Yukawa Corrections to the Newtonian Gravitational Potential: Finite Size Effects in a Recent Experiment

Abstract

We provide a formalism to calculate the effect of the finite size of the sample on hypothetical Yukawa-like corrections to the Newtonian gravitational potential. It is explicitly shown that finite size effect contributions are negligible when the extent of the sample is larger than the range of the Yukawa term. In particular we show that these contributions are small in the configuration of a recent experiment. In the experiment a gold coated sphere was moved across the interface between two materials with different mass densities. In view of these new experimental results, we analyze the situation when the error on the hypothetical correction could result to be significant.

New Constraints on Yukawa-Type Interactions From the Casimir Effect

“…The reason is that the density of Si is smaller that the density of Au deposited on both test bodies in the respective experiments. 20,21,[26][27][28] It is pertinent to note, however, that line 7 in Fig. 3(b) is in a very good qualitative agreement with lines 2 and 3 and, thus, provides confirmation to previously obtained constraints.…”

“…1 we present constraints on the parameters of Yukawa-type interaction (λ, α) following 21,25 from measurements of the normal Casimir force between Au surfaces by means of an AFM 19 (line 1), from measuring the gradient of the Casimir force between similar surfaces (or, equivalently, the Casimir pressure between two Au-coated parallel plates) by means of a micromachined oscillator 20,21 (line 2), and from the Casimir-less experiment, [26][27][28] where the contribution of the Casimir force was made equal to zero (line 3). The constraints for larger λ were obtained 29 from the torsion-pendulum experiment 30 Fig.…”

Stronger Limits on Hypothetical Yukawa Interactions in the 30–8000 nm Range

“…The experiments were performed at P ≃ 10 −5 torr and the motion of the plate was detected by the change in capacitance between the plate and the underlying electrodes as in Refs. [13,16,17]. Calibration of the MTO was performed by using the electrostatic interaction between the Au-coated test and source masses [16].…”

“…(1) over the geometry of the sample is necessary. Disregarding finite size effects across the width of the ring [17], the expected difference when the sphere is over a Au or Si sector is [13,19] where K t ðK s Þ is a term associated only with the layered structure of the test (source) mass, ρ S , ρ Cr , ρ Au , and ρ Si are the sapphire, Cr, Au, and Si densities, respectively. The setup is optimized to select the first harmonic of the force associated with the angular distribution of the sample.…”

“…Gluing the sphere reduced the MTO's natural frequency of oscillation from f 0 = 708.23 ± 0.05 Hz to f r = 307.34 ± 0.05 Hz, and it reduced the oscillator's quality factor from Q ∼ 9000 to Q 7200 for a pressure P ≤ 10 −5 torr. The experiments were performed at P 10 −5 torr and the motion of the plate was detected by the change in capacitance between the plate and the underlying electrodes as in [13,17,18]. Calibration of the MTO was performed by using the electrostatic interaction between the Au-coated test and source masses [17].…”