Правка разделов «Statement», «Answer»
en/13.2.23.md
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| ### Statement | |||
| − | $13.2.23.$ [Insert the problem statement] | ||
| + | $13.2.23.$ 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 | |||
|  | |||
| 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. | |||
| @@ -10,4 +10,4 @@Solution | |||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | $x/n$ | ||
| @@ -1,6 +1,6 @@ | |||
| ### Statement | ### Statement | ||
| $13.2.23.$ [Insert the problem statement] | $13.2.23.$ 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 | ### Solution | ||
|  |  | ||
| 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. | 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. | ||
| @@ -10,4 +10,4 @@Solution | |||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | $x/n$ | ||