Short-range test of the universality of gravitational constant $G$ at the millimeter scale using a digital image sensor

The composition dependence of gravitational constant $G$ is measured at the millimeter scale to test the weak equivalence principle, which may be violated at short range through new Yukawa interactions such as the dilaton exchange force. A torsion balance on a turning table with two identical tungsten targets surrounded by two different attractor materials (copper and aluminum) is used to measure gravitational torque by means of digital measurements of a position sensor. Values of the ratios ${\tilde{G}}_{Al-W}\mathrm{/}{\tilde{G}}_{Cu-W}-1$ and ${\tilde{G}}_{Cu-W}\mathrm{/}{G}_N-1$ were $(0.9 \pm 1.1_{$\mathrm{s}\mathrm{t}\mathrm{a}$} \pm 4.8_{$\mathrm{s}\mathrm{y}\mathrm{s}$}) \times 10^{-2}$and$ (0.2 \pm 0.9_{$\mathrm{s}\mathrm{t}\mathrm{a}$} \pm 2.1_{$\mathrm{s}\mathrm{y}\mathrm{s}$}) \times 10^{-2}$ , respectively; these were obtained at a center to center separation of 1.7 cm and surface to surface separation of 4.5 mm between target and attractor, which is consistent with the universality of $G$. A weak equivalence principle (WEP) violation parameter of $\eta_{Al-Cu}(r\sim 1: $\mathrm{c}\mathrm{m}$)=(0.9 \pm 1.1_{$\mathrm{s}\mathrm{t}\mathrm{a}$} \pm 4.9_{$\mathrm{s}\mathrm{y}\mathrm{s}$}) \times 10^{-2} $ at the shortest range of around 1 cm was also obtained.