Earnshaw’s theorem states that a collection of point charges cannot be maintained in a stable Earnshaw’s theorem forbids magnetic levitation in many common situations. If the materials are not hard, Braunbeck’s extension shows that. The electromagnets on the underside of the train pull it up to the ferromagnetic stators on the track and levitate the train. The magnets on the side keep the train . Safety of High Speed Magnetic Levitation Transportation Systems: German Published Date: Language: English. Filetype [PDF MB].

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As it turns out, the equations for magnetic fields and forces that I was hoping to plot in Mathematica are too complicated for Mathematica to handle. However, while most of the magnetic field is expelled from the superconductor, some of it still passes through, thus causing the superconductor to be both attracted to and repelled away from the track.

As the top coil is pulled downwards by gravity, the two fields will interact much more, increasing the repulsion force, and thus levitating the train containing the superconducting coils. One equation that I am considering using to model the behavior of a magnetically levitating object is that for the magnetic field or magnetic flux density above the supporting magnet, given by:.

Although the levitation force in the EMS system is entirely attractive and not very dependent upon the speed of the moving train, the EDS system relies upon repulsive forces and speed drives the levitation. About Help Contact Us. From now on, I will focus on finding the relationships between the speed of the train or, rather, the superconducting coils inside of itthe distance between the train and the tracks, and the magnetic lifting force and drag force that allow the EDS system to function.

For a particle to be in a stable equilibrium, small perturbations “pushes” on the particle in any direction should not break the equilibrium; the particle should “fall back” to its previous position.

One method for dealing with this invokes the fact that, in addition to the divergence, the curl of any electric field in free space is also zero in the absence of any magnetic currents.

The first, electromagnetic leviitation EMSinvolves electromagnets on the levitating object that are oriented toward the rail from below.

## Earnshaw’s theorem

The second method, called electrodynamic suspension EDS levitatioh, requires that both the levitating object and the track exert a magnetic field, so that the object is levitated by the repulsive force between the two fields. This report, one in a series of planned reports on maglev safety, That is, paramagnetic materials can be unstable in all directions but not stable in all directions and diamagnetic materials can be stable in all directions but not unstable in all directions.

Then, as the train moves forward some distance, a new image is created underneath the track, and the previous image begins to move away from the track, further into the ground. The challenge lies in transferring the experience of Germany to the U. These can thus seem to be exceptions, though in fact they exploit the constraints of the theorem.

As a result, due to the loss in current during bthere is a net negative current the shaded region flowing in the lower loop after the upper loop passes. A stable equilibrium of the particle cannot exist and there must be an instability in some direction.

However, the main component of this system that changes over time is the current flowing through the electromagnet, which would be extremely difficult to study analytically, and so I will focus on the other main maglev method.

It is proven here that the Laplacian of each individual component of a magnetic field is zero. Recommendations are also included for FRA consideration.

Graphs of the repulsion force and resulting position of the object with respect to the track will be necessary to analyze the relationship between the two variables. By using this site, you agree to the Terms of Use and Privacy Policy.

Protected by Akismet Blog with WordPress. These requirements were developed by a: See Maxwell’s equations for a more detailed discussion of these properties of magnetic levitaion. In order to quantify the lifting force, I attempted to use the following equation, given in Jayawant: Ideally, then, I could derive an expression for filetjpe ideal height the height that maximizes of the maglev train at any given speed.

Intuitively, though, it’s plausible that if the theorem holds for a single point charge then it would also hold for two opposite point charges connected together. Maglev trains are one application. Earnshaw’s theorem was originally formulated for electrostatics point charges to show that there is no stable configuration of a collection of point charges.

Diamagnetic materials are excepted because they exhibit only repulsion against the magnetic field, whereas the theorem requires materials that have both repulsion and attraction. In order to easily visualize this effect, I generalized the magnetic flux density by replacing it with a constant multiple times the inverse of distance cubed which is the relationship between magnetic field strength and distanceproducing the following graph in Mathematica:.

This report presents the results of a systematic review of the safety requirements selected for the German Transrapid: The German safety requirements were reviewed in terms of safety related functional areas of the following areas of the following seven maglev system elements: These requtrements have been compared and assessed to identify similarities to and differences from equivalent U. The dipole will only be stably levitated at points where the energy has a minimum.

### Earnshaw’s theorem – Wikipedia

This works because the theorem shows only that there is some direction in which there will be an instability. Alternatively, one can use the identity. Levitattion Rules and Safety. A simplified model of the interaction between the fi,etype opposing magnetic fields is given in the following vector plot. As a practical consequence, this theorem also states that there is no possible static configuration of ferromagnets that can stably levitate an object against gravity, even when the magnetic forces are stronger than the gravitational forces.

However, I took a slight deviation to begin my studies with a simpler equation to model, the equation for the magnetic force between ,evitation plates of some finite area, given by: Here are the various references that I consulted while researching the methods of magnetic levitation:.

Thus, I followed an assumption suggested by Kraftmakherand replaced this equation for the lifting force on a rectangular coil with the simplified equation for the lifting filrtype on a magnetic dipole, given by Reitz as:. This shows the need to invoke the properties of magnetic fields that the divergence of a magnetic field is always zero and the curl of a magnetic field is zero in free space.

It is usually referenced to magnetic fieldsbut was first applied to electrostatic fields.