Earnshaw's theorem
Encyclopedia
Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium
configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw
in 1842. It is usually referenced to magnetic fields, but originally applied to electrostatic fields. It applies to the classical inverse-square law
force
s (electric
and gravitational
) and also to the magnetic
forces of permanent magnets and paramagnetic materials or any combination, (but not diamagnetic materials).
. For a particle to be in a stable equilibrium, small perturbations ("pushes") on the particle in any direction should not break the equilibrium; the particle should "fall back" to its previous position. This means that the force field lines around the particle's equilibrium position should all point inwards, towards that position. If all of the surrounding field lines point towards the equilibrium point, then the divergence
of the field at that point must be negative (i.e. that point acts as a sink). 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
deriving from a potential 
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 point
s. A stable equilibrium of the particle cannot exist and there must be an instability in at least one direction.
To be completely rigorous, strictly speaking, the existence of a stable point does not require that all neighboring force vectors point exactly toward the stable point; the force vectors could spiral in towards the stable point, for example. 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).
This theorem also states that there is no possible static configuration of ferromagnets which 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, as diamagnetism constitutes a (small) repulsive force, but no attraction.
There are, however, several exceptions to the rule's assumptions which allow magnetic levitation
.
. However, moving ferromagnets, certain electromagnetic systems, pseudo-levitation and diamagnetic materials are areas to which Earnshaw's theorem doesn't apply and thus can seem to be exceptions, though in fact these exploit the constraints of the theorem.
Spinning ferromagnets (such as the Levitron
) can—while spinning—magnetically levitate using only permanent ferromagnets. Note, since this is spinning, this is not a non-moving ferromagnet.
Switching electromagnets polarity allows for keeping a system levitating by the continuous expenditure of energy. An example of this is maglev trains
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 for levitation with fewer than the full 3 dimensions available for movement (note that the theorem is proven for 3 dimensions, not 1D or 2D).
Diamagnetic
materials are excepted because they exhibit only repulsion against the magnetic field, whereas the theorem requires materials that have both repulsion and attraction. A fun example of this is the famous levitating frog (see diamagnetism
).
These questions eventually pointed the way to quantum mechanical
explanations of the structure of the atom, and it turns out that the Pauli exclusion principle
is responsible for holding bulk matter in a rigid shape.
The energy U of a magnetic dipole
with a magnetic dipole moment M in an external magnetic field B is given by
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
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,
This is proved at the very end of this article as it is central to understanding the overall proof.
where
, 
and 
are constant. In this case the Laplacian of the energy is always zero,
so the dipole can have neither an energy minimum or 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
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
which, combined with the constant 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
so the energy will be given by
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 Levitron
).
The energy U of the magnetic dipole M in the external magnetic field B is given by
The Laplacian will be
Expanding and rearranging the terms (and noting that the dipole M is constant) we have
but the Laplacians of the individual components of a magnetic field are zero in free space (not counting electromagnetic radiation) so
which completes the proof.
Expanding and rearranging terms,
but since the Laplacian of each individual component of the magnetic field is zero,
and since the square of a magnitude is always positive,
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.
of a magnetic field is always zero and the 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
Because the curl of B is zero,
and 
so we have
But since
is continuous, the order of differentiation doesn't matter giving
The divergence of B is zero,
, so
The Laplacian of the y component of the magnetic field
field and the Laplacian of the z component of the magnetic field 
can be calculated analogously. Alternatively, one can use the identity 
, where both terms in the parentheses vanish.
Mechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw
Samuel Earnshaw
Samuel Earnshaw was an English clergyman and mathematician, noted for his contributions to theoretical physics, especially "Earnshaw's Theorem"....
in 1842. It is usually referenced to magnetic fields, but originally applied to electrostatic fields. It applies to the classical inverse-square law
Inverse-square law
In physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....
force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s (electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and gravitational
Gravitational field
The gravitational field is a model used in physics to explain the existence of gravity. In its original concept, gravity was a force between point masses...
) and also to the magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
forces of permanent magnets and paramagnetic materials or any combination, (but not diamagnetic materials).
Explanation
Informally, the case of a point charge in an arbitrary static electric field is a simple consequence of Gauss's lawGauss's law
In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss's law states that:...
. For a particle to be in a stable equilibrium, small perturbations ("pushes") on the particle in any direction should not break the equilibrium; the particle should "fall back" to its previous position. This means that the force field lines around the particle's equilibrium position should all point inwards, towards that position. If all of the surrounding field lines point towards the equilibrium point, then the divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of the field at that point must be negative (i.e. that point acts as a sink). 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
deriving from a potential will always be divergenceless (satisfy Laplace's equationLaplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
):
Therefore, there are no local minima or maxima of the field potential in free space, only saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
s. A stable equilibrium of the particle cannot exist and there must be an instability in at least one direction.
To be completely rigorous, strictly speaking, the existence of a stable point does not require that all neighboring force vectors point exactly toward the stable point; the force vectors could spiral in towards the stable point, for example. 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).
This theorem also states that there is no possible static configuration of ferromagnets which 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, as diamagnetism constitutes a (small) repulsive force, but no attraction.
There are, however, several exceptions to the rule's assumptions which allow magnetic levitation
Magnetic levitation
Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended with no support other than magnetic fields...
.
Loopholes
Earnshaw's theorem has no exceptions for unmoving permanent ferromagnetsFerromagnetism
Ferromagnetism is the basic mechanism by which certain materials form permanent magnets, or are attracted to magnets. In physics, several different types of magnetism are distinguished...
. However, moving ferromagnets, certain electromagnetic systems, pseudo-levitation and diamagnetic materials are areas to which Earnshaw's theorem doesn't apply and thus can seem to be exceptions, though in fact these exploit the constraints of the theorem.
Spinning ferromagnets (such as the Levitron
Levitron
Levitron is a brand of levitating toys and gifts in science and educational markets marketed by Creative Gifts Inc. and Fascination Toys & Gifts. The Levitron top device is a commercial toy that displays the phenomenon known as spin stabilized magnetic levitation...
) can—while spinning—magnetically levitate using only permanent ferromagnets. Note, since this is spinning, this is not a non-moving ferromagnet.
Switching electromagnets polarity allows for keeping a system levitating by the continuous expenditure of energy. An example of this is maglev trains
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 for levitation with fewer than the full 3 dimensions available for movement (note that the theorem is proven for 3 dimensions, not 1D or 2D).
Diamagnetic
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the...
materials are excepted because they exhibit only repulsion against the magnetic field, whereas the theorem requires materials that have both repulsion and attraction. A fun example of this is the famous levitating frog (see diamagnetism
Diamagnetism
Diamagnetism is the property of an object which causes it to create a magnetic field in opposition to an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the...
).
Impact on physics
Earnshaw’s theorem, in addition to the fact that configurations of classical charged particles orbiting one another are also unstable due to electromagnetic radiation, mean that even dynamic systems of charges are unstable, long term. This, for quite some time led to the puzzling question of why matter stays together as much evidence was found that matter was held together electromagnetically.These questions eventually pointed the way to quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
explanations of the structure of the atom, and it turns out that the Pauli exclusion principle
Pauli exclusion principle
The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions may occupy the same quantum state simultaneously. A more rigorous statement is that the total wave function for two identical fermions is anti-symmetric with respect to exchange of the particles...
is responsible for holding bulk matter in a rigid shape.
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
Magnetic dipole
A magnetic dipole is the limit of either a closed loop of electric current or a pair of poles as the dimensions of the source are reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not complete. In particular, a magnetic...
with a magnetic dipole moment M in an external magnetic field B is given by
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
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,
This is proved 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 bywhere
, and are constant. In this case the Laplacian of the energy is always zero,so the dipole can have neither an energy minimum or 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
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
which, combined with the constant 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
so the energy will be given by
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 Levitron
Levitron
Levitron is a brand of levitating toys and gifts in science and educational markets marketed by Creative Gifts Inc. and Fascination Toys & Gifts. The Levitron top device is a commercial toy that displays the phenomenon known as 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 magnetics dipoles because they are formulated in terms of energy which is additive. A rigorous treatment of this topic, however, is currently beyond the scope of this article.Fixed-orientation magnetic dipole
It will be proven that at all points in free spaceThe energy U of the magnetic dipole M in the external magnetic field B is given by
The Laplacian will be
Expanding and rearranging the terms (and noting that the dipole M is constant) we have
but the Laplacians of the individual components of a magnetic field are zero in free space (not counting electromagnetic radiation) so
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 byExpanding and rearranging terms,
but since the Laplacian of each individual component of the magnetic field is zero,
and since the square of a magnitude is always positive,
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 divergenceDivergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
of a magnetic field is always zero and the 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
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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
Because the curl of B is zero,
and so we haveBut since
is continuous, the order of differentiation doesn't matter givingThe divergence of B is zero,
, soThe Laplacian of the y component of the magnetic field
field and the Laplacian of the z component of the magnetic field can be calculated analogously. Alternatively, one can use the identity , where both terms in the parentheses vanish.External links
- "Is magnetic levitation possible?", a discussion of Earnshaw's theorem and its consequences for levitation, along with several ways to levitate with electromagnetic fields
- Biography and other information about Samuel EarnshawSamuel EarnshawSamuel Earnshaw was an English clergyman and mathematician, noted for his contributions to theoretical physics, especially "Earnshaw's Theorem"....