Engineering & Science
Casimir Effect Quantum Vacuum Pressure (1/d^4) & MEMS Plate Force Formula
Calculate the attractive quantum-electrodynamic Casimir pressure P_C = (PI()^2 * hbar * c) / (240 * d^4) in Pascals and stiction force F_C in microNewtons between parallel uncharged conducting plates separated by d nanometers.
Feel empty space push two uncharged mirrors together with 1/d^4 MEMS stiction!
Place two perfectly uncharged, grounded parallel mirrors a distance d apart inside an ultra-high-vacuum chamber at 0 Kelvin—classical physics predicts ZERO force! Yet Hendrik Casimir (1948) proved that because only discrete standing electromagnetic zero-point modes fit INSIDE the narrow cavity whereas a continuum of vacuum modes pushes from OUTSIDE, quantum vacuum fluctuations exert a pure attractive pressure scaling inversely with the FOURTH POWER of gap separation: P_C = ((PI() ^ 2) * hbar * c) / (240 * (d ^ 4))!
In nanomechanical MEMS/NEMS switches and accelerometers, Casimir attraction causes permanent "pull-in stiction" whenever gaps drop below ~100 nm: at d = 100 nm, Casimir pressure is 13.001 Pa; halving the gap to d = 50 nm multiplies pressure 2^4 = 16-fold to 208.020 Pa; and at d = 10 nm, empty quantum vacuum squeezes the plates together at 130,012.58 Pa (~1.28 atmospheres of pressure from nothingness!)!
| Parallel Plate Gap d (nm) | 1 / d^4 Scaling Jump | Casimir Pressure P_C (Pascals) | Force on 100x100 um^2 Pad (uN) |
|---|---|---|---|
| d = 100 nm (Sub-micron MEMS) | Baseline 1x | 13.001 Pa | 0.1300 uN |
| d = 50 nm (Half Gap) | 2^4 = 16x Jump! | 208.020 Pa | 2.0802 uN |
| d = 10 nm (NEMS Contact) | 10^4 = 10,000x Jump! | 130,012.577 Pa (~1.28 atm!) | 1,300.1258 uN |
Attractive Quantum Vacuum Casimir Pressure Magnitude P_C (Pascals)
Evaluates ((PI()^2) * 1.054571817e-34 * 299792458) / (240 * (d_nm * 10^-9)^4) in Pascals (N/m^2).
=IF(B2 <= 0, "Invalid Gap Separation", ROUND(((PI() ^ 2) * 1.054571817e-34 * 299792458) / (240 * ((B2 * 1e-9) ^ 4)), 3))Attractive Casimir Force F_C in MicroNewtons (uN) on Pad Area B3 (um^2)
Multiplies P_C (N/m^2) by pad area B3 * 10^-12 m^2 and converts Newtons to uN (* 10^6 -> net * 10^-6).
=IF(OR(B2 <= 0, B3 <= 0), "Invalid Gap Separation", ROUND((((PI() ^ 2) * 1.054571817e-34 * 299792458) / (240 * ((B2 * 1e-9) ^ 4))) * B3 * 0.000001, 4))Try the example
Use the original values or change them to test your understanding.