Casimir Ideal-Plate Scaling

Worked Example

Consider two ideal, perfectly conducting parallel plates separated by a gap of a = 1 μm with an area of A = 4 mm². Let's calculate the attractive Casimir pressure and force. We use the frozen constants ħ = 1.054571817 × 10⁻³⁴ J·s and c = 299792458 m/s.

a = 1 × 10⁻⁶ m A = 4 × 10⁻⁶ m² ħc = (1.054571817 × 10⁻³⁴) * 299792458 ≈ 3.1615264 × 10⁻²⁶ J·m P = - (π² ħ c) / (240 a⁴) P = - (π² * 3.1615264 × 10⁻²⁶) / (240 * (10⁻⁶)⁴) P = - (31.1989 * 3.1615264 × 10⁻²⁶) / (240 × 10⁻²⁴) P ≈ -0.00130012577 Pa F = P * A F = -0.00130012577 * (4 × 10⁻⁶) F ≈ -5.2005 × 10⁻⁹ N

The calculated attractive pressure is approximately -0.00130012577 Pa and the resulting force is -5.2005 × 10⁻⁹ N.

Experiment

In the interactive module, you can investigate the Casimir scaling behavior by adjusting the physical parameters. Follow these steps:

1. Select a value from the "Gap micrometers" dropdown to set the distance a between the plates. Notice how the pressure scales as 1/a⁴.
2. Select a value from the "Area square millimeters" dropdown to set the surface area A. This will linearly scale the total force and interaction energy.
3. Observe the signed energy curve and the attractive pressure derivative on the diagram to see how changes to these controls directly impact the interaction.

Try the interactive Casimir Ideal-Plate Scaling module to visualize these calculations.

Limitations

This model relies on an ideal approximation that assumes perfectly conducting plates at absolute zero temperature (T = 0 K). In real physical systems, several factors significantly alter the interaction:

• Finite Conductivity: Real metals have finite conductivity, which reduces the Casimir force compared to the ideal case, especially at shorter gaps.
• Temperature Corrections: At non-zero temperatures, thermal fluctuations add a contribution that becomes dominant at large separations (typically a > 1 μ m).
• Surface Roughness: Microscopic irregularities on the plate surfaces can lead to corrections that are significant when the gap is small.
• Edge Effects: The derivation assumes infinite parallel plates; finite lateral dimensions introduce edge effects that are omitted here.

Crucially, the theoretical formulas are valid only for a gap a much smaller than the lateral dimensions and strictly when a > 0. The equations diverge at a = 0, representing an unphysical state, and this model must never be extended to a = 0 or be interpreted as usable free energy.

References

The theoretical derivation and physical context of the Casimir effect are described in detail in educational literature. For the underlying principles used in this model, see the seminar notes on the Casimir effect:

• Casimir Effect (MIT) - Overview of the vacuum attraction between ideal conductors.