Casimira

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.