to the track does not increase the kinetic energy indefinitely. For a given track geometry (specific K and L values) there is a maximum kinetic energy achievable equal to the work done placing the slider at x=0 in the repulsive orientation. This will probably occur for about 10 magnet pairs. Extending the track beyond this will not add to the kinetic energy that can be produced.

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2) Suppose we have a series of identical tracks placed in line separated by free space. Work would be done loading the first and this would be converted to kinetic energy of the slider. If there were no friction losses and no energy was removed from the system the slider would have just enough kinetic energy to overcome the barrier to entering the second track. Its kinetic energy would be converted to potential energy at the track entrance and then back to kinetic energy as it escapes. No additional energy is obtained so the slider does not speed up indefinitely. If the track were infinite and completely frictionless the slider could continue forever. But no additional energy to that originally given to the system has been produced. In practise friction losses cannot be completely removed and so the slider will come to rest when its KE is smaller than the entrance PE barrier of an array. Any attempt to harness the energy of the slider will cause it to stop at the next array it comes to.

3) Same problem as 2.4) Same problem as 2!

Appendix

Here are potential energy UN(x)/c curves for various K,L,N ( c= 21/2A = 21/2o m1m23/(4) ).Each plot has N= 4, 8, 16, 21, 25 magnet pairs in the fixed array. The following curves are plotted:

L=1 K= 1.0, 1.5, 1.75, 2.0, 2.5, 3.0, 4.0, 6.0, 8.0, 12.0

L=0.5 K= 0.5, 0.8, 1.0, 1.5, 2.0, 12.0

L=0.25 K= 0.5, 2.0, 4.0

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