13.2.23. A thin beam of light passing through a hemisphere of glass with a refractive index of $n$ is collected at a distance of $x$ from the convex surface. At what distance from the flat surface of the hemisphere will the rays gather if the beam is launched from the reverse side?
Solution
For problem $13.2.23$
When light rays parallel to the principal axis are incident on the flat surface of the hemisphere, its thickness is irrelevant, and we may treat it as a thin lens with one flat surface and one convex surface of radius $R$ and with focal length $x$. By the lens maker formula, $1/x=(n-1)/R$, so $R=(n-1)x$. Consequently, when light rays parallel to the principal axis are incident on the convex surface of the thin lens, they converge to Point $A$ which is at the distance $x$ from the convex surface. If a sufficiently long piece of glass is placed next to the flat surface of the thin lens, these light rays will be refracted to Point $B$ which is at the distance $nx$ from the convex surface of the thin lens. Clearly, $nx>R$. Thus, if a piece of glass of length $R$ is placed next to the flat surface of the thin lens instead, light rays leaving the glass will be refracted to Point $F$ which is at the distance $OB/n=(nx-R)/n=x/n$ from the left surface of the glass.
