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This force is only measurable when the distance between the two plates is extremely small, on the order of several atomic diameters. This attraction is called the Casimir effect.

The Casimir effect is caused by the fact that space is filled with vacuum fluctuations, virtual particle-antiparticle pairs that continually form out of nothing and then vanish back into nothing an instant later. Alternatively, the Casimir energy (and force) may be calculated from the zero-point energy of the Fourier modes of the electromagnetic field between the plates. The gap between the two plates restricts the range of wavelengths possible for these virtual particles, and so fewer of them are present within this space. This results in a lower energy density between the two plates than is present in open space; in essence, there is less than "nothing" between the two plates, creating negative energy and pressure, which pulls the plates together.

The narrower the gap, the more restricted the wavelength of the virtual particles, the more negative the energy and pressure, the more restricted the vacuum modes and the smaller the vacuum energy density, and thus the stronger is the attractive force.

Similarly, fluctuations in the electronic structure of molecules cause transient dipoles which lead to the Van der Waals forceIn chemistry, the term van der Waals force originally referred to all forms of intermolecular forces; however, in modern usage it tends to refer only to London forces: those forces which arise from induced rather than permanent dipoles. The forces are nam.

The Casimir effect has recently been measured by Steve K. Lamoreaux of Los Alamos National LaboratoryLos Alamos National Laboratory (LANL is a United States Department of Energy (DOE) national laboratory, managed by the University of California, located in Los Alamos, New Mexico. The Laboratory is one of the largest multidisciplinary institutions in the and by Umar Mohideen of the University of California at Riverside and his colleague Anushree Roy .

The Casimir force per unit area for idealized, perfectly conducting plates with vacuum between them is

where

(hbar, ℏ) is the reduced Planck constant (sometimes known as the Dirac constant),

is the speed of lightCherenkov effect in a "swimming pool" nuclear reactor. The effect is due to electrons moving faster than the speed at which light moves in water. The speed of light (denoted as c reputedly from the Latin celeritas "speed", and also known as Einstein's con,

is Archimedes's constantOr pi with a lower-case p . minuscule, or lower-case, pi The mathematical constant (written as pi when the Greek letter is not available) is ubiquitous in mathematics and physics. In Euclidean plane geometry, π may be defined as either the ratio of a c, the ratio of the circumference of a circle to its diameter, and

is the distanceFor distance between people, see proxemics. Distance between two points The distance between two points is the length of a straight line between them. In the case of two locations on Earth, usually the distance along the surface is meant: either " as the between the two plates.

This shows that the Casimir force per unit area is very small.

The calculation shows that the force happens to be proportional to the sum where the numbers represent the frequencies of standing waveA standing wave also known as a stationary wave is a wave that remains in a constant position. This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interferences between the plates; each possible standing wave behaves as a quantum harmonic oscillator whose ground state energy equal to contributes to the total potential energy; the force then equals minus the derivative of the potential energy with respect to the distance.

The series (the sum of integers) is divergent and needs to be regularized. A useful tool is provided by the Riemann zeta function because the sum can be formally written as which equals . Although it may sound strange (and even though more rigorous ways to obtain the same result exist), the correct result for the sum of positive integers is . The same sum also appears in string theory.

It has since been shown (see Ref.1) that, with materials of certain permittivity and permeability, the Casimir effect can be repulsive instead of attractive.