Earnshaw's theorem

Earnshaw's theorem shows that simultaneous focusing in two directions at once is impossible.

Even without those losses, Earnshaw's theorem means that dynamic systems of charges are unstable over long periods.

Earnshaw's theorem seems to preclude the possibility of static magnetic levitation.

Earnshaw's theorem shows that any combination of static magnets cannot be in a stable equilibrium.

Earnshaw's theorem has no exceptions for non-moving permanent ferromagnets.

Earnshaw's theorem does not apply to diamagnets.

However, Earnshaw's theorem does not apply to moving ferromagnets, certain electromagnetic systems, pseudo-levitation and diamagnetic materials.

However, the various levitation systems achieve stable levitation by violating the assumptions of Earnshaw's theorem.

Due to Earnshaw's theorem, no static arrangement of classical electrostatic fields can be used to stably levitate a point charge.

Earnshaw's theorem proved conclusively that it is not possible to levitate stably using only static, macroscopic, paramagnetic fields.

In 1939, he disproved Earnshaw's theorem by showing that there are magnetic fields in which small diamagnetic bodies can float in a stable position.

Earnshaw's theorem states that a collection of point charges cannot be maintained in an equilibrium configuration solely by the electrostatic interaction of the charges.

Earnshaw's theorem was originally formulated for electrostatics (point charges) to show that there is no stable configuration of a collection of point charges.

From Earnshaw's theorem at least one stable axis must be present for the system to levitate successfully, but the other axes can be stabilised using ferromagnetism.

If an F quadrupole and a D quadrupole are placed immediately next to each other, their fields completely cancel out (in accordance with Earnshaw's theorem).

Earnshaw's theorem assumes that the magnets are static and unchanging in field strength and that the relative permeability is constant and greater than unity everywhere.

However, Earnshaw's theorem only applies to objects with positive susceptibilities, such as ferromagnets (which have a permanent positive moment) and paramagnets (which induce a positive moment).

Earnshaw's theorem does not allow for a static configuration of permanent magnets to stably levitate another permanent magnet or materials that are paramagnetic or ferromagnetic against gravity.

Passive magnetic bearings (PMBs) use permanent magnets and, therefore, do not require any input power but are difficult to design due to the limitations described by Earnshaw's theorem.

An example of this - that has bypassed the Earnshaw's theorem issues - is the homopolar electrodynamic bearings (EDB) invented by Dr Torbjörn Lembke.

Charged particles cannot be trapped in 3D just by electrostatic forces because of Earnshaw's theorem, since Laplace's equation for electrostatics does not allow confining potentials in all three orthogonal directions.

Emile Bachelet applied Earnshaw's theorem and the Braunbeck extension and stabilized magnetic force by controlling current intensity and turning on and off power to the electromagnets at desired frequencies.

Earnshaw's theorem explains why a system of electrons is not stable and was invoked by Niels Bohr in his atom model of 1913 when criticizing J. J. Thomson's atom.

Earnshaw's theorem applies to classical inverse-square law forces (electric and gravitational) and also to the magnetic forces of permanent magnets, if the magnets are hard (the magnets do not vary in strength with external fields).

Earnshaw's theorem proves that using only ferromagnetic or paramagnetic materials it is impossible to stably levitate against gravity, but servomechanisms, the use of diamagnetic materials, superconduction, or systems involving eddy currents permit this to occur.