Respuesta :
Given:
The mirror is concave.
The distance of the image is
[tex]d_0=20\text{ cm}[/tex][tex]d_0=20\text{ cm}[/tex]The image is magnified by a factor of two.
[tex]m=2[/tex]Required: the radius of curvature of the concave mirror.
Explanation:
we know that the magnification is given by
[tex]m=-\frac{d_i}{d_0}[/tex]substitute the value of d0 and m in the above equation, and we get:
[tex]\begin{gathered} 2=-\frac{d_i}{-20.\text{ cm}} \\ d_i=40\text{ cm} \end{gathered}[/tex]now, we will calculate the focal length of the mirror.
the mirror formula is,
[tex]\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_0}[/tex]Plugging all the values in the above, we get:
[tex]\begin{gathered} \frac{1}{f}=\frac{1}{40\text{ cm}}+\frac{1}{20\text{ cm}} \\ \frac{1}{f}=\frac{1+2}{40} \\ f=\frac{40}{3} \\ f=\text{ 13.33 cm} \end{gathered}[/tex]now, we calculate the radius of curvature.
we know that,
[tex]f=\frac{R}{2}[/tex]Substitute the value of f in the above relation, we get:
[tex]\begin{gathered} 13.33\text{ cm=}\frac{R}{2} \\ R=2\times13.33\text{ cm} \\ R=26.66\text{ cm} \end{gathered}[/tex]Thus, the radius of the curvature is
[tex]26.66\text{ cm}[/tex]