Quantum Levitation: the teacher’s guide to superconductivity

Superconductivity in high school?

Since its discovery in 1911, superconductivity was only discussed at the high-school physics level as an interesting topic or an anecdote. The phenomenon couldn’t be observed in class because it occurred only at extremely low temperatures — a few degrees above absolute zero (0 K).

During the late 1980s, the rapid succession of newly discovered high-temperature superconductors, which operate at liquid-nitrogen temperature (77 K), turned the tables. Superconductivity was now well within the reach of high-school students. It became possible to perform classroom demonstrations of magnetic levitation and to observe quantum phenomena using relatively cheap liquid nitrogen.

Quantum Levitation demonstrations always capture students’ attention. They become entranced by an upside-down levitated magnet, they wonder how it works and predict what it can be used for — scientific inquiry has begun. Students’ curiosity will be limited only by their imagination.

Superconductivity is widely regarded as one of the great scientific discoveries of the 20th century and has played a central role in at least five Nobel Prizes in Physics (1913, 1972, 1973, 1987 and 2003). Nevertheless, the history of superconductors is only just beginning. The possible discovery of room-temperature superconductors could bring superconducting devices into our everyday lives; superconductivity is already applied in transportation, power production, medicine and more.

The questions students ask first

Short answers to the questions that come up in every class (and in every search box). Each one is expanded in the chapters below.

A.1 Zero resistance at low temperatures

It had been known for many years that the resistance of metals fell gradually when cooled below room temperature, but it was not known what limiting value the resistance would approach if the temperature were reduced to very close to absolute zero.

The era of low-temperature physics began in 1908 when Dutch physicist Heike Kamerlingh Onnes first liquefied helium, which boils at 4.2 K. Three years later, Onnes passed a current through a very pure mercury wire and measured its resistance as he steadily lowered the temperature. Much to his surprise, there was no levelling off of the resistance until the temperature reached 4.2 K, at which point the resistance suddenly vanished. Current was flowing through the mercury wire and nothing was stopping it; the resistance was zero. Onnes called this new state of zero resistance superconductivity.

In 1913, Onnes was awarded the Nobel Prize in Physics for the study of matter at low temperatures and the liquefaction of helium. Soon afterwards, many other metals were found to exhibit zero resistance when their temperatures were lowered below a certain characteristic temperature, called the critical temperature, or T_c.

The importance of this discovery to the scientific community, as well as its commercial potential, was clear: an electrical conductor with no resistance could carry a direct current with no resistive losses at all.

A.2 Expulsion of magnetic fields — the Meissner effect

The magnetic properties of superconductors are as dramatic as their complete lack of resistance. In 1933, Walther Meissner and Robert Ochsenfeld studied the magnetic behaviour of superconductors and found that when certain ones are cooled below their critical temperature, they expel a magnetic field. A superconductor will not allow a magnetic field to penetrate its interior. It achieves this by producing a “magnetic mirror”: surface currents which produce a magnetic field that exactly counters the external field. The expulsion of magnetic fields from the interior of a superconductor is known as the Meissner effect.

A good comparison to electricity is that a good conductor expels static electric fields by moving charges to its surface; the surface charges produce an electric field that exactly cancels the externally applied field inside the conductor. In a similar manner, a superconductor expels magnetic fields by forming surface currents. At ordinary temperatures these currents decay almost instantaneously because of the finite resistivity of the conductor. When the superconductor is cooled below T_c, however, persistent surface currents are induced and produce a magnetic field that exactly cancels the applied field inside the superconductor.

Levitation of a magnet above a cooled superconductor can be explained by the Meissner effect. If a superconductor is cooled below its critical temperature while in a magnetic field, the field surrounds but does not penetrate the superconductor. The magnet induces currents in the superconductor which create a counter-magnetic force that causes the two to repel. The induced currents are due to the presence of the external field itself — not to flux changes, as is the case in ordinary metals (Faraday’s and Lenz’s laws).

The complete expulsion described so far holds only while the external field is weak. What happens as the field grows depends on the type of superconductor. In a type-I superconductor (most pure metals, such as the mercury of Onnes’s experiment) there is a single critical field, B_c: below it the field is fully excluded, above it the field floods in and superconductivity is lost.

YBCO, like all high-temperature superconductors, is type II, and has two critical fields. Below the lower one, B_c1, it behaves as a perfect Meissner diamagnet. Between B_c1 and the much larger upper field, B_c2, the material is in the mixed state: the field enters as thin quantised filaments (vortices) while the rest of the sample stays superconducting. Only above B_c2 does superconductivity disappear. A levitator over a neodymium magnet sits comfortably in the mixed state — for YBCO B_c1 is a few hundredths of a tesla and B_c2 is over 100 T — which is why the vortex physics of section A.5 is essential to every demonstration in this guide.

A.3 High-temperature superconductors

It has long been a dream of scientists working in the field to find a material that becomes a superconductor at room temperature. About half of the metallic elements and a large number of alloys superconduct at very low temperatures, which requires handling liquid helium — a complex and expensive task. A great deal of effort has therefore been directed towards finding new superconductors with higher critical temperatures.

Early in 1986, Georg Bednorz and Karl Alex Müller made a remarkable discovery that revolutionised the field: an oxide of lanthanum, barium and copper became superconducting at about 30 K. Inspired by this, scientists worldwide worked intensively to discover materials with even higher T_c. A year later, a ceramic material, YBa₂Cu₃O_7−δ (YBCO; the δ records a small, adjustable oxygen deficiency on which T_c depends), was found to superconduct at 92 K. This was a milestone because the transition temperature is above the boiling point of liquid nitrogen (77 K) — a coolant that is readily available, far safer, inexpensive and much easier to handle than liquid helium. Bednorz and Müller were awarded the 1987 Nobel Prize in Physics.

The superconductor encapsulated inside the Quantum Levitator is YBCO, a compound of yttrium, barium, copper and oxygen. Its atoms are arranged in an orthorhombic crystal structure (a cuboid unit cell). The material is a poor electrical conductor at room temperature and becomes a superconductor below 92 K. The superconducting layer inside the levitator is only 1–3 microns thick, a crystal grown on a metallic substrate (Hastelloy / stainless steel) and protected by a silver layer.

A.4 Applications of superconductors

Electrical power

The ability of superconductors to conduct electricity with zero resistance can be exploited in transmission lines. Currently a substantial fraction of electricity is lost as heat through the resistance of conventional conductors such as copper or aluminium. If transmission lines could be made superconducting, these resistive losses would be greatly reduced. They would not vanish entirely: real cables still have joints, need refrigeration, and in AC systems suffer small hysteresis losses in the superconductor itself — engineering constraints that today’s demonstration projects are designed to quantify.

High-temperature superconductor (HTS) technologies have developed rapidly in the three decades since 1987; today the HTS industry has advanced to full-scale power equipment prototypes and demonstration projects. The foundation is a new generation of wire, capable of carrying on the order of 100 times more current than a copper wire of the same dimensions, with zero or negligible resistive losses. These wires promise a revolution in the way electricity is generated, delivered and consumed — much as optical fibre did for telecommunications.

Transportation

Magnetically levitated trains employing superconducting electromagnets literally “fly” to their destination, floating above the guideway. Superconducting electromagnets on the train induce currents in levitation and guidance coils set into the guideway sidewalls; those currents create magnetic forces that lift and centre the train, while separately powered propulsion coils push it forward. In 2015 the SCMaglev in Japan reached speeds in excess of 600 km/h.

Superconductivity can improve electrified transport of all kinds, from high-speed trains to advanced ship propulsion: better efficiency and performance, lower weight and fuel consumption, longer range. One can envision a future of vehicles of all sorts gliding above a freeway on superconducting magnets.

MRI

The first large-scale commercial application of superconductivity was magnetic resonance imaging (MRI). The intense magnetic fields these instruments need are a perfect application for superconductors. Normal electromagnets would dissipate a great deal of heat and have huge power requirements; superconducting magnets need almost no power apart from cooling. Once current flows in the superconducting coil, the power supply can be switched off and, because the wire forms a loop, the current persists indefinitely as long as the temperature stays below T_c.

Most conventional high-field clinical scanners (1.5 T and 3 T) are built around a superconducting magnet, because it delivers a strong, extremely uniform and stable field — essential for the resolution, precision and speed of clinical imaging — at a fraction of the running cost of a resistive electromagnet. Lower-field, open and portable MRI systems using permanent or resistive magnets also exist, and are an active area of development.

A.5 The physics behind superconductivity — a quantum phenomenon

Superconductivity is a purely quantum phenomenon. Not surprisingly, a successful theoretical explanation had to wait almost 50 years for the foundations of quantum mechanics to be consolidated. A full understanding requires quantum mechanics beyond the scope of this guide; here the fundamental terms and phenomena are discussed qualitatively.

BCS theory: conventional superconductors

According to classical physics, part of the resistance of a metal is due to collisions between free electrons and the vibrations of the crystal lattice (phonons); part is due to scattering from defects or impurities. Soon after the discovery of superconductivity, scientists recognised that this classical model could never explain the superconducting state: electrons always suffer some collisions, so resistivity can never be zero.

In 1957, three American physicists at the University of Illinois — John Bardeen, Leon Cooper and Robert Schrieffer — developed a model that has since stood as a good mental picture of why superconductors behave as they do. Its central feature is that two electrons in the superconductor can form a bound pair, a Cooper pair, if they experience an attractive interaction. At first sight this seems counter-intuitive, since electrons repel one another. But a net attraction can arise if the electrons interact via the motion of the crystal lattice: the lattice is momentarily deformed by a passing electron, and the second electron is attracted by the displaced positive ions.

The interaction between a Cooper pair is transient. Each electron goes on to form a Cooper pair with other electrons, and the process continues, so that in the end each electron in the solid is attracted to every other electron, forming a large network of interactions.

The model is expressed in the advanced language of quantum mechanics, but its main idea is that electrons in a superconductor condense into a quantum ground state and travel together, collectively and coherently. In 1972, Bardeen, Cooper and Schrieffer received the Nobel Prize in Physics for their theory, now known as the BCS theory.

Flux pinning — the physics of “quantum locking”

Type-II superconductors — and every high-temperature superconductor is type II — admit the magnetic field partially once it exceeds the lower critical field B_c1 (section A.2). The penetration takes the form of thin filaments, called flux lines or vortices. Each vortex is a cylindrical swirl of current surrounding a core that lets some flux through. The vortices repel each other and arrange themselves in an orderly array known as a flux lattice.

Depending on the quality of the superconductor, the vortices may be free to move (clean samples) or strongly pinned to defects (dirty samples). In practice, high-temperature superconductors have defects — missing or misplaced atoms, impurity atoms — in their crystal lattices. These defects and grain boundaries stop the motion of the vortices: flux pinning. Pinning the field lines also means stopping their motion relative to the magnet. Pinning sites can be introduced deliberately by adding impurities or through radiation damage.

B.1 Questions

1. Discuss the problems that scientists must overcome before superconductors can be effectively used in our daily lives.
2. In your own words, explain the Meissner effect.
3. Why was the discovery of YBCO so important? What made it different from the other superconductors known at the time?
4. Superconductivity has been central to at least five Nobel Prizes in Physics: 1913, 1972, 1973, 1987 and 2003. Name the laureates and the prize motivation in each case. (Two of the five — 1973 and 2003 — are not covered in this guide and make a good research task.)
5. List two applications of superconductors in use today and describe the role of the superconductor in each.

B.2 Problems

6. What is the resistance of a superconductor at room temperature if 500 mA of current pass through the sample and 3.5 mV are measured across the voltage probes?
   Answer

   R = V / I = 0.0035 / 0.5 = 0.007 Ω

7. What is the resistivity of the superconductor in problem 6 at room temperature, assuming the sample is rectangular? The sample is 2.5 mm wide and 3.4 mm high, and the distance between the probes is 2.5 cm.
   Answer

   Use ρ = R·A / L with A = 2.5 mm × 3.4 mm = 8.5×10⁻⁶ m² and L = 0.025 m: ρ = 0.007 × 8.5×10⁻⁶ / 0.025 = 2.38×10⁻⁶ Ω·m (238×10⁻⁶ Ω·cm).

8. What is the temperature in kelvin, and what state of matter will nitrogen be in, at T = −319 °F (assume atmospheric pressure)? The conversion from Fahrenheit to kelvin is T_K = (T_°F + 459.6) × 5/9.
   Answer

   (−319 + 459.6) × 5/9 = 78 K, which is above nitrogen’s boiling point (77 K). The nitrogen is a gas.

B.3 Graph analysis: the resistive transition of YBCO

9) A student ran an experiment to collect data on a YBCO sample. A 100 mA current flowed through the sample while the student measured the voltage across it. The measured voltage and temperature are in the table below.

1. Complete the table above using Ohm’s law.
2. Using the data, plot resistance as a function of temperature.
3. Calculate the slope of the graph where the temperature is above T = −173 °C. (T(K) = T(°C) + 273)
4. Estimate the critical temperature from the slope of the graph and the data in the table. Explain how you determined it.

Answers

a) The completed resistance column is R = V / 0.1 A; the filled-in values are shown in the table of the answer key.

b) The graph is shown on the right: a flat, gently sloping normal-state resistance above ~93 K, then a sharp drop to zero.

c) Slope of the normal-state region (100 K and above): a least-squares fit to the eleven points gives α ≈ 5.4×10⁻⁵ Ω/K; taking just the end points (100 K, 118.2 K) gives 4.7×10⁻⁵ Ω/K. Anything in the range 4.5–5.5×10⁻⁵ Ω/K is acceptable — the scatter in the data is a good discussion point.

d) T_c ≈ 91.5 K = −181 °C, read where the resistance departs from the linear trend and collapses to zero.

Show table answers

B.4 Graph analysis: the critical magnetic field

10) For a type-I superconductor the critical magnetic field (B_c) is the external field that destroys superconductivity (the four elements below are all type I; YBCO, being type II, has two critical fields, see A.2). The formula for B_c as a function of temperature T is empirical — based on experimental evidence rather than a solid theoretical foundation:

B_c(T) = B₀ [ 1 − (T / T_c)² ]

Here T_c is the critical temperature in the absence of an external field, and B₀ is B_c at 0 K. The table gives four metals, their critical temperature and their critical field at 0 K.

Element	Tc (K)	B0 (T)
Al	1.20	0.01
Ga	1.08	0.0058
Hg	4.15	0.041
Nb	9.26	0.1991

1. Use the formula to plot B_c (y-axis) against T (x-axis) for each element, from 0 K to T_c.
2. Is mercury (Hg) a superconductor when an external field of 0.03 T is applied and it is cooled to 3 K? Explain.
3. Is aluminium (Al) a superconductor when an external field of 0.002 T is applied and it is cooled to 1 K? Explain.
4. Which of the four elements is least sensitive to an external magnetic field? Explain.

Answers

a) The four curves are shown on the right; each falls from B₀ at 0 K to zero at T_c.

b) No. At 3 K mercury’s critical field is 0.041 × [1 − (3/4.15)²] ≈ 0.020 T; 0.03 T is above it, so the point lies in the non-superconducting region of the plot.

c) Yes. At 1 K aluminium’s critical field is 0.01 × [1 − (1/1.2)²] ≈ 0.0031 T; 0.002 T is below it.

d) Niobium. The area under its curve is by far the largest — it tolerates the highest field at every temperature.

Each activity lists the equipment needed, describes the demonstration, and gives a physical explanation using the theory students have already met in Chapter A. Together they build theoretical knowledge and let students experience the practical side of superconductors.

C.1 How does quantum levitation work? Two effects at once

The phenomenon of quantum levitation is composed of two different effects that occur simultaneously: the Meissner effect and quantum locking. The rationale of this teaching sequence is to separate the two, so students understand the role each plays. After the levitation is demonstrated, a series of POE (Predict, Observe, Explain) activities follows: first focusing on the Meissner effect, then on the special quantum properties of locking, and finally creating nearly frictionless motion through symmetry in the magnetic field.

C.2 The Meissner effect

C.3 Quantum locking

In these activities students meet phenomena that the Meissner effect cannot explain. They can only be explained by flux pinning — quantum locking.

C.4 Final demonstration: the maglev track

This is the most impressive demonstration of quantum levitation.

D.1 Experiment 1: measuring the levitation force

Objective

Measure the magnetic force between the levitator and the magnet array, and observe the hysteresis caused by flux pinning.

Equipment

Handheld magnetic device, Quantum Levitator, plastic tweezers, liquid nitrogen and a force-sensor balance stand.

Method

1. Place the Handheld magnetic device on the force sensor and tare the sensor to 0.
2. Cool the levitator in liquid nitrogen and hold it with the tweezers while you start recording.
3. Bring the levitator towards the magnets until it is a few millimetres away, then continuously pull it back up, increasing the distance from the magnet in a smooth motion.
4. Repeat until you obtain a clean, continuous graph.

Data analysis

If the experiment was performed correctly, the graph should look like Figure D1. As the levitator approaches the magnet (Region 1), the repulsion between them increases and so does the net force the magnet applies to the sensor. Both the Meissner effect (always repulsive) and the pinning force (resisting the increase in trapped flux) act to repel here, and the sensor reads their sum: roughly 1 N. Note that the sensor never measures the two separately — it always reads the total magnetic force. What tells the two effects apart is how that force depends on the history of the motion.

By Newton’s third law the levitator experiences the same force upwards, against gravity. The repulsion between magnet and levitator can therefore carry as much as 100 g. This is remarkable given that the levitator weighs about 3 g and, more strikingly, that the superconducting film inside it (a disc 4.5 cm across and about 2 µm thick, density 6.3 g/cm³) has a mass of only about 0.02 g: the film supports several thousand times its own weight.

The most interesting part occurs as the levitator moves away (Region 2). The force decreases with distance as expected, but instead of falling gradually to zero the net force on the magnets becomes negative — something is pulling the magnets upwards. The Meissner effect, which can only repel, cannot explain this. That leaves flux pinning, which opposes any change to the flux trapped inside the superconductor. On the way in, the field was increasing, so pinning added to the repulsion; on the way out the trapped vortices try to hold on to the field they already have, so pinning pulls the levitator back down towards the magnets — and, by Newton’s third law, pulls the magnets up. If that pull exceeds the Meissner repulsion, the total force is attractive and the sensor goes negative. The difference between the approach and withdrawal curves is hysteresis, and it is the signature of pinning: a pure Meissner interaction would trace the same curve in both directions.

You have just measured a quantum phenomenon with macroscopic, quantitative tools.

D.2 Experiment 2: measuring the critical temperature using magnetic levitation

Objective

Measure the critical temperature of the superconductor with a resistance thermometer.

Equipment

Levitator with a resistance thermometer attached, multimeter, Handheld magnetic device, tweezers and liquid nitrogen.

Phase A — calibrating the resistance thermometer

A platinum resistance thermometer (a Pt100: 100 Ω at 0 °C) is thermally coupled to the superconductor in the levitator. Its resistance falls smoothly as it cools, and platinum is the standard for accurate thermometry from −200 °C to +850 °C — but the relationship is not linear over that range. The international standard (IEC 60751) describes it with the Callendar–Van Dusen equation, which for temperatures below 0 °C reads

R(T) = R₀ [ 1 + A·T + B·T² + C·(T − 100)·T³ ]

with T in °C, R₀ = 100 Ω, A = 3.9083×10⁻³, B = −5.775×10⁻⁷ and C = −4.183×10⁻¹². Around T_c this gives 26.7 Ω at 92 K (−181 °C) and 20.2 Ω at 77 K (−196 °C). Use the equation, or the manufacturer’s resistance table for your sensor, to convert every reading.

Phase B — measuring T_c

1. Connect the resistance thermometer to the multimeter and cool the levitator in liquid nitrogen.
2. Place the levitator horizontally above the magnetic matrix of the Handheld device and record the temperature at which it stops levitating and lands completely on the magnet. The disappearance of levitation is an excellent indicator of the transition: above T_c the material is in its normal state, the Meissner response and flux pinning disappear, and the field simply passes through it.

Data analysis

Compare the result with the accepted critical temperature of YBCO (about 92 K). What could explain a difference between the two? (Thermal lag between the thermometer and the film, the finite width of the transition, the oxygen content of the YBCO, and the fact that levitation needs a minimum pinning force — so it fails a little below T_c — are all worth discussing.)

D.3 Experiment 3: superconductivity as a tool for classical mechanics

Quantum levitation and nearly frictionless motion give an accessible, easy-to-use tool for investigating classical phenomena that involve motion. With no mechanical contact, the only losses are air drag and small magnetic hysteresis losses, which yields much cleaner measurements of quantities such as energy and speed — and lets students quantify how far a real system departs from the ideal model.

Harmonic motion

Objective

Investigate the harmonic motion of the levitator at different values of the restoring force. Students measure the frequency of the motion as a function of the slope angle of the maglev track.

Equipment

The maglev track, a stand to change its slope angle, a stopwatch, liquid nitrogen and a levitator.

Method

1. Cool the levitator in liquid nitrogen.
2. Place the cooled levitator on the tilted track, displace it a small angular distance from the lowest point (no more than 10–15° around the track) and release it. Larger amplitudes break the small-angle approximation used below.
3. The levitator moves in simple harmonic motion along the circular track. Gravity has two components: one perpendicular to the surface, cancelled by the levitation force, and one parallel to the surface along the track — the restoring force.
4. Change the angle of the track by adjusting the height of the stand.
5. With the stopwatch, measure the time the levitator takes to complete one oscillation.
6. Take several measurements at each height to reduce the error.

Data analysis

The motion can be modelled as a simple pendulum on a string of length L (here, the radius of the track). With the small-angle approximation, the period of a simple gravity pendulum is

T = 2π √(L / g)

In our case the effective gravity is only the component parallel to the track’s plane, g·sin α. For a levitator displaced by an angle θ around the track from its lowest point, the restoring force along the rail is mg·sin α·sin θ ≈ mg·sin α·θ for small θ, which gives the relation between the period T and the slope angle α:

T = 2π √( L / (g · sin α) )

Plot T² against 1 / sin α to obtain a straight line whose slope is 4π²L/g. Ask students to check whether the period drifts as the oscillation dies away — a measure of the small damping that remains.

Conservation of mechanical energy

Objective

Investigate the conservation of mechanical energy in a nearly frictionless system. Students find the relationship between the velocity of the levitator and the height it was released from on the track, and compare it with the ideal prediction.

Equipment

The maglev track, a stand to set its slope, levitator, liquid nitrogen, a long ruler, a photogate or a high-speed camera, and Tracker video-analysis software.

Method

1. Cool the levitator in liquid nitrogen.
2. Measure the initial height from which the levitator is released, relative to the lowest point of the track.
3. Set up the photogate or camera to measure the levitator’s velocity at the bottom of the track.
4. With a photogate, place it at the bottom of the track and make sure the levitator passes through it horizontally, sides parallel to the track.
5. With a camera, film the levitator at the bottom of the track and analyse the video in Tracker to obtain the velocity.
6. Repeat at different heights, taking several velocity measurements at each; measure the height every time it is changed.

Data analysis

Mechanical energy is conserved when the work of all non-conservative forces is zero. There is no contact friction here, and the magnetic force is perpendicular to the motion, so it does no work. In the ideal case the initial potential energy is converted entirely into kinetic energy at the bottom of the track:

½ m v² = m g h   ⇒   v = √(2 g h)

Plot v² against h. The ideal model predicts a straight line through the origin with slope 2g = 19.6 m/s². Ask students why the measured slope may come out a little lower (air drag, magnetic hysteresis where the field is not perfectly uniform, the levitator not being released from rest) and how they could estimate the energy lost per run. Turning the approximation into a measured discrepancy is the real experiment.