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Casimira

The Casimir Effect Pixel

\[\frac{F}{A} = -\frac{\pi^2\hbar c}{240\,d^4}\]
$4.99 USD
$4.99 USD
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Casimira, an armoured knight-mage, presses two upright plates together with a sliver of gap between them while violet vacuum light crackles all around.

The Physics

Put two uncharged, perfectly conducting parallel plates a small distance a apart in vacuum and they attract. Between the plates only electromagnetic modes that fit the boundary conditions are allowed; outside, every mode is. Summing the zero-point energies ½ħω of the modes, and subtracting the free-space value, leaves a finite energy that falls as the plates approach. Hendrik Casimir derived it in 1948.

P = π²ħc / (240 a⁴)

  • P: attractive pressure, force per unit plate area (Pa)
  • ħ: reduced Planck constant, h/2π = 1.054 571 817 × 10⁻³⁴ J s
  • c: speed of light, 299 792 458 m s⁻¹
  • a: gap between the plates (m)
  • π²/240: numerical factor from summing the allowed modes

At a = 1 µm, P ≈ 1.3 × 10⁻³ Pa (1.3 mPa); halve the gap and the pressure rises sixteen-fold. S. K. Lamoreaux first measured the force accurately in 1997, using a sphere and a flat plate (Phys. Rev. Lett. 78, 5); Bressi and colleagues measured it between true parallel plates in 2002 (Phys. Rev. Lett. 88, 041804). Source for the theory: H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948).

Honest caveat: The formula is for ideal mirrors at zero temperature; real metals, roughness and temperature change it. Whether the force “proves” vacuum energy is debated: R. L. Jaffe (Phys. Rev. D 72, 021301, 2005) showed it can be derived as a relativistic van der Waals force between the plates’ charges, without reference to zero-point energy.

In character

Leave me nothing but a gap and I will still push. The emptiness outside is fuller than the emptiness within, and that difference is my strength.

On the sticker

Printed text: CASIMIRA with the line P = π²ħc/(240a⁴). Pixel-art kiss-cut vinyl sticker in 2″, 3″, 4″ and 6″ sizes. See more physicist stickers or quantum designs.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size3″ × 3″

3″ × 3″ kiss-cut sticker.

Shipping

Shipping calculated at checkout by destination. Printed to order for you.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

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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