A 2.0 cm high object is placed at a distance of 20 cm from a concave mirror. A real image is formed 40 cm from the mirror. Calculate the focal length of the mirror and the size of the image.

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Solution

Step 1: Given that:
Height of object( $h_{o}$ )= $2 c m$
Object distance( $u$ )= $- 20 c m$
Image distance( $v$ )= $- 40 c m$
( As, the image is real, it means the image is formed in front of the mirror.)
Step 2: Formula used:
Mirror formula is given as;
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
Where, $f$ is the focal length of the mirror.
Magnification of the mirror is given by;
$m = \frac{h_{i}}{h_{o}} = - \frac{v}{u}$
Step 3: Calculation of focal length of the concave mirror:
$\frac{1}{f} = \frac{1}{- 40 c m} + \frac{1}{- 20 c m}$
$\frac{1}{f} = - \frac{1}{40} - \frac{1}{20}$
$\frac{1}{f} = \frac{- 1 - 2}{40}$
$\frac{1}{f} = \frac{- 3}{40}$
$f = \frac{- 40}{3} c m$
Step 4: Calculation of size of the image:
$\frac{h_{i}}{2 c m} = - \frac{- 40 c m}{- 20 c m}$
$h_{i} = - 2 \times 2$
$h_{i} = - 4 c m$
(Negative sign indicates that the image is inverted.)
Hence,
The focal length of the concave mirror= $\frac{40}{3} c m$
Size of the image= $4 c m$

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