Editing Earnshaw's theorem (section)

==Proofs for magnetic dipoles==

===Introduction===
While a more general proof may be possible, three specific cases are considered here. The first case is a magnetic dipole of constant magnitude that has a fast (fixed) orientation. The second and third cases are magnetic dipoles where the orientation changes to remain aligned either parallel or antiparallel to the field lines of the external magnetic field. In paramagnetic and diamagnetic materials the dipoles are aligned parallel and antiparallel to the field lines, respectively.

===Background===
The proofs considered here are based on the following principles.

The energy U of a [[magnetic dipole]] with a [[magnetic dipole moment]] '''M''' in an external magnetic field '''B''' is given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -(M_x B_x + M_y B_y + M_z B_z).</math>

The dipole will only be stably levitated at points where the energy has a minimum. The energy can only have a minimum at points where the Laplacian of the energy is greater than zero. That is, where
<math display="block">\nabla^2 U = \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} + \frac{\partial^2 U}{\partial z^2} > 0.</math>

Finally, because both the divergence and the curl of a magnetic field are zero (in the absence of current or a changing electric field), the Laplacians of the individual components of a magnetic field are zero. That is,
<math display="block">\nabla^2 B_x = \nabla^2 B_y = \nabla^2 B_z = 0.</math>

This is proven at the very end of this article as it is central to understanding the overall proof.

===Summary of proofs===
For a magnetic dipole of fixed orientation (and constant magnitude) the energy will be given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -(M_x B_x + M_y B_y + M_z B_z),</math>
where ''M<sub>x</sub>'', ''M<sub>y</sub>'' and ''M<sub>z</sub>'' are constant. In this case the Laplacian of the energy is always zero,
<math display="block">\nabla^2 U = 0,</math>
so the dipole can have neither an energy minimum nor an energy maximum. That is, there is no point in free space where the dipole is either stable in all directions or unstable in all directions.

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. In these cases the energy will be given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -k\mathbf{B}\cdot\mathbf{B} = -k \left (B_x^2 + B_y^2 + B_z^2 \right ),</math>
where ''k'' is a constant greater than zero for paramagnetic materials and less than zero for diamagnetic materials.

In this case, it will be shown that
<math display="block">\nabla^2 \left (B_x^2 + B_y^2 + B_z^2 \right ) \geq 0,</math>
which, combined with the constant {{math|''k''}}, shows that paramagnetic materials can have energy maxima but not energy minima and diamagnetic materials can have energy minima but not energy maxima. 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. Of course, both materials can have saddle points.

Finally, the magnetic dipole of a ferromagnetic material (a permanent magnet) that is aligned parallel or antiparallel to a magnetic field will be given by
<math display="block">\mathbf{M} = k{\mathbf{B} \over |\mathbf{B}|},</math>

so the energy will be given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -k\frac{\mathbf{B}\cdot\mathbf{B}}{ |\mathbf{B}|} = -k\frac{|\mathbf{B}|^2 }{ |\mathbf{B}|} = -k\left (B_x^2 + B_y^2 + B_z^2 \right )^{\frac{1}{2}};</math>

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. There are, however, no known configurations of permanent magnets that stably levitate so there may be other reasons not discussed here why it is not possible to maintain permanent magnets in orientations antiparallel to magnetic fields (at least not without rotation—see [[spin-stabilized magnetic levitation]].

===Detailed proofs===
Earnshaw's theorem was originally formulated for electrostatics (point charges) to show that there is no stable configuration of a collection of point charges. The proofs presented here for individual dipoles should be generalizable to collections of magnetic dipoles because they are formulated in terms of energy, which is additive. A rigorous treatment of this topic is, however, currently beyond the scope of this article.

===Fixed-orientation magnetic dipole===
It will be proven that at all points in free space
<math display="block"> \nabla \cdot (\nabla U) = \nabla^2 U = {\partial^2 U \over {\partial x}^2} + {\partial^2 U \over {\partial y}^2} + {\partial^2 U \over {\partial z}^2} = 0.</math>

The energy ''U'' of the magnetic dipole '''M''' in the external magnetic field '''B''' is given by
<math display="block">U = -\mathbf{M}\cdot\mathbf{B} = -M_x B_x - M_y B_y - M_z B_z.</math>

The Laplacian will be
<math display="block">\nabla^2 U = -\frac{\partial^2}{{\partial x}^2} \left(M_x B_x + M_y B_y + M_z B_z\right) - \frac{\partial^2}{{\partial y}^2} \left(M_x B_x + M_y B_y + M_z B_z\right) - \frac{\partial^2}{{\partial z}^2} \left(M_x B_x + M_y B_y + M_z B_z\right)</math>

Expanding and rearranging the terms (and noting that the dipole '''M''' is constant) we have
<math display="block">\begin{align}
  \nabla^2 U &= -M_x\left({\partial^2 B_x \over {\partial x}^2} + {\partial^2 B_x \over {\partial y}^2} + {\partial^2 B_x \over {\partial z}^2}\right) - M_y\left({\partial^2 B_y \over {\partial x}^2} + {\partial^2 B_y \over {\partial y}^2} + {\partial^2 B_y \over {\partial z}^2}\right) - M_z\left({\partial^2 B_z \over {\partial x}^2} +{\partial^2 B_z \over {\partial y}^2} +{\partial^2 B_z \over {\partial z}^2}\right)\\[3pt]
             &= -M_x \nabla^2 B_x - M_y \nabla^2 B_y - M_z \nabla^2 B_z
\end{align}</math>

but the Laplacians of the individual components of a magnetic field are zero in free space (not counting electromagnetic radiation) so
<math display="block">\nabla^2 U = -M_x 0 - M_y 0 - M_z 0 = 0,</math>

which completes the proof.

===Magnetic dipole aligned with external field lines===
The case of a paramagnetic or diamagnetic dipole is considered first. The energy is given by
<math display="block">U = -k|\mathbf{B}|^2 = -k \left (B_x^2 + B_y^2 + B_z^2 \right ).</math>

Expanding and rearranging terms,
<math display="block">\begin{align}
\nabla^2 |\mathbf{B}|^2 &= \nabla^2 \left (B_x^2 + B_y^2 + B_z^2 \right ) \\
                        &= 2\left( |\nabla B_x|^2 + |\nabla B_y|^2 + |\nabla B_z|^2 +B_x\nabla^2 B_x + B_y\nabla^2 B_y + B_z\nabla^2 B_z \right)
\end{align}</math>

but since the Laplacian of each individual component of the magnetic field is zero,
<math display="block">\nabla^2 |\mathbf{B}|^2 = 2\left( | \nabla B_x |^2 + | \nabla B_y |^2 + | \nabla B_z |^2 \right);</math>

and since the square of a magnitude is always positive,
<math display="block">\nabla^2 |\mathbf{B}|^2 \geq 0.</math>

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).

Further, because the energy for a dipole of fixed magnitude aligned with the external field will be the square root of the energy above, the same analysis applies.

===Laplacian of individual components of a magnetic field===
It is proven here that the Laplacian of each individual component of a magnetic field is zero. This shows the need to invoke the properties of magnetic fields that the [[divergence]] of a magnetic field is always zero and the [[curl (mathematics)|curl]] of a magnetic field is zero in free space. (That is, in the absence of current or a changing electric field.) See [[Maxwell's equations]] for a more detailed discussion of these properties of magnetic fields.

Consider the Laplacian of the x component of the magnetic field
<math display="block">\begin{align}
\nabla^2 B_x &= \frac{\partial^2 B_x}{\partial x^2} + \frac{\partial^2 B_x}{\partial y^2} + \frac{\partial^2 B_x}{\partial z^2} \\
&= \frac{\partial}{\partial x} \frac{\partial B_x}{\partial x}  + \frac{\partial}{\partial y} \frac{\partial B_x}{\partial y} + \frac{\partial}{\partial z} \frac{\partial B_x}{\partial z}
\end{align}</math>

Because the curl of '''B''' is zero,
<math display="block">\frac{\partial B_x}{\partial y} = \frac{\partial B_y}{\partial x},</math>
and
<math display="block">\frac{\partial B_x}{\partial z} = \frac{\partial B_z}{\partial x},</math>
so we have
<math display="block">\nabla^2 B_x = \frac{\partial}{\partial x} \frac{\partial B_x}{\partial x} + \frac{\partial}{\partial y}\frac{\partial B_y}{\partial x} + \frac{\partial}{\partial z}\frac{\partial B_z}{\partial x}.</math>

But since ''B<sub>x</sub>'' is continuous, the order of differentiation doesn't matter giving
<math display="block">\nabla^2 B_x = {\partial \over \partial x}\left({\partial B_x \over \partial x} +{\partial B_y \over \partial y} +{\partial B_z \over \partial z} \right) = {\partial \over \partial x}(\nabla \cdot \mathbf{B}).</math>

The divergence of '''B''' is zero,
<math display="block">\nabla \cdot \mathbf{B} = 0,</math>
so
<math display="block">\nabla^2 B_x = {\partial \over \partial x}(\nabla \cdot \mathbf{B}) = 0.</math>

The Laplacian of the ''y'' component of the magnetic field ''B<sub>y</sub>'' field and the Laplacian of the ''z'' component of the magnetic field ''B<sub>z</sub>'' can be calculated analogously.  Alternatively, one can use the [[vector calculus identities|identity]]
<math display="block">\nabla^{2}\mathbf{B} = \nabla \left(\nabla \cdot \mathbf{B}\right) - \nabla \times \left( \nabla \times \mathbf{B} \right),</math>
where both terms in the parentheses vanish.

=== Loopholes ===
Earnshaw's theorem has no exceptions for non-moving permanent [[ferromagnetism|ferromagnets]].  However, Earnshaw's theorem does not necessarily apply to moving ferromagnets,<ref name=":0">{{cite journal |last1=Simon |first1=Martin D. |last2=Heflinger |first2=Lee O. |last3=Ridgway |first3=S.L. |date=1996 |title=Spin stabilized magnetic levitation |journal=American Journal of Physics |volume=65 |issue=4 |pages=286–292 |doi=10.1119/1.18488}}</ref> certain electromagnetic systems, pseudo-levitation and diamagnetic materials. These can thus seem to be exceptions, though in fact they exploit the constraints of the theorem.

[[Spin-stabilized magnetic levitation]]: Spinning ferromagnets (such as the [[Levitron]]) can, while spinning, magnetically levitate using only permanent ferromagnets, the system adding gyroscopic forces.<ref name=":0" /> (The spinning ferromagnet is not a "non-moving ferromagnet"). 

Switching the polarity of an electromagnet or system of electromagnets can levitate a system by continuous expenditure of energy. [[Maglev (transport)|Maglev trains]] are one application.

[[Pseudo-levitation]] constrains the movement of the magnets usually using some form of a tether or wall.  This works because the theorem shows only that there is some direction in which there will be an instability.  Limiting movement in that direction allows levitation with fewer than the full 3 dimensions available for movement (note that the theorem is proven for 3 dimensions, not 1D or 2D).

[[Diamagnetism|Diamagnetic]] materials are excepted because they exhibit only repulsion against the magnetic field, whereas the theorem requires materials that have both repulsion and attraction.  An example of this is the famous [[Magnetic levitation#Direct diamagnetic levitation|levitating frog]] (see [[Diamagnetism]]).