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LEVITATION
PRITHVIR P. JHAVERI, DUKE UNIVERSITY 2019
Abstract
This paper was written with the purpose of exposing my Professor Hubert Bray*,
and the rest of my class**, to the various kinds of levitation that exist and their
respective applications. It’s possible to levitate a frog and a grasshopper using
direct- diamagnetic levitation. Could we actually apply the same principles and
be able to levitate a human? Or will we have to use some different kind of
levitation? Furthermore, could levitation help us in the real world? Can we use
levitation to propel cars, or even trains?
Introduction
The claim that we could levitate a frog using direct diamagnetic levitation was
investigated using calculations that went from utilizing newton’s principles of
force and motion to the upward derivative of magnetic energy. These
calculations were applied to see if we could levitate a human using direct
diamagnetic levitation. However, we concluded that this was not possible. We
were forced to apply the Casimir effect to the levitation of humans, and this
proved to be theoretically viable. We tried to apply the concept of Quantum
Locking and the Meissner Effect to the real world. We actually proved how
Maglev cars and trains are able to function at high-speeds while levitating.
*
Hubert Bray is a Professor of Mathematics and Physics at Duke University
This class is a seminar on introductory astrophysics called Math 89S at Duke University
**
Direct Diamagnetic Levitation
A diamagnetic substance is one that repels in a strong magnetic field. The
electrons in diamagnetic substances rearrange their orbits slightly, creating
small persistent currents which oppose the external magnetic field. Many
common materials such as animals, wood, plants, animals, diamonds, fingers
actually have weak magnetic properties. Water is a strong diamagnetic
substance. Since these materials have high water content, they are considered
diamagnetic in nature. The forces created by diamagnetism are extremely
weak. In fact, these forces are about a million times weaker than the forces
between ferromagnetic substances like iron. However, in certain carefully
arranged situations, the influence of diamagnetic materials can produce
startling effects such as levitation.
Dutch physicist Sir Andre Geim actually won the Ig Nobel in Physics in 2000
for levitating a live frog with magnets. Most thought that water’s magnetism
was not strong enough to counter gravity. However, his experimental
demonstration showed its true force*.
Frogs, like everything around and inside us, are made up of millions and
billions of atoms. Each of these atoms contain electrons that whizz around a
central nucleus. But, when atoms are in a magnetic field, the electrons shift
their orbits slightly. These shifts give the atoms their own magnetic field so
when a frog is put in a very strong magnetic field, it is essentially made up of
lots of tiny magnets. And there’s nothing special about frogs. All materials –
*
Here’s a video showing Greim’s experimental demonstration
https://www.youtube.com/watch?v=KlJsVqc0ywM
including strawberries, water and gold – are ‘diamagnetic’ to some extent, but
some are more convenient to levitate than others.
Frogs are convenient not only because they have a high water content, which
is a good diamagnetic material, but also because they fit easily inside a tubeshaped Bitter electromagnet. Bitter electromagnets* use a very large electric
current to create an extremely strong magnetic field which magnetizes the
frog in such a way that its magnetization is in the opposite direction to the
applied field. This means that the magnetized frog is pushed up from a region
of high magnetic field into one of lower field, and levitates.
A bitter electromagnet has a maximum field of 20 tesla (T) (this is 400000
times the earth magnetic field). It consists of two concentric electrical coils.
These coils are electrically connected in series. The magnet is cooled by deionized water of about 10°C.
The main condition that really needs to be satisfied for the frog to levitate and
remain in equilibrium is that the magnetic force exerted on the frog should be
able to balance the gravitational force exerted on it. Given by the following
equation:
Gravitational Force <= Magnetic Force;
We calculate the gravitational force on the frog using the equation F = m*g;
where m is the mass of the frog which is usually around 22.7g or 0.0227 kg,
and g is the acceleration due to gravity (usually 9.8 m/s2). So, Gravitational
Force comes out to be 0.22246 N, and since the frog has a density (similar to
water) of about 103 kg/m3, the force per unit meter cubed comes out to be
*here’s
more information on bitter electromagnets
http://www.ru.nl/hfml/research/levitation/diamagnetic/bitter-solenoid/
104N/m3. So to balance the force of gravity, we'll need an upward magnetic
force per unit volume of about 104 N/m3.
In a magnetic field B, the magnetic energy of the frog per unit volume will be
-moMH=-MB. Magnetization M=cH=cB/mo, and since c=-10-5, and mo is about
10^^6, M=-10B, and the magnetic energy per unit volume -MB will be about
10B2. Of course there will be no magnetic force in a uniform magnetic field.
The magnetic force will equal the upward derivative of the magnetic energy,
i.e., 10 d(B2)/dx, and this must equal the gravitational force to produce
levitation. So we'll need d(B2)/dx=2B(dB/dx) =103T2/m (10T2/cm) to levitate
a frog - or a person. As discussed, Magnetic fields of 10T or more can be
produced only with bitter electromagnets. With B=10T, you'll need a field
gradient dB/dx of about 0.5 T/cm.
Bitter magnets have that sort of field gradient over volumes as big as that of
a frog, or even a grasshopper (which has a similar density as a frog but a mass
of around o.5g, even less than, and thus easier to levitate). But, bitter
electromagnets cannot produce enough magnetic field to balance a human’s
downward gravitation force*. This leads us to try to see if the Casimir effect
can be used to levitate humans.
Casimir Effect: The Casimir effect is a small force that acts between two close
parallel uncharged conducting plates. It is due to quantum vacuum
fluctuations** of the electromagnetic field. Between two plates, only those
virtual photons whose wavelengths fit a whole number of times into the gap
should be counted when calculating the vacuum energy. The energy density
decreases as the plates are moved closer, which implies that there is a small
force drawing them together.
The Casimir force between two plates of area A separated by a distance L can
be calculated to be:
πhc
F = ---------A
480 L4
*
A human’s density is about 1020 kg/m3. Thus, they’d have a downward gravitational force
per unit meter cubed of about 1020 * 9.8 = 9996 N/m3.
** A quantum fluctuation is the temporary appearance of energetic particles out of empty
space, as allowed by the uncertainty principle. The uncertainty principle states that for a pair
of conjugate variables such as position/momentum or energy/time, it is impossible to have a
precisely determined value of each member of the pair at the same time. For example, a
particle pair can pop out of the vacuum during a very short time interval.
where h is Planck's constant (6.626176 x 10-34 Js), and c is the speed of light (3
x 108 m/s).
Thus, the Casimir force would come out to be
 3.14159 * 6.626176 x 10-34 * 3 * 108 * A / (480 * L4).
 1.3 A/L4.
Thus, if we were to place a human on top of a plate, such that another plate
is parallel to it on the ground, such that there is vacuum between the two
plates, let’s see what dimensions these plates would have to be.
The main condition that really needs to be satisfied for the human to levitate
and remain in equilibrium is:
Gravitational Force <= Magnetic Force;
Say, an average human weighs 70 kg. So, the gravitational force (using F = m*g,
where g is the acceleration due to gravity of 9.8 m/s2) comes out to be 686N.
 so, we would need an A to L4 ratio of about 686/1.3 or 527.69.
 say, we wanted to levitate the human by about 1 meter. We would need
square plates of length √ (527.69) = 22.9 meters
 say, we wanted to levitate the human by about 10 meters. We would
need square plates of length √ (10*10*10*10*527.69) or about 2.29 km!
In conclusion, yes we can theoretically levitate humans. However, we would
have to first make major advancements in our ability to create and sustain
large moving vacuums.
Quantum Levitation
Quantum levitation is a process by which a superconductor* is levitated over
a magnetic source.
The reason this works is due to something called the Meissner effect. The
Meissner effect dictates that a superconductor in a magnetic field will always
expel the magnetic field inside of it, and thus bend the magnetic field around
it. This happens because electrons flow through superconductors with no
resistance. When magnetic fields get close to a superconducting material, the
superconductor forms small currents on its surface, cancelling out the
incoming magnetic field. The result is that the magnetic field intensity inside
the the surface of the superconductor is precisely zero.
So, if we were to suspend a superconductor on a magnetic track, the
superconductor would actually remain above the track, essentially being
pushed away by the strong magnetic field right at the track's surface. There is
a limit to how far above the track it can be pushed, of course, since the power
of the magnetic repulsion has to counteract the force of gravity.
Maglev Trains
The notion that we could propel a superconductor forward while placed in a
magnetic track is actually being used purposefully in the real world. A few
countries have been able to develop high-speed trains, called maglev trains.
Maglev, quite obviously, is short for magnetic levitation.
Superconductors are materials that conduct electricity with zero resistance below a certain
temperature. Superconductivity relies on electrons not repelling one another as they do in
ordinary materials, but instead forming delicate couples, known as Cooper pairs, that can
flow through superconductors effortlessly.
*
The magnetized coil running along the track, called a “guideway”, repels the
large magnets on the train's undercarriage, allowing the train to levitate
between 0.39 and 3.93 inches (1 to 10 centimeters) above the guideway. Once
the train is levitated, power is supplied to the coils within the guideway walls
to create a unique system of magnetic fields that pull and push the train along
the guideway. The electric current supplied to the coils in the guideway walls
is constantly alternating to change the polarity of the magnetized coils. This
change in polarity causes the magnetic field in front of the train to pull the
vehicle forward, while the magnetic field behind the train adds more forward
thrust.
Maglev trains float on a cushion of air, eliminating friction. This lack of friction
and the trains' aerodynamic designs allow these trains to reach unprecedented
ground transportation speeds of more than 310 mph (500 km/h), or twice as
fast as Amtrak's fastest commuter train. In comparison, a Boeing-777
commercial airplane used for long-range flights can reach a top speed of about
562 mph (905 km/h). Developers say that maglev trains will eventually link
cities that are up to 1,000 miles (1,609 kilometers) apart. At 310 mph, you
could travel from Paris to Rome in just over two hours.
Japanese engineers are developing a competing version of maglev trains that
use an electrodynamic suspension (EDS) system, which is based on the
repelling force of magnets. The key difference between Japanese and German
maglev trains is that the Japanese trains use super-cooled, superconducting
electromagnets. This kind of electromagnet can conduct electricity even after
the power supply has been shut off. In the EMS system, which uses standard
electromagnets, the coils only conduct electricity when a power supply is
present. By chilling the coils at frigid temperatures, Japan's system saves
energy. However, the cryogenic system uses to cool the coils can be expensive.
Another difference between the systems is that the Japanese trains levitate
nearly 4 inches (10 centimeters) above the guideway. One potential drawback
in using the EDS system is that maglev trains must roll on rubber tires until
they reach a liftoff speed of about 62 mph (100 km/h). Japanese engineers say
the wheels are an advantage if a power failure caused a shutdown of the
system.
In conclusion, it is only a matter of time till we see human’s floating around
on real life hover-boards that utilize the Casimir Effect. And, it is only a matter
of time till roads have Maglev capacities, and thus we will soon see cars and
bikes levitating as a result of the Meissner effect. As with nearly all futuristic
science predictions, we cannot, however, predict how many years these
contraptions will take, and how efficient/ sustainable they will be.
References:
1. H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetensch. B51, 793 (1948)
2. S. Lamoreaux, Phys. Rev. Lett. 78, 5 (1996)
3. Munday, J. N., Capasso, F. & Parsegian, V. A. Nature 457, 170–173 (2009).
4. Meissner, W.; R. Ochsenfeld (1933). "Ein neuer Effekt bei Eintritt der
Supraleitfähigkeit"
5. Lev D. Landau; Evgeny M. Lifschitz (1984). Electrodynamics of
Continuous Media
6. David J. E. Callaway (1990). "On the remarkable structure of the
superconducting intermediate state"
7. J.E Hirsch (2012). "The origin of the Meissner effect in new and old
superconductors"
8. P. W. Higgs (1966). "Spontaneous Symmetry Breakdown without
Massless Bosons"
9. Wilczek, F. (2000). "The recent excitement in high-density QCD"
10.Weinberg, S. (1986). "Superconductivity for particular theorists"
11.Meissner Effect (Science from scratch) Short video from Imperial College
London about the Meissner effect and levitating trains of the future.
12.Introduction to superconductivity Video about Type 1 Superconductors:
R=0/Transition
effect/Energy gap
temperatures/B
is
a
state
variable/Meissner
13.http://www.slate.com/articles/business/how_failure_breeds_success/
2014/05/nobel_prize_in_physics_andre_geim_went_from_levitating_fr
ogs_to_science.html
14. http://www.spellmaker.com/frog.htm
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tm
16.http://physics.about.com/od/physicsmtop/g/MeissnerEffect.htm
17.http://physics.about.com/od/glossary/g/superconductor.htm
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