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Discuss Bill Compton's answer to: Dependence of an induced magnetic dipole on the field that induces it.

Can you show mathematically that diamagnetism alone exerts a repulsive force. Thus, diamagnetism can produce levitation. Note that a magnetic field induces a magnetic dipole with an opposite field to ...

Bill Compton Answered:

Your question can be divided into three parts which we shall discuss separately. However, the mathematical rationale is a bit complex, which your illustration fails to demonstrate.

The nature of diamagnetism. Note that the magnetic field is applied in one magnet and induced in another showing the same polarity, hence we have repulsion. We need to bring both into a superconductive state for the action to be significant: see the Meissner Effect.

The Lorentz Effect is wholly different from the Meissner Effect in that the Lorentz Effect operates at ninety degrees from the applied force and is sequential, hence we suffer specific related inefficiency of the Meissner Effect.

The Bose Effect is a paramagnetic vector that operates between the Lorentz Effect and Meissner Effect, rendering yet further inefficiency.

(Courtesy Wikipedia):

Theory of diamagnetism

The Bohr–van Leeuwen theorem proves that there cannot be any diamagnetism or paramagnetism in a purely classical system. Yet the classical theory for Langevin diamagnetism gives the same prediction as the quantum theory.[11] The classical theory is given below.

Langevin diamagnetism

The Langevin theory of diamagnetism applies to materials containing atoms with closed shells (see dielectrics). A field with intensity B, applied to an electron with charge e and mass m, gives rise to Larmor precession with frequency ? = eB / 2m. The number of revolutions per unit time is ? / 2?, so the current for an atom with Z electrons is (in SI units)[11]

$I = -\frac{Ze^2B}{4 \pi m}.$

The magnetic moment of a current loop is equal to the current times the area of the loop. Suppose the field is aligned with the z axis. The average loop area can be given as $\scriptstyle \pi\left\langle\rho^2\right\rangle$ , where $\scriptstyle \left\langle\rho^2\right\rangle$ is the mean square distance of the electrons perpendicular to the z axis. The magnetic moment is therefore

$\mu = -\frac{Ze^2B}{4 m}\langle\rho^2\rangle.$

If the distribution of charge is spherically symmetric, we can suppose that the distribution of x,y,z coordinates are independent and identically distributed. Then $\scriptstyle \left\langle x^2 \right\rangle \;=\; \left\langle y^2 \right\rangle \;=\; \left\langle z^2 \right\rangle \;=\; \frac{1}{3}\left\langle r^2 \right\rangle$ , where $\scriptstyle \left\langle r^2 \right\rangle$ is the mean square distance of the electrons from the nucleus. Therefore $\scriptstyle \left\langle \rho^2 \right\rangle \;=\; \left\langle x^2\right\rangle \;+\; \left\langle y^2 \right\rangle \;=\; \frac{2}{3}\left\langle r^2 \right\rangle$ . If $N$ is the number of atoms per unit volume, the diamagnetic susceptibility is

$\chi = \frac{\mu_0 N \mu}{B} = -\frac{\mu_0 N Z e^2}{6 m}\langle r^2\rangle.$

Posted 1 year ago
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Anonymous Comment

Rocmike you forgot to use the word Atheist.

Anonymous Comment

Rocmike give it a break.

Paul Wittenberg Jr

This is a bit over my head as I usually deal in fuels chemistry. However we use a magnetic pre-filter here to remove metal chunks before they get to the centrifuge where we remove any traces of water. They haul out a ton of metal scrap every week removed from the fuel.

I know how magnetic fields attract iron, but until now I was unaware how the same field repelled aluminum pieces in the same stream of jet fuel. I will read more on it. That is really fascinating!

Thank you for the insight!

Paul Wittenberg Jr

BTW, Bill, the math didn't show up in your post.

Anonymous Comment

Rocmike aka Paul aka Bill GIVE IT A BREAK.

harry Thinks this answer is Helpful:

Hello Bill,

Thank you for your reply. If i understand Langevin's theory of diamagnetism correctly, it describes a modification of orbital motion in atoms by applying a torque due to a magnetic field. This adds a precession to the orbital motion, at the Larmor frequency. My question was about the production of an orbital motion at the gyration or cyclotron frequency by the action (Lorentz's force) of a magnetic field on a charge moving on a straight line. I would like to know whether the magnetic dipole moment resulting from this induced gyration would be repelled by the magnetic field inducing it.

Bill Compton

harry, let us gain a bit more basic understanding of the relationship between paramagnetism, diamagnetism, and ferromagnetism.

As earlier shown, PARA-magnetic substances are ATTRACTED to magnetic fields, and DIA-magnetic substances are REPELLED by magnetic fields. FERRO-magnetic substances RETAIN magnetic fields in temporary DOMAINS. These domains lose their relationships when we heat the substance to the TRANSITION POINT.

Let us take the simplest molecule, H(2) and determine if it is paramagnetic or diamagnetic. Then let us examine O(2). Then let us examine Neodynium, a frequently used rare earth magnetic metal.

Look at the 1S orbital: one photon working between the (p+/p+)--(e-/e-). There are no unpaired electrons. Adding another electron renders the molecule unstable, the (p+/p+) cannot hold more and the (e-/e-) repel the incoming (e-) with 1ev. H(2) therefore is diamagnetic.

O(2) has two openings on the outer shell, however it fills those orbitals with a weak molecular bond. As we apply a field, we squirt in a large number of (e-) which find a welcome (but temporary) home in the outer orbital of the O atom. It is therefore attracted to the field. O(2) is therefore paramagnetic.

Nedoynium has a complex electron structure in that it changes very rapidly, and holds other Nd atoms loosely in a METALLIC bond, which is neither covalent nor ionic. Let's discuss metallic bonding separately.

These bonds form almost randomly until we re-arrange the outer orbital by application of a magnetic field. When we do so, we form DOMAINS, where one Nd atom hands off an electron to another Nd atom and so forth, setting up a permanent magnetic field.

If we apply a field polarized opposite to the field of the permenent magnet that is weaker than the permenent field, we do not change the polarity of the permanent magnet. To change the field we apply another field at 90 degrees to the polarization of the permanent magnet above the Maxwell Limit, 7/2 that of the existing field.

Bill Compton

Paul, if you would like, I will continue the discussion of magnetic interaction with you by email. It will be a very stimulating discussion, and we will cover (then discover) a few things that the average whitecoat does not know!

Bill Compton

Thanks for the email Paul but why do you ask me to display it here? Suit yourself, I suppose.

Note Dirac's Third Law correlating diamagnetism and capacitance: C=2I/r/C\tK. Here we see that dynamic fields interact INVERSELY to static fields dependent on temperature, accounting for the Meissner Effect.

Left: A schematic view of how an assembly of microscopic dipoles produces opposite surface charges as shown at top and bottom. Right: How an assembly of microscopic current loops add together to produce a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancelation occurs.

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The Meissner Effect only becomes effective at the lower transition point of metals. Nonmetals have a third transition point. I concur with your findings.

I might wish to take issue with you about magnetism affecting gravity: we understand precious little of gravity but who knows? You might have a Nobel if you pursue it to conclusion.

The man without a name Thinks this answer is Helpful:

I like your explanation of magnetic flux but the math in your Wiki is a little old. Still it is better than going into college level material for a high schooler like Harry.

We just got done with a magnetic separator system for a landfill out here. They are recovering 92% of their aluminum waste now. Your explanation of diamagnetism makes a great deal of sense.