# Casimir effect

In 1948 Dutch physicist Hendrik B. G. Casimir of Philips Research Labs predicted that two uncharged parallel metal plates will be subject to a force pressing them together. 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**. It is related to Van der Waals force.

## Explanation

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. 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. As a result, there is a lower energy density between the two plates than in open space; in essence, there are fewer virtual particles between the two plates than on the other side of them, creating a pressure difference which some mistakenly call negative energy but which is actually only due to a higher virtual particle pressure outside the plates than between them, which pulls the plates together. The narrower the gap, the more restricted the wavelength of the virtual particles, the larger the pressure difference between the outside and inside of the plates, the more restricted the vacuum modes and the smaller the vacuum energy density, and thus the stronger the attractive force. A more complete explanation can be found at PhysicsWeb.

Since the Casimir effect is small and falls off as the fourth power of distance, its effect is greatest on small objects that are close together. It can become an important consideration in studies of the interaction of molecules, together with other small scale effects, such as the fluctuations in the electronic structure of molecules causing transient dipoles which lead to Van der Waals forces.

## Analogies

A similar analysis can be used to explain Hawking radiation that causes the slow "evaporation" of black holes (although this is generally explained as the escape of one particle from a virtual particle-antiparticle pair, the other particle having been captured by the black hole).

An effect analogous to the Casimir effect was observed by 18th century French sailors. Where two ships are rocking from side to side in conditions with a strong swell but light wind, and the ships come closer together than roughly 40 m, destructive interference eliminates the swell between the ships. The calm sea between the ships has a lower energy density than the swell to either side of the ships, creating a pressure that can push the ships closer together. If they get too close together, the ships' rigging can become entangled. As a countermeasure, a handbook from the early 1800s recommends that each ship should send out a boat rowed by 10 to 20 sailors to physically pull the ships apart.

## Calculation

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 Casimir force per unit area **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_c / A}**
for idealized, perfectly conducting plates with vacuum between them is

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {F_c \over A} = {\hbar c \pi^2 \over 240 d^4}}**

where

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar}**(hbar, ℏ) is the reduced Planck constant (sometimes known as the Dirac constant),**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}**is the speed of light,**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d}**is the distance between the two plates.

This shows that the Casimir force per unit area **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_c / A}**
is very small.

The calculation shows that the force happens to be proportional to the sum **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+2+3+4+5+\dots}**
where the numbers **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,2,3,4,5,\dots}**
represent the frequencies of standing waves between the plates; each possible standing wave behaves as a quantum harmonic oscillator whose ground state energy equal to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar\omega/2}**
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 renormalized. A useful tool is provided by the Riemann zeta function because the sum can be formally written as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \zeta(-1)}**
which equals **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1/12}**
. 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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1/12}**
. The same sum also appears in string theory.

## Measurement

The Casimir effect was measured in 1997 by Steve K. Lamoreaux of Los Alamos National Laboratory and by Umar Mohideen of the University of California at Riverside and his colleague Anushree Roy. In practice, rather than using two parallel plates, which would require phenomenally accurate alignment to ensure they were parallel, the experiments use one plate that is flat and another plate that is a part of a sphere with a large radius of curvature.

Further research has shown that, with materials of certain permittivity and permeability, or with a certain configuration, the Casimir effect can be repulsive instead of attractive.

## References

- A. Lambrecht,
*The Casimir effect: a force from nothing*, Physics World, September 2002 - M. Bordag, U. Mohideen, V.M. Mostepanenko,
*New Developments in the Casimir Effect*, quant-ph/0106045 (275 page review paper) *Repulsive Casimir forces*, O. Kenneth, I. Klich, A. Mann and M. Revzen, Department of Physics, Technion - Israel Institute of Technology, Haifa, February 2002- S. K. Lamoreaux, "Demonstration of the Casimir Force in the 0.6 to 6 µm Range",
*Phys. Rev. Lett.*78, 5–8 (1997)

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