The Thermal Fork in Sphere-Plate Casimir Measurements: Two Material Models and the Size of Their Disagreement

Precision Casimir experiments do not use parallel plates, which cannot be kept parallel well enough; they measure the force between a sphere and a plate and convert with the proximity-force approximation. For a hundred-micrometre sphere the force at one micrometre is 2.723e-13 newtons, comfortably measurable, but at the wide separations where theory disagreements live it falls steeply. The disagreement is thermal: at room temperature, two standard treatments of the metal's low-frequency response, the Drude model and the plasma model, predict forces differing by a fraction that grows with separation, reaching tens of percent near the thermal wavelength of 7.63 micrometres. This paper computes both the force and the fork across five separations, showing that the absolute difference the experiments must resolve peaks in the few-micrometre range. The episode is a useful discipline for vacuum accounting generally: here is a vacuum prediction that is uncertain at the tens-of-percent level for stated material reasons, beside a gravitational remainder known only within orders of magnitude. Computed from stated models, not measured.