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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 15.http://physics.about.com/od/quantumphysics/f/QuantumLevitation.h tm 16.http://physics.about.com/od/physicsmtop/g/MeissnerEffect.htm 17.http://physics.about.com/od/glossary/g/superconductor.htm 18.https://en.wikipedia.org/wiki/Meissner_effect 19.https://en.wikipedia.org/wiki/Magnetic_levitation 20.https://www.google.com/webhp?sourceid=chromeinstant&ion=1&espv=2&ie=UTF-8#q=dimensions%20of%20joules 21.https://www.google.com/webhp?sourceid=chromeinstant&ion=1&espv=2&ie=UTF-8#q=value%20of%20planck%20constant 22.http://math.ucr.edu/home/baez/physics/Quantum/casimir.html 23.https://www.google.com/search?q=Ig&oq=Ig&aqs=chrome..69i57j0l5.4 66j0j1&sourceid=chrome&ie=UTF-8#safe=off&q=density+of+a+human 24.http://www.nature.com/news/2009/090107/full/news.2009.4.html 25.http://science.howstuffworks.com/transport/enginesequipment/maglev-train1.htm 26.http://science.howstuffworks.com/transport/enginesequipment/maglev-train.htm 27.http://science.howstuffworks.com/transport/enginesequipment/maglev-train2.htm 28.https://www.google.com/webhp?sourceid=chromeinstant&ion=1&espv=2&ie=UTF-8#q=superconductor