Правка разделов «Statement», «Solution», «Answer»
en/13.3.24.md
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| ### Statement | |||
| − | $13.3.24.$ [Insert the problem statement] | ||
| + | $13.3.24.$ A concave spherical mirror of radius $R$ gives an image of the source that coincides with the source itself. When some liquid was poured into the mirror, a second image appeared between the source and the mirror, located at a distance $l$ from the source, $l<R$. Find the refractive index of the liquid. | ||
| ### Solution | |||
| − | Originally, the source and its image are at the same position, which means that the source is at the center of curvature of the concave mirror. After some liquid is poured | ||
| + | Originally, the source and its image are at the same position, which means that the source is at the center of curvature of the concave mirror. After some liquid is poured into the mirror, it becomes a thin convex lens with one flat surface whose focal length is $R/(n-1)$, where $n$ is the refractive index of the liquid. A new image is formed after light rays from the source are refracted by the lens, reflected by the mirror, and refracted by the lens again. Thus the focal length $f$ of the compound lens-mirror-lens satisfies the equation | ||
| \[\frac{1}{f}=\frac{n-1}{R}+\frac{2}{R}+\frac{n-1}{R}=\frac{2n}{R}.\] | |||
| Since the new image is at the distance $l$ from the source (towards the mirror), we have | |||
| \[\frac{1}{f}=\frac{1}{R}+\frac{1}{R-l}=\frac{2R-l}{R(R-l)},\] | |||
| − | so $n=(2R-l)/[2(R-l)]$ | ||
| + | so $n=(2R-l)/[2(R-l)]$. | ||
| #### Answer | |||
| − | [Insert a concise answer or boxed result] | ||
| + | $\frac{2R-l}{2(R-l)}$ | ||
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| ### Statement | ### Statement | ||
| $13.3.24.$ [Insert the problem statement] | $13.3.24.$ A concave spherical mirror of radius $R$ gives an image of the source that coincides with the source itself. When some liquid was poured into the mirror, a second image appeared between the source and the mirror, located at a distance $l$ from the source, $l<R$. Find the refractive index of the liquid. | ||
| ### Solution | ### Solution | ||
| Originally, the source and its image are at the same position, which means that the source is at the center of curvature of the concave mirror. After some liquid is poured |
Originally, the source and its image are at the same position, which means that the source is at the center of curvature of the concave mirror. After some liquid is poured into the mirror, it becomes a thin convex lens with one flat surface whose focal length is $R/(n-1)$, where $n$ is the refractive index of the liquid. A new image is formed after light rays from the source are refracted by the lens, reflected by the mirror, and refracted by the lens again. Thus the focal length $f$ of the compound lens-mirror-lens satisfies the equation | ||
| \[\frac{1}{f}=\frac{n-1}{R}+\frac{2}{R}+\frac{n-1}{R}=\frac{2n}{R}.\] | \[\frac{1}{f}=\frac{n-1}{R}+\frac{2}{R}+\frac{n-1}{R}=\frac{2n}{R}.\] | ||
| Since the new image is at the distance $l$ from the source (towards the mirror), we have | Since the new image is at the distance $l$ from the source (towards the mirror), we have | ||
| \[\frac{1}{f}=\frac{1}{R}+\frac{1}{R-l}=\frac{2R-l}{R(R-l)},\] | \[\frac{1}{f}=\frac{1}{R}+\frac{1}{R-l}=\frac{2R-l}{R(R-l)},\] | ||
| so $n=(2R-l)/[2(R-l)]$ | so $n=(2R-l)/[2(R-l)]$. | ||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | $\frac{2R-l}{2(R-l)}$ | ||