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Casimira

Casimir Effect Physics

\[\frac{F}{A} = -\frac{\pi^2\hbar c}{240\,d^4}\]
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Can empty space pull mirrors together? Two uncharged, parallel conducting plates a gap d apart attract with a pressure of π²ħc/(240 d⁴), about 1.3 millipascals at 1 µm. Hendrik Casimir predicted it in 1948, and Lamoreaux measured it in 1997. Whether it shows vacuum energy is debated, since Jaffe derived it without that picture in 2005. A 3" pixel-art sticker; equation, symbols and sources in the physics panel below.

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Size3″ × 3″

3″ × 3″ kiss-cut sticker.

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The physics

Quantum vacuum fluctuations between two close, parallel, uncharged metal plates are restricted, so the plates are pulled together. The pressure falls off as one over the fourth power of their separation.

Casimira stands between two uncharged mirrors and braces them apart, because they pull toward each other.

For ideal parallel conducting plates a gap d apart, the attraction per unit area is π²ħc/(240 d⁴). At d = 1 µm that is about 1.3 mPa; at 100 nm it is 10,000 times larger.

On the sticker the cyan wiggles outside are field modes of every wavelength; between the plates only modes that fit the gap survive (amber). The magenta arrows show the net push inward.

The force has been measured, starting with Lamoreaux in 1997. Calling it a push from vacuum energy is the popular picture; Jaffe (2005) showed it can also be derived without it, so the interpretation is debated.

Nothing pushes in.

Equations

\[\frac{F}{A} = -\frac{\pi^2\hbar c}{240\,d^4}\]
SymbolMeaningUnit
\(F/A\) force per unit area between the plates (negative = attractive) \(\mathrm{Pa}\)
\(\hbar\) reduced Planck constant \(\mathrm{J s}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(d\) gap between the plates \(\mathrm{m}\)

Sources

  1. Casimir, H. B. G. (1948) On the attraction between two perfectly conducting plates, Proc. K. Ned. Akad. Wet. 51, 793
  2. Lamoreaux, S. K. (1997) Demonstration of the Casimir force in the 0.6 to 6 µm range, Phys. Rev. Lett. 78, 5
  3. Jaffe, R. L. (2005) Casimir effect and the quantum vacuum, Phys. Rev. D 72, 021301
  4. Wikipedia, Casimir effect

Source Casimir, H. B. G. (1948). On the attraction between two perfectly conducting plates. Proceedings of the Section of Sciences, Koninklijke Nederlandsche Akademie van Wetenschappen, 51, 793–795.

Field notes

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