SUBDUING EARNSHAW THEOREM

FOR PERMANENT MAGNETS IN

MAGNETIC LEVITATION

by

AMAL MADHU M

ETAKEPE002

• Levitation means a process by which an object is suspended against gravity by

using any physical force.

• Magnetic levitation is the process of levitating an object by exploiting

magnetic fields. The magnetic pressure associated with a magnetic field is

exploited here.

• Either magnetic force of attraction or force of repulsion are used for levitating

objects.

• Magnetic materials and systems are able to attract and repel each other with

the force dependent on their magnetic field; in turn with their magnetic

pressure.

• Primarily applied magnetic levitation principles are electro-

magnetic suspension, meissner effect, electrodynamic

suspension

e.t.c

Earnshaw Theorem

• It was the major challenge faced by scientists working on magnetic levitation.

• It states that

a magnetic body cannot be supported in a stable manner in

the field

produced by passive magnetic poles or permanent magnet.

• The static force as a function of position F(x) acting on any body in vacuum due

to

static fields will always be divergence-less; div F =0 .

• Divergence of a vector field is the extent to which the vector field flow behaves

like a source or sink at a given point.

The exceptions to Earnshaw’s theorem are as follows:

• QUANTUM THEORY - At the atomic level there is a type of levitation occurring

through forces of repulsion between particles. As these forces are weak, they

are not considered• DIAMAGNETISM - Earnshaw’s theorem doesn’t apply to diamagnetic materials,

because they have a relative permeability less than one. They always repel

magnetic flux and hence are not considered as magnetic.

• MEISNNER EFFECT - In conductors cooled to below their critical temperature

(typically close to 0 K). Below this temperature, they become superconductors,

with an internal resistance of zero. They attain a relative permeability of zero,

making them the perfect diamagnetic material. This allows them to maintain

their repelling magnetic field as long as a foreign source of magnetic flux is

present. • IN ELECTROMAGNETS – Use of feedback and oscillation systems help in attaining

a stable levitation in electromagnets.

• It is possible to levitate an object using permanent magnets , defying the basic

idea off Earnshaw theorem.

• It doesn’t mean that divergence theorem is false, but possible to levitate an

object overcoming the constraints….

HOW ???

1. Spin stabilized magnetic levitation

2. Halbach array arrangement

LEVITRON

The spinning top is balanced by two forces:

• Magnetic fields

• Rotation of the top

• The Levitron or spinning top is a small rotating magnetic dipole which is held at

its position by the magnetic field of the base component.

• Dipole field and base field push off one another, both fields have the same

magnetic polarity working against each other, i.e; N-N or S-S arrangement of

poles.

• The top stays at that position where its gravity force is counteracted exactly by

the magnetic repelling force.

• Principle is that two similar poles repel and two unlike poles attract; and the net

effect determined by the poles which are closer. There are four magnetic forces

on the top:On its north pole,

1. repulsion from base’s north pole and

2. Attraction from base’s south pole.

On its south pole,

1. Attraction from base’s south pole and

2. Repulsion from base’s north pole.

• To prevent the spinning top from overturning it needed to be in spinning motion.

• Magnetic field of the base tend to turn the top’s axis of spin by a torque and

hence, if the top is not spinning, Opposite pole of the top come near the base

magnet’s vicinity; and the initial repulsive force gets reversed into attractive

force, This leads to falling of the levitron.

• When the top rotates, gyroscopic action tends to make it rotate in a somewhat

vertical direction and thus preventing overturning. This nearly vertical rotation is

called as Precession.

• For the levitron to spin, the essential condition is not simple

equilibrium,

but stable equilibrium. Hence, if any force tend to move the levitron

horizontally or vertically, the top’s axis is slightly deviated.

• Small range of heights for which the levitron remains

stable is

1.25 to 1.75 inches.

• The levitron is completely unstable above 40 rps and below 18 rps

• The force that acts on the spinning top to hinder its rotation is AIR FRICTION

• Due to air friction, the levitron rotation slows down below 18 rps and eventually

falls.• Spin lifetime of levitron is enhanced by placing it in vacuum. Laboratory

experiments shows evidence of levitron spinning for 30 minutes.

• Reasons for falling of levitron in vacuum could be factors like temperature

induced in rotation, vibrations of the vacuum equipment etc.

APPLICATIONS

• The spin stabilized magnetic levitation is applied in studies of microscopic

particles. For example, neutrons are held or trapped in magnetic field

generated by a system of coils. Here neutrons are spinning magnetic particles

and the base magnet is analogical to magnetic field of coils.

• A spin stabilized magnetic levitation device is marketed in kit-form with

required attachments to the device, popularly known as Levitron- The Physics

Toy

A Halbach array (HA) is a specific arrangement of permanent magnets that

concentrates the magnetic field on one side of the array while cancelling the

field to almost zero on the other side.

• It consists of two sets of magnets, i.e. main magnets and transit magnets as

shown in Figure

• The superposition of the magnetic flux caused by the main magnets and the

transit magnets concentrates the magnetic field to one side of the Halbach array

(the active side)

• Assuming the magnetic field strength generated by the main magnets is identical

to that generated by the transit magnets, the active-side of the Halbach array

will have double the magnetic field strength compared with the normal

configuration.

• The other side of the Halbach array (the quiet-side) will have a negligible

magnetic field.

Linear halbach array

Cylindrical halbach array

• The pattern for rotation of magnets is like ← ↑ → ↓ ; i.e.; Left-Up-Right-

Down and this pattern can be repeated indefinitely to have the same effect.

N S

S

N

S

N

S

N

N

S

N

S

initially

Resultant

Magnetic Field in Halbach array

Augmente

d

Cancelled

• The augmented magnetic field of Halbach array is periodic in nature.

APPLICATIONS

• INDUCTRACK MAGLEV

A passive induced-current system employing permanent magnets on the moving

vehicle, the Inductrack maximizes levitation forces by a combination of two

elements.

First, the permanent magnets on the vehicle are arranged in a Halbach

array, a magnet configuration that optimally produces a periodic magnetic

field below the array, while canceling the field above the array.

Second, the track is made up of close-packed shorted electrical circuits.

levitating forces of order 40 metric tonnes per square meter of Halbach array

can be generated, using NdFeB magnets whose weight is a few percent of the

levitated weight.

• Initial thrive to their invention was for particle guiding in particle accelerators .

Thank you