Question #43893

A large tank is filled with a transparent liquid of density $\rho$, surface tension $T$, and refractive index $n$. A narrow vertical capillary tube of inner radius $r$ is connected to the bottom of the tank and bends upwards, forming a U-tube system. The open top of the capillary tube is at a vertical distance $H$ below the free surface of the liquid in the large tank. The liquid bulges out of the top of the capillary tube, forming a convex spherical meniscus pinned at the tube's edge. A small luminous object is placed inside the capillary tube at a depth $d$ below the pole of the meniscus. When viewed from above along the axis of the tube, the light rays emerging from the meniscus into the air are parallel. Assuming $H < \frac{2T}{r \rho g}$ and all quantities are in consistent SI units, what is the required depth $d$?