Image positions of a vertical rod in a liquid-filled cylindrical container

We describe a rather simple optical experiment, which many students can easily carry out, but the theoretical explanation of which requires far-from-simple mathematical analysis and application of numerical methods. A thin vertical rod, partially immersed in a liquid-filled transparent cylindrical container, is moved from the centre toward the wall and observed laterally. As it moves, the part seen through the air and the part seen through the water start to separate, so that at a certain distance from the centre the lower part of the rod becomes invisible. We show that this happens at the distance for which the refracted ray that reaches the observer's eye is along the tangent to the surface of a cylinder. We derive the expression for this distance as a function of the index of refraction and ratio d/r, where d is the distance of the observer from the surface and r is the radius of the container. The locus of image positions is determined by evaluating the intersections of pairs of close rays from the rod which reach the observer's eye after refraction at the surface of the container