NationMaster - Encyclopedia: Earnshaw's theorem

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 Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but originally applied to electrostatic fields, and, in fact, applies to any classical inverse-square law force or combination of forces (such as magnetic, electric, and gravitational fields). This page is a candidate for speedy deletion. ... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... Electrostatics is the branch of physics that deals with the force exerted by a static (i. ... This diagram shows how the law works. ... In physics, force is an influence that may cause an object to accelerate. ... Magnetic field lines shown by iron filings In physics, a magnetic field is a solenoidal vector field in the space surrounding moving electric charges and magnetic dipoles, such as those in electric currents and magnets. ... In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... A gravitational field is a model used within physics to explain how gravity exists in the universe. ...

1 Explanation
2 Proofs for magnetic dipoles
3 References
4 External links

Explanation

This follows from Gauss's law. Intuitively, 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 any of the field lines pointed outwards, then if the particle moved along those field lines, it would leave its equilibrium, which would therefore not be stable.) Because all of the surrounding field lines point towards the equilibrium point, the divergence of the field at that point must be nonzero (in fact, negative). However, Gauss's Law says this is impossible: the force acting on an object F(x) (as a function of position) due to a combination of inverse-square law forces (forces deriving from a potential which satisfies Laplace's equation) will always be divergenceless ( ·F = 0) in free space. Therefore, there is no point in empty space where the force due to the field points inward from all directions, and a stable equilibria of particles cannot exist. There are no local minima or maxima of the field potential in free space, only saddle points. In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... Del symbol (also known as nabla; used in mathematical physics). ... The largest and the smallest element of a set are called extreme values, or extreme records. ... The largest and the smallest element of a set are called extreme values, or extreme records. ... Plot of y = x3 with a saddle-point at (0,0). ...

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. There are, however, several exceptions to the rule's assumptions which allow magnetic levitation. A ferromagnet is a piece of ferromagnetic material, in which the microscopic magnetized regions, called domains, have been aligned by an external magnetic field (e. ... Levitation is the process by which an object is suspended against gravity, in a stable position, by a force without physical contact. ... Levitating pyrolytic carbon Magnetic levitation, maglev, or magnetic suspension is a method by which an object is suspended above another object with no support other than magnetic fields. ...

Earnshaw’s theorem, in addition to the fact that configurations of classical charged particles orbiting one another are also unstable due to electromagnetic radiation, pointed the way to quantum mechanical explanations of the structure of the atom.

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 fixed (unchanging) orientation. The second and third cases are magnetic dipoles where the orientation changes to remain aligned either parallel or anti-parallel to the field lines of the external magnetic field. In paramagnetic and diamagnetic materials the dipoles are aligned parallel and anti-parallel to the field lines, respectively.

Background

The proofs considered here are based on the following principles.

The energy U of a magnetic dipole M in an external magnetic field B is given by

$U = -mathbf{M}cdotmathbf{B} = -(M_x B_x + M_y B_y + M_z B_z)$

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

$nabla^2 U = {partial^2 U over partial x^2} + {partial^2 U over partial y^2} + {partial^2 U over partial z^2} > 0$

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

$nabla^2 B_x = 0, nabla^2 B_y = 0, nabla^2 B_z = 0$

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 by

$U = -mathbf{M}cdotmathbf{B} = -(M_x B_x + M_y B_y + M_z B_z)$

where $M x$ , $M y$ and $M z$ are constant. In this case the Laplacian of the energy is always zero

$nabla^2 U = 0$

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 anti-parallel 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

$U = -mathbf{M}cdotmathbf{B} = -kmathbf{B}cdotmathbf{B} = -k (B_x^2 + B_y^2 + B_z^2)$

Where k is constant greater than zero for paramagnetic materials and less than zero for diamagnetic materials.

In this case, it will be shown that

$nabla^2 (B_x^2 + B_y^2 + B_z^2) geq 0$

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 anti-parallel to a magnetic field will be given by

$mathbf{M} = k{mathbf{B} over |mathbf{B}|}$

so the energy will be given by

$U = -mathbf{M}cdotmathbf{B} = -k{mathbf{B} over |mathbf{B}|}cdotmathbf{B} = -k{(B_x^2 + B_y^2 + B_z^2) over (B_x^2 + B_y^2 + B_z^2)^{1/2}} = -k(B_x^2 + B_y^2 + B_z^2)^{1/2}$

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.

It should be noted, however, there are 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 anti-parallel to magnetic fields (at least not without rotational motion - see Levitron). Levitron is a registered trademark of Creative Gifts, Inc. ...

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 space

$nabla cdot (nabla U) = nabla^2 U = {partial^2 U over {partial x}^2} + {partial^2 U over {partial y}^2} + {partial^2 U over {partial z}^2} = 0$

The energy U of the magnetic dipole M in the external magnetic field B is given by

$U = -mathbf{M}cdotmathbf{B} = -(M_x B_x + M_y B_y + M_z B_z)$

The Laplacian will be

$nabla^2 U = -left( {partial^2 (M_x B_x + M_y B_y + M_z B_z) over {partial x}^2} + {partial^2 (M_x B_x + M_y B_y + M_z B_z) over {partial y}^2} + {partial^2 (M_x B_x + M_y B_y + M_z B_z) over {partial z}^2} right)$

Expanding and rearranging the terms (and noting that the dipole M is constant) we have

$nabla^2 U = -left( M_xleft({partial^2 B_x over {partial x}^2} + {partial^2 B_x over {partial y}^2} + {partial^2 B_x over {partial z}^2}right) + M_yleft({partial^2 B_y over {partial x}^2} + {partial^2 B_y over {partial y}^2} + {partial^2 B_y over {partial z}^2}right) + M_zleft({partial^2 B_z over {partial x}^2} + {partial^2 B_z over {partial y}^2} + {partial^2 B_z over {partial z}^2}=right) right)$

$nabla^2 U = -(M_x nabla^2 B_x + M_y nabla^2 B_y + M_z nabla^2 B_z)$

but the Laplacians of the individual components of a magnetic field are zero in free space (not counting electromagnetic radiation) so

$nabla^2 U = -(M_x 0 + M_y 0 + M_z 0) = 0$

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

$U = -k (B_x^2 + B_y^2 + B_z^2).$

Expanding and rearranging terms,

but since the Laplacian of each individual component of the magnetic field is zero

$nabla^2 (B_x^2 + B_y^2 + B_z^2) = 2[ | nabla B_x |^2 + | nabla B_y |^2 + | nabla B_z |^2 ]$

and since the square of a magnitude is always positive

$nabla^2 (B_x^2 + B_y^2 + B_z^2) geq 0.$

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 proved 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 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. In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...

Consider the Laplacian of the x component of the magnetic field

$nabla^2 B_x = {partial^2 B_x over partial x^2} + {partial^2 B_x over partial y^2} + {partial^2 B_x over partial z^2} = {partial over partial x} {partial over partial x} B_x + {partial over partial y} {partial over partial y} B_x + {partial over partial z} {partial over partial z} B_x.$

Because the curl of B is zero,

${partial B_x over partial y} = {partial B_y over partial x}$

and

${partial B_x over partial z} = {partial B_z over partial x}$

so we have

$nabla^2 B_x = {partial over partial x} {partial over partial x} B_x + {partial over partial y} {partial over partial x} B_y + {partial over partial z} {partial over partial x} B_z$

but since $B x$ is continuous the order of differentiation doesn't matter giving

$nabla^2 B_x = {partial over partial x}left( {partial B_x over partial x} + {partial B_y over partial y} + {partial B_z over partial z} right) = {partial over partial x}(nabla cdot mathbf{B}).$

The divergence of B is constant (zero, in fact) so

$nabla^2 B_x = {partial over partial x}(nabla cdot mathbf{B} = 0) = 0.$

The Laplacian of the y component of the magnetic field $B y$ field and the Laplacian of the --z-- component of the magnetic field $B z$ can be calculated analogously.

References

Earnshaw, S., On the nature of the molecular forces which regulate the constitution of the luminiferous ether., 1842, Trans. Camb. Phil. Soc., 7, pp 97-112.

External links

http://www.hfml.kun.nl/levitation-possible.html - 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 Earnshaw

Categories: Electrostatics | Levitation | Physics theorems

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Contents

Proofs for magnetic dipoles

Introduction

Background

Summary of proofs

Detailed proofs

Fixed-orientation magnetic dipole

Magnetic dipole aligned with external field lines

Laplacian of individual components of a magnetic field

References

External links