6. (10 points) A beam of light with a diameter of $d_{1}=5 \mathrm{~cm}$ falls on a thin diverging lens with an optical power of $D_{p}=-6$ Diopters. On a screen positioned parallel to the lens, a bright spot with a diameter of $d_{2}=20 \mathrm{~cm}$ is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens.
# Answer: 10 Dptr

Solution. The optical scheme corresponding to the condition:

(3 points)

![](https://cdn.mathpix.com/cropped/2024_05_06_be79a21ed2ca3a6294abg-10.jpg?height=825&width=1170&top_left_y=73&top_left_x=520)

The path of the rays after the diverging lens is shown in black, and after the converging lens in red. We have $C D=\left|\frac{1}{D_{p}}\right|=\frac{1}{6}$.

From the similarity of triangles, it follows that: $\frac{C D}{C F}=\frac{B K}{A L}=\frac{1}{4}$,

then $D F=C F-C D=4 C D-C D=3 C D=\frac{1}{2}$.

From the similarity of triangles, it follows that: $\frac{D E}{F E}=\frac{B K}{A L}$.

We get that: $D E=\frac{1}{5} D F=\frac{1}{10}$.

As a result, the optical power of the converging lens: $D_{c}=\frac{1}{D E}=10$ Dptr.

(2 points)