Vacuum modes matter
Only standing modes that fit between the plates contribute. The mismatch with free space modes gives the net attractive pressure.
Boundary conditions
Casimir Effect Force Calculator
The weird vacuum pressure: F = π²ħcA / (240 a⁴). Pick your plate spacing a and area A to see the attraction between perfect conductors.
Inputs
Results
Force magnitude
—
Attractive; |F| = π²ħcA / (240 a⁴)
Pressure
—
Equivalent to F / A
Atmospheres
—
Pressure / 1 atm
Weight equivalent
—
Force / g, shown in grams
Scaling vs a⁻⁴
—
Casimir quick facts
a⁻⁴ is brutal
Double the gap → 16× weaker force. Going from 100 nm to 1 μm reduces force by 10,000×.
Scaling
Real plates differ
The ideal formula assumes perfect conductors at 0 K. Real surfaces, roughness, finite conductivity, and temperature change the numbers.
Reality check
It was measured in 1997
Lamoreaux’s torsion pendulum experiment nailed the effect within a few percent, confirming the vacuum-fluctuation prediction.
Experiment
Repulsion is possible
In special materials or with a fluid in between, geometry and dielectric contrasts can flip the sign, giving a repulsive Casimir force.
Repulsive cases
How this is calculated
- Formula: F = π²ħcA / (240 a⁴). Pressure is the same without A. We report the magnitude (attractive).
- Units: Separation converts to meters. Area converts to m². Force is Newtons; pressure is Pascals.
- Scaling bar: We show your pressure as a fraction of the pressure at 100 nm. Handy for seeing how quickly it fades.
- Assumptions: Perfectly parallel, perfectly conducting plates in vacuum at 0 K. Real experiments can differ by tens of percent.
The Casimir effect is one of the wildest demonstrations of vacuum fluctuations—you are calculating energy in “empty” space pushing metal together.
Why the Casimir effect matters
The Casimir effect connects abstract quantum field theory to a tangible, measurable force. Intro courses teach that the vacuum teems with zero-point fluctuations, but that idea feels hand-wavy until you see that changing boundary conditions—two parallel plates—actually changes the vacuum energy and produces pressure you can measure with a torsion balance. This calculator keeps the math simple (π²ħc/240a⁴) yet lets learners feel the brutality of the a⁻⁴ dependence: doubling a 100 nm gap cuts the pressure by 16×. That single knob drives home how strongly short-range phenomena scale and why nanofabrication tolerances matter in MEMS design.
Educators can also use the pressure-in-atmospheres output to anchor intuition. A 100 nm gap over 1 cm² produces a pressure that is tiny compared to 1 atm, but not infinitely tiny—enough to matter in nanoscale actuators. The weight-equivalent readout translates an exotic quantum force into grams, an everyday unit students can grasp. By toggling between 100 nm and 1 μm presets, classes can see a 10,000× drop without touching a single equation, reinforcing how physical effects often depend on higher powers than the linear or inverse-square scaling they know from classical mechanics and electromagnetism.
The tool also sparks discussions about idealizations. It explicitly notes that real metals, temperature, and surface roughness shift the result. That opens a window into finite-temperature Casimir-Lifshitz theory, dielectric contrasts that can even yield repulsion, and experimental finesse—like Lamoreaux’s 1997 measurement—that turned a thought experiment into data. Because the calculator is client-side, students can safely explore without signups or telemetry, making it suitable for homework, outreach demonstrations, or lab pre-reads. As a bridge between textbook vacuum fluctuations and practical nanoscale engineering, it delivers both conceptual clarity and a visceral sense of scale.