Earnshaw theorem

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

Explanation

A point at which a test specimen should assume a stable equilibrium position must be a minimum of the potential . If the specimen is moved away from this minimum, this costs work . A driving force towards the minimum acts clearly on the test specimen .

The statement of the theorem can be deduced directly from the Maxwell equations . In the source-free space , the divergence is equal to 0 for magnetic and electric fields, as well as for the gravitational field and other fields. With divergence disappearing everywhere, there are saddle points at best . There is therefore at least one direction in which the test specimen does not experience any restoring force. Even with an arbitrarily small deflection in this direction, the specimen will no longer return to the saddle point. ${\displaystyle {\tfrac {1}{r^{2}}}}$

An interesting application of the theorem is the proof of the impossibility of creating stable floating constructions using only permanent magnets . For so-called magnetic levitation one therefore needs either actively regulated, dynamic fields or, as Werner Braunbek had shown in 1939, diamagnetic materials.

proof

The theorem can be shown with the help of multi-dimensional function analysis. Let the electrical potential be. A necessary condition for an extremum in the point is that is. Another necessary condition for an extremum is that the Hessian matrix is not indefinite in the point . Furthermore, it is required that not all are eigenvalues , otherwise there is a saddle point. In addition, there should be no charge in an epsilon environment of the extremum, because it is a matter of a stable equilibrium achieved solely by electrostatic fields . ${\displaystyle \varphi (\mathbf {r} )}$ ${\displaystyle \mathbf {r_{0}} }$ ${\displaystyle \nabla \varphi (\mathbf {r} )|_{\mathbf {r_{0}} }=0}$ ${\displaystyle H}$ ${\displaystyle \mathbf {r_{0}} }$ ${\displaystyle E_{i}=0}$

${\displaystyle H(\varphi (r))|_{\mathbf {r_{0}} }=\left.{\begin{pmatrix}\partial _{x}^{2}\varphi &\partial _{x}\partial _{y}\varphi &\partial _{x}\partial _{z}\varphi \\\partial _{x}\partial _{y}\varphi &\partial _{y}^{2}\varphi &\partial _{z}\partial _{y}\varphi \\\partial _{x}\partial _{z}\varphi &\partial _{y}\partial _{z}\varphi &\partial _{z}^{2}\varphi \end{pmatrix}}\right|_{\mathbf {r_{0}} }}$

Using linear algebra and Maxwell's equations, with no charge it follows from that ${\displaystyle H(\varphi (r))|_{\mathbf {r_{0}} }=0}$

${\displaystyle \sum _{i}E_{i}={\text{Spur}}(H|_{\mathbf {r_{0}} })=\Delta \varphi |_{\mathbf {r_{0}} }=0.}$

From this it follows that if not all eigenvalues ​​are equal to zero, the Hessian matrix is ​​indefinite and therefore no extremum can exist.

example

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

${\displaystyle \Delta \varphi ={\vec {\nabla }}\cdot {\vec {E}}=0}$

A simple example of a hypothetical potential that would be attractive in all three spatial directions ( , and ) is: ${\displaystyle \varphi }$ ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

${\displaystyle \varphi =ax^{2}+by^{2}+cz^{2}}$

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

${\displaystyle a+b+c=0}$

In order for this equation to be fulfilled, at least one of the three constants must be less than zero . This means that the potential must be repulsive in at least one of the three spatial directions. However, this contradicts the assumption that there is a potential that is attractive in all three spatial directions.

Practical meaning

In experimental physics, structures are needed that can catch particles. Due to the Earnshaw theorem, more complex methods than static fields must be used.

Ions can e.g. B. be caught in an ion trap using alternating electric fields . An example of this is the Paul trap . In this, a restoring force acts on ions (but also on electrically neutral particles such as neutral atoms or neutrons ) due to ponderomotive forces with small deflections.

literature

• Samuel Earnshaw: On the nature of the molecular forces which regulate the constitution of the luminiferous ether . In: Transactions of the Cambridge Philosophical Society . tape 7 , 1842, ZDB -ID 208399-1 , p. 97-112 .