Question

A concave spherical mirror is produced 18.00mm inverted image of object of object 9.00mm tall and distance between object and the mirror is 15.0cm. find radius of curvature of the mirror. 

Answer 1

First, let's find the focal length of the mirror. Starting from the mirror and lens equation:

$$\dfrac{1}{d_o} + \dfrac{1}{d_i} = \dfrac{1}{f}$$

where $d_o = 0.15m$ is the distance between object and the mirror, $d_i$ is the distance between the image and the mirror and $f$ is the focal length of the mirror, we can express $f$:

$$f = \dfrac{d_od_i}{d_o+d_i}$$

To find the $d_i$ we can use the espression for the magnification:

$$m = \dfrac{h_i}{h_o} = \dfrac{d_i}{d_o}$$

where $h_i = 18\times10^{-3}m$ is the sizo of the image and $h_o = 9\times 10^{-3}m$ is the size of the object.

Thus:

$$d_i = \dfrac{h_id_o}{h_o} = 0.3m$$

and the focal length:

$$f = \dfrac{0.15\cdot0.3}{0.15+0.3} = 0.1m$$

The radius of curvature of the mirror is equal to:

$$R = 2f = 2\cdot 0.1 = 0.2m$$

Answer. 0.2 m.