Earnshaw's theorem

The Earnshaw 's theorem is a theorem in electrodynamics. It states that there is no static magnetic or electric field can hold the objects in a stable equilibrium. It is named after Samuel Earnshaw, who proved it in 1842.

Explanation

A point at which a specimen is to assume a stable position of equilibrium, a minimum of the potential need be. If the specimen of this minimum moves away, can be arranged for work. Clearly acting on the test body a restoring force to the minimum down.

The statement of the theorem can be deduced directly from the Maxwell equations. In the source-free space for magnetic and electric field, and also for the gravitational field and other fields, the divergence equal to 0 with vanishing divergence everywhere but there are at best saddle points. Therefore there are at least a direction in which the specimen undergoes no restoring force. Also at an arbitrarily small deflection in this direction of the sample body is no longer return to the saddle point.

An interesting application of the theorem is to prove the impossibility to create stable floating structures with permanent magnets. For the so-called magnetic levitation therefore requires either active -controlled, dynamic fields or diamagnetic materials.

Evidence

The theorem can be shown by means of multi-dimensional feature analysis. Be to the electric potential. A necessary condition for an extremum in the point is that is. Another necessary condition is that the Hessian matrix at the point is not indefinite. Furthermore, it is required that not all eigenvalues ​​, otherwise there is a saddle point and that in the Epsilonumgebung the extremum no charge is to be present.

It follows that, if not all the eigenvalues ​​are equal to zero, the Hessian matrix is indefinite, and thus there may be no extremum.

Example

This example illustrates the statement of Earnshaw 's theorem. The Laplace equation or the first Maxwell equation in source-free space is:

A simple example of a hypothetical potential that would be attractive in all three spatial directions ( and ), is:

With the three constants, a, b, c > 0 (all three constants greater than zero ). Insertion into the Laplace equation yields

Thus, this equation can be satisfied, but at least one of the three constants must be less than zero. This means that the potential of at least one of the three spatial directions has to be repellent. However, this contradicts the assumption that there is a potential that is attractive in all three spatial directions.

Countermeasures

In experimental physics structures are needed to catch the particles. Due to the Earnshaw theorem elaborate methods must be used as static fields.

Ions can be trapped by the use of electric fields to an ion trap, for example. An example of this is the Paul trap. This acts on ions ( as well as to electrically neutral particles such as neutral atoms or neutrons) with ponderomotive forces at low deflections, a restoring force.