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Lenzor

Lenz’s Law / Eddy-Current Braking Physics

\[\mathcal{E} = -\frac{d\Phi_B}{dt}\]
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Lenzor is a Pixelated Physics law character, one of our forces designs. Lenz’s law: a changing magnetic flux drives eddy currents that push back, so a magnet falls slowly through a copper pipe. This is a 3" × 3" pixel-art sticker. The physics panel below gives the equation, what each symbol means and the sources.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

A changing magnetic flux induces a voltage, and the induced current always flows so as to oppose the change. A magnet dropped through a copper pipe therefore drives eddy currents that brake it, so it drifts down at a steady terminal speed.

Lenzor is a copper pipe with a contrarian streak.

Drop a strong magnet down him and it does not fall like a stone. It drifts down as if through syrup, though copper is not magnetic and nothing touches the magnet. The answer is induction. As the magnet approaches a ring of the pipe, the magnetic flux through that ring grows; as it leaves, the flux shrinks. Michael Faraday found in 1831 that a changing flux drives a current. In 1834 the Russian physicist Heinrich Lenz gave the rule for its direction: the induced current always flows so as to oppose the change that made it. That opposition is the minus sign in the first line on this page.

So Lenzor pushes back. Below the falling magnet, a ring builds a pole that repels it; above, a ring circulating the other way clings on and pulls. Eddy currents swirl round the pipe just ahead of the magnet and just behind it.

But he can’t stop it. His push exists only while the magnet moves, and it grows with the speed. The magnet speeds up until the drag matches its weight, then falls at a slow, steady terminal speed: the second line. The energy it loses on the way down does not vanish and is not stored; it ends up as heat in the copper. In 1993 MacLatchy, Backman and Bogan turned this magnetic braking into a quantitative student experiment.

The rule is energy bookkeeping. If the induced current helped the change instead of opposing it, the magnet would be pulled along for free while the copper warmed, and energy would not be conserved. Opposition is his whole personality, which is exactly what the minus sign means.

Change the flux. I push back.

Equations

\[\mathcal{E} = -\frac{d\Phi_B}{dt}\]
\[m g = F_{\text{eddy}}(v_t) \propto \sigma\,v_t\]
SymbolMeaningUnit
\(\mathcal{E}\) EMF induced around one ring of the pipe \(\mathrm{V}\)
\(\Phi_B\) magnetic flux through that ring \(\mathrm{Wb}\)
\(t\) time \(\mathrm{s}\)
\(m\) mass of the falling magnet \(\mathrm{kg}\)
\(g\) gravitational acceleration \(\mathrm{m/s^2}\)
\(F_{\text{eddy}}\) upward drag from the eddy currents; it grows with the magnet's speed \(\mathrm{N}\)
\(v_t\) terminal speed \(\mathrm{m/s}\)
\(\sigma\) electrical conductivity of the pipe (copper) \(\mathrm{S/m}\)

Sources

  1. Lenz (1834) Ueber die Bestimmung der Richtung der durch elektrodynamische Vertheilung erregten galvanischen Ströme, Ann. Phys. 107, 483–494
  2. MacLatchy, Backman & Bogan (1993) A quantitative magnetic braking experiment, Am. J. Phys. 61, 1096
  3. Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction
  4. Encyclopaedia Britannica — Faraday’s law of induction (Faraday’s observations of 1831)
  5. Encyclopaedia Britannica — Lenz’s law (deduced in 1834 by the Russian physicist H. F. E. Lenz; conservation of energy)

Source Lenz, E. (1834). Ueber die Bestimmung der Richtung der durch elektrodynamische Vertheilung erregten galvanischen Ströme. Annalen der Physik 107, 483–494.doi.org/10.1002/andp.18341073103

Field notes

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