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Magnetron Twins

Electromagnetism / Polarity Lock Physics

\[U = -\frac{\mu_0}{4\pi r^{3}}\left[3(\vec m_1\cdot\hat r)(\vec m_2\cdot\hat r) - \vec m_1\cdot\vec m_2\right]\]
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Size guide S–5XL · inches

By classical physics, there should be no magnets. Magnetism in matter comes mainly from electron spin, about one Bohr magneton each; in iron, the exchange interaction lines the spins up. Two dipoles attract nose to tail, and side by side, pointing the same way, they repel. Magnetron Twins on a pixel-art unisex tee; equation, symbols and sources in the panel below.

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

Pour l'échelle : la taille M mesure 20 in (50.80 cm) de large, soit environ 3.143 × 1034 longueurs de Planck (lP = 1.616255 × 10−35 m). Un mètre ruban est plus rapide.

Measurements in inches, ±1.5 in tolerance.

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The physics

The energy of two small magnets depends on both their distance and how they are oriented relative to the line joining them. Aligned end to end they attract, and side by side and parallel they repel, with the energy falling off as one over r cubed.

The Magnetron Twins should not exist. According to classical physics, there are no magnets.

Feynman says so in Chapter 34 of the second volume of his Lectures: the magnetic effects of materials cannot be understood in any honest way from classical physics. A classical theorem shows that charges moving around in thermal equilibrium produce no net magnetism at all. Magnets are quantum objects.

The source is the electron itself. Besides any orbital motion, every electron carries an intrinsic angular momentum, spin, and with it a magnetic moment of about one Bohr magneton, 9.27 × 10⁻²⁴ joules per tesla. It is not a tiny charge spinning on an axis; it is simply a property the electron has. In iron, cobalt and nickel, a quantum effect called the exchange interaction makes neighbouring electron spins line up, and vast numbers of them pointing together make a fridge magnet. Most of the magnetism in ordinary matter comes from electron spin, not from charges circling in orbits.

The formula on this page is what two such magnets do to each other: the energy of two dipoles depends on their separation, falling as one over the cube of the distance, and on how each points relative to the line joining them. For the twins: placed nose to tail along the same line, pointing the same way, they attract. Placed side by side pointing the same way, they repel; flip one and they attract. The formula tracks every angle in between.

There are no single magnetic poles in this picture. Cut a magnet in half and you get two magnets, each with both ends. Nobody has ever found an isolated north pole.

The Magnetron Twins are always a pair. Physics has never offered either of them the alternative.

Equations

\[U = -\frac{\mu_0}{4\pi r^{3}}\left[3(\vec m_1\cdot\hat r)(\vec m_2\cdot\hat r) - \vec m_1\cdot\vec m_2\right]\]
SymbolMeaningUnit
\(U\) interaction energy \(\mathrm{J}\)
\(\mu_0\) vacuum permeability \(\mathrm{N/A²}\)
\(\vec m_1, \vec m_2\) magnetic dipole moments \(\mathrm{A·m²}\)
\(r\) separation \(\mathrm{m}\)
\(\hat r\) unit vector between dipoles \(\mathrm{1}\)

Sources

  1. Feynman Lectures on Physics Vol. II Ch. 34: The Magnetism of Matter
  2. Feynman Lectures on Physics Vol. II Ch. 36: Ferromagnetism
  3. Encyclopaedia Britannica — Dipole
  4. NIST CODATA 2022 — vacuum magnetic permeability μ0
  5. NIST CODATA 2022 — Bohr magneton

Source Maxwell, J. C. (1873). A Treatise on Electricity and Magnetism, Vol. II, Part III (Magnetism). Oxford: Clarendon Press.

Field notes

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