A theorem due to Earnshaw proves that it is not possible to achieve static levitation using any combination of fixed magnets and electric charges. Static levitation. The answer is no, and this fact is referred to as the Earnshaw’s theorem. We will prove this assuming $q \gt 0$, but the proof is similar for $q \lt. PDF | A classical electrodynamical results known as Earnshaw theorem forbids the stable static levitation in stationary fields. Even though, permanent magnets.

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Speaking well about Homer is not a thing you have mastered, it’s a divine power that moves you, as a “Magnetic” stone moves iron rings. That’s what Euripides called it; most people call it “Heraclian”. This stone not thworem pulls those rings, if they are iron, it also puts power in the rings, so they in turn can do just what the stone does – pull other rings – so that there is sometimes a very long chain of iron pieces, hanging from one another.

And the power in all of them depends on this stone. Magnetism has been known since ancient times. The magnetic property of lodestone Fe 3 O 4 was mentioned by the Greek philosopher Thales c. It was also found in the nearby province of Heraclia, which is presumably why Socrates says that most people called the stone “Heraclian”. Apparently we have the great dramatist Euripides to thank for not having to theorsm the electro-heraclian field.

About AD the Chinese began to use theorme as a compass for finding directions on land, and soon afterwards Muslim sailors were using compasses to navigate at sea. Europeans began using magnetic compasses for navigation around AD, probably bringing the idea back from the Crusades. The first scientific study of magnets was apparently by the English physician William Gilbert inwho is credited with “discovering” that the Earth itself is a magnet. After Gilbert, the subject languished for almost years, as the attention of most scientists turned to gravitation earnzhaw working out the consequences of Newton’s great synthesis of dynamics and astronomy.

Not until was the subject taken up again, first by the Frenchman Charles Coulomb, then by Poisson, Oersted, Ampere, Henry, Faraday, Weber, and Gauss, culminating in Maxwell’s classical synthesis of electromagnetic theory in However, despite the great achievements of these scientists, no satisfactory understanding of the various kinds of magnetic behavior exhibited by different materials was achieved. Only with the advent of quantum mechanics in the ‘s did it become possible to give a coherent account of the main magnetic properties of materials.

It’s a surprisingly complex subject, but we can give a broad outline of the modern explanations of magnetic phenomena. The three main types of magnetic behavior exhibited by material substances are called diamagnetism, paramagnetism, and ferromagnetism. The first two can be explained in terms of the magnetic fields produced by the orbital motions of the electrons in an atom.

Each electron in an atom can be regarded as having some “orbital” motion about the nucleus, and this moving charge represents an electric current, which sets up a magnetic field for the atom, as shown below. Many atoms have essentially no net magnetic dipole field, because the electrons orbit the nucleus about different axes, so their fields cancel out.

Thus, whether or not an atom has a net dipole field depends on the structure of the electron shells surrounding the nucleus. In broad terms, diamagnetism and paramagnetism are different types of responses to an externally applied magnetic field.

Diamagnetism is a natural consequence of Lenz’s law, according to which the electric current resulting from an applied field will be in the direction that opposes the applied field. In other words, the induced current will flow in the direction that creates a field opposite to the applied field, as illustrated below.

Conservation of energy implies that a force is required to push the magnet through the ring, thereby setting up the flow of current in the opposite direction of the electron motion.

Hence there is a repulsive force between the magnet and the conducting ring. Likewise when an atom is subjected to an applied magnetic field, there is a tendency for the orbital motions of the electrons to change so as to oppose the field.

As a result, the atom is repelled from any magnetic field. Notice that this is true regardless of the polarity of the applied field, because the induced “currents” i. This phenomenon is present in all substances to some degree, but it is typically extremely small, so it is not easily noticed.

It is most evident for elements whose atoms have theorm or no net magnetic moment absent an externally applied field. Among all the elements at ordinary room temperatures, bismuth has the strongest diamagnetism, but even for bismuth the effect is extremely theorme, because the currents that can be established by the electron orbital motions are quite small. It’s possible, however, to construct a perfect diamagnet using superconductivity. A superconductor is, in many respects, like a quantum-mechanical atom, but on a macroscopic scale, and it can support very large currents.


In earnshaq with Lenz’s Law, these currents oppose any applied field, so it’s actually possible to achieve stable levitation of a permanent magnet over earnzhaw superconductor. In view of Lenz’s Law, it might seem surprising that any material could actually be attracted to a magnetic field, but in fact there are many such substances.

Earnshaa is due to the phenomena called paramagnetism. Unlike the atoms of diamagnetic materials, the electrons of atoms in paramagnetic materials are arranged in such a way that there is a net magnetic dipole due to the orbital motions of the electrons around the nucleus.

Is Magnetic Levitation Possible?

Thus, each atom is a small permanent magnet, but the poles tend to be oriented randomly, so a macroscopic sample of the substance usually has no net magnetic field. When such a substance is subjected to an external magnetic field, there is as always a small diamagnetic effect on the orbital motions of the electrons, tending to cause a repulsion as explained abovebut there is also a tendency for the individual atomic dipoles to become aligned with the imposed field, rather than being oriented randomly.

This gives the substance an overall net magnetic dipole in the same direction as the applied field, so if the substance is located in a non-uniform magnetic field, it will be attracted in the direction of increasing field strength. This paramagnetic attraction effect is much stronger than the diamagnetic repulsion, so paramagnetism usually masks the effect of diamagnetism for such substances. However, even paramagnetism is so weak that it’s often not noticed, because the thermal agitation of the atoms at room temperature tends to disrupt the alignment.

The last major category of magnetic behavior is called ferromagnetism. This is the phenomenon responsible for the strong magnetic properties of iron, and for the existence of permanent magnets, i. Many of the early researchers in the science of magnetism thought this was nothing but a strong and persistent form of paramagnetism, but the strength and persistence of ferromagnetism show that it is the result of a fundamentally different mechanism, an effect that is absent in merely paramagnetic substances.

Whereas both diamagnetism and paramagnetism are essentially due to the atomic fields resulting from the orbital motions of the electrons about the nucleus, ferromagnetism is due almost entirely to alignment of the intrinsic spin axes of the individual electrons. An individual electron possesses a quantum property known as “spin”, which is somewhat analogous to the spin of a macroscopic object.

This analogy is not exact, and can be misleading in some circumstances, but it’s useful for gaining an intuitive understanding of the magnetic properties of materials.

Earnshaw’s theorem

According to this view, an electron’s charge is distributed around its surface, and the surface is spinning about some axis, so there is a tiny current loop, setting up a magnetic field as illustrated below. The contribution of the nucleus itself to the magnetic field of an atom is typically negligible compared with that of the electrons.

In most elements the spin axes of the electrons point in all different directions, so there is no significant net magnetic dipole. However, in ferromagnetic substances, the intrinsic spins of many of the electrons are aligned, both within atoms and between atoms. The key question is what causes all these dipoles to be aligned, especially in the absence of an external field.

It can be shown that the dipole interaction itself is not nearly strong enough to achieve and maintain alignment of the electron spin axes at room temperatures, so some other factor must be at work.

Earnshaw’s theorem – Wikipedia

Quantum mechanics furnishes the explanation: For particular arrangements of certain kinds of atoms in the lattice structure of certain solids, the inter-electron distances within atoms and between neighboring atoms are small enough that the wave functions of the thdorem overlap significantly. As a result, there is a very strong effective “coupling force” between them due to their indistinguishability.

This is called an “exchange interaction”, and is purely a quantum-mechanical phenomenon. There is no classical analogy. In essence, quantum mechanics tells us that there is a propensity for the identities of neighboring electrons to be exchanged, and this locks the spin orientations of the electrons together. This is actually true only under certain circumstances.

It’s also possible for exchange interactions to lock the spins of neighboring electrons in opposite directions, in which case the behavior is called anti-ferromagnetism. In order for the exchange interaction to operate, the inter-electron distances must be just right, and these distances are obviously affected by the temperature, so there is a certain temperature, called the Curie temperature, above which ferromagnetism breaks down. Only five elements have electron shell structures that support ferromagnetism, namely, iron, cobalt, nickel, gadolinium, and dysprosium.

In addition, many compounds based on these elements are also ferromagnetic. One example is the compound Fe 3 O 4also called eanrshaw, which the ancient Greeks found lying around in Magnesia. These are all “transition elements”, with partially populated 3d inner electron shells. When magnetized, the theorfm axes of all the electrons in the 3d shells are aligned, not only for one atom, but for neighboring atoms as well. This gives the overall lattice of atoms a very strong net magnetic dipole.


It’s worth noting that this is due to the intrinsic spins of the individual electrons, not due to the orbital motions of the electrons as is the case with diamagnetism and paramagnetism.

Recall that, for paramagnetic substances, the alignment of atomic dipoles is maintained only as long as the external field is applied. As soon as the field is removed, the atomic dipoles tend to slip back into random orientations.

This is because the ordinary dipole field is not nearly strong enough to resist thermal agitation at room temperatures.

In contrast, after a ferromagnetic substance has been magnetized, and the externally applied field is removed, a significant amount of magnetization remains. This effect is called hysteresis. In general, the earnzhaw spins of earnshas the atoms with a suitable lattice will be locked in alignment, with or without an external field, but a real large-scale piece of a substance typically cannot be a single perfectly coherent lattice.

Instead, it consists of many small regions of pure lattices, within which the exchange interaction keeps all the electron spins aligned, but the exchange interaction does not extend across the boundaries between domains.

In effect, these boundaries are imperfections in the lattice. As a result, although each small domain is perfectly magnetized, the domains in an ordinary piece of iron are not aligned, so it has no significant net magnetic field.

However, when subjected to an external field, there is enough extra impetus to trigger a chain reaction of alignment across the boundaries of the individual regions in the iron, causing the overall object to become a magnet. This is the phenomenon described by Socrates, when he explained how a Magnet has the power not only to attract iron, but to convey that power to the iron.

He was describing a purely quantum mechanical effect, by which an applied magnetic field causes the intrinsic spin axes of individual electrons in the 3d shells of transition elements such as iron to become aligned – although he presumably wasn’t thinking about it in those terms. When the external field is removed, the various regions in the iron object will tend to slip back to their natural orientations, given the imperfections in the lattice structure, so much of magnetism of the object will be lost.

However, there will be typically have been some structural re-organization of the lattice depending on the strength of the applied field, and the temperature of the ironso that a higher percentage of the domains are aligned, and this re-structuring of the lattice persists even after the external field is removed.

This accounts for the hysteresis effect, by which a piece of iron acquires some permanent magnetism after having been exposed to a strong field. In order to create a strong permanent magnet, a piece of ferrous material is heated to a molten state, and then placed in a strong magnetic field and allowed to cool.

This creates a lattice structure with very few magnetic imperfections in the lattice, so the electron spins are naturally locked in alignment throughout the material. Not surprisingly, if a magnetized piece of iron is struck with a hammer, it’s possible to scramble the domains and thereby de-magnetize the object.

In summary, the three main kinds a magnetism are illustrated schematically in the figures below. One of the most common questions about permanent magnets is whether there exist a stable and static configuration of permanent magnets that will cause an object to be levitated indefinitely. Obviously the levitation itself is not a problem, because many magnets have fields strong enough to lift their own weight.

Equilibrium is also not a problem, because there is obviously a configuration at the boundary between falling and rising. The problem is stability. In order to have stability, there must be a restorative force counter-acting any displacement away from the equilibrium point. We need to be careful when considering this question, because, as discussed above, there are several kinds of magnetic behavior exhibited by different substances in different circumstances.

We can certainly achieve stable levitation with a superconductor, which is really just a perfect diamagnet. In fact, even at room temperatures, it is possible to use the diamagnetic property of a substance like bismuth to achieve marginal stability for magnetic levitation. Of course, in such a case, the paramagnet is too weak to do the actual levitating; it just provides a small window of stability for an object that is actually being lifted by ferromagnetic effects.

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