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 $\tilde{G}_{Al-W}/\tilde{G}_{Cu-W} -1$ and $\tilde{G}_{Cu-W}/G_{N} -1$ were $(0.9 \pm 1.1_{\mathrm{sta}} \pm 4.8_{\mathrm{sys}}) \times 10^{-2}$ and $ (0.2 \pm 0.9_{\mathrm{sta}} \pm 2.1_{\mathrm{sys}}) \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{cm})=(0.9 \pm 1.1_{\mathrm{sta}} \pm 4.9_{\mathrm{sys}}) \times 10^{-2} $ at the shortest range of around 1 cm was also obtained.