Question #43557

An astronomical telescope is modified by replacing its standard objective lens with a solid glass sphere of radius $R = 10 \text{ cm}$ and refractive index $\mu = 1.5$. To collect more light, a thin converging lens $L_1$ of focal length $f_1 = 30 \text{ cm}$ is placed coaxially in front of the sphere such that the center of the sphere is at a distance $d = 20 \text{ cm}$ from $L_1$. Parallel light rays from a distant star enter $L_1$, pass through the glass sphere, and are finally viewed through an eyepiece (a thin converging lens of focal length $f_e = 5 \text{ cm}$). If the entire system is in normal adjustment (final image at infinity), what is the distance between the center of the glass sphere and the eyepiece?