Earnshaw's theorem forbids magnetic levitation in many common situations.
Informally, the case of a point charge in an arbitrary static electric field is a simple consequence of Gauss's law.
However, Gauss's law says that the divergence of any possible electric force field is zero in free space.
In mathematical notation, an electrical force F(r) deriving from a potential U(r) will always be divergenceless (satisfy Laplace's equation):
Therefore, there are no local minima or maxima of the field potential in free space, only saddle points.
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).
It is also possible to prove this theorem directly from the force/energy equations for static magnetic dipoles (below).
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.
Earnshaw's theorem has even been proven for the general case of extended bodies, and this is so even if they are flexible and conducting, provided they are not diamagnetic,[2][3] as diamagnetism constitutes a (small) repulsive force, but no attraction.
But it is sometimes more natural to work in a rotating reference frame that contains a fictitious centrifugal force that violates the assumptions of Earnshaw's theorem.
For example, in the restricted three-body problem, the effective potential from the fictitious centrifugal force allows the Lagrange points L4 and L5 to lie at local maxima of the effective potential field even if there is only negligible mass at those locations.
(Even though these Lagrange points lie at local maxima of the potential field rather than local minima, they are still absolutely stable in a certain parameter regime due to the fictitious velocity-dependent Coriolis force, which is not captured by the scalar potential field.)
For quite some time, Earnshaw's theorem posed a startling question of why matter is stable and holds together, since much evidence was found that matter was held together electromagnetically despite the proven instability of static charge configurations.
Since Earnshaw's theorem only applies to stationary charges, there were attempts to explain stability of atoms using planetary models, such as Nagaoka's Saturnian model (1904) and Rutherford's planetary model (1911), where the point electrons are circling a positive point charge in the center.
Yet, the stability of such planetary models was immediately questioned: electrons have nonzero acceleration when moving along a circle, and hence they would radiate the energy via a non-stationary electromagnetic field.
Bohr's model of 1913 formally prohibited this radiation without giving an explanation for its absence.
Eventually this led the way to Schrödinger's model of 1926, where the existence of non-radiative states in which the electron is not a point but rather a distributed charge density resolves the above conundrum at a fundamental level: not only there was no contradiction to Earnshaw's theorem, but also the resulting charge density and the current density are stationary, and so is the corresponding electromagnetic field, no longer radiating the energy to infinity.
At a more practical level, it can be said that the Pauli exclusion principle and the existence of discrete electron orbitals are responsible for making bulk matter rigid.
The first case is a magnetic dipole of constant magnitude that has a fast (fixed) orientation.
In paramagnetic and diamagnetic materials the dipoles are aligned parallel and antiparallel to the field lines, respectively.
For a magnetic dipole of fixed orientation (and constant magnitude) the energy will be given by
Magnetic dipoles aligned parallel or antiparallel to an external field with the magnitude of the dipole proportional to the external field will correspond to paramagnetic and diamagnetic materials respectively.
but this is just the square root of the energy for the paramagnetic and diamagnetic case discussed above and, since the square root function is monotonically increasing, any minimum or maximum in the paramagnetic and diamagnetic case will be a minimum or maximum here as well.
but the Laplacians of the individual components of a magnetic field are zero in free space (not counting electromagnetic radiation) so
As discussed above, this means that the Laplacian of the energy of a paramagnetic material can never be positive (no stable levitation) and the Laplacian of the energy of a diamagnetic material can never be negative (no instability in all directions).
See Maxwell's equations for a more detailed discussion of these properties of magnetic fields.
But since Bx is continuous, the order of differentiation doesn't matter giving
However, Earnshaw's theorem does not necessarily apply to moving ferromagnets,[4] certain electromagnetic systems, pseudo-levitation and diamagnetic materials.
Spin-stabilized magnetic levitation: Spinning ferromagnets (such as the Levitron) can, while spinning, magnetically levitate using only permanent ferromagnets, the system adding gyroscopic forces.
Pseudo-levitation constrains the movement of the magnets usually using some form of a tether or wall.