SUBDUING EARNSHAW THEOREM FOR PERMANENT MAGNETS IN MAGNETIC LEVITATION by AMAL MADHU M ETAKEPE002
Jul 18, 2015
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