Question

An object is placed at a distance of $25\mathrm{cm}$ from a spherical mirror and its image is formed behind the mirror at a distance of $5\mathrm{cm}$. Find focal length. Is it a concave or convex mirror?

Answer 1

Step 1: Given data:

Object distance($\mathrm{u}$) = $-25\mathrm{cm}$

Image distance($\mathrm{v}$) = $+5\mathrm{cm}$

(Note: Sign convention:

1. All the distances taken in front of a mirror that is on the left of the pole mirror are negative($-$) and the distances taken on the right of the pole or behind the mirror are positive($+$).
2. Therefore the object distance($\mathrm{u}$) is taken as negative.
3. And the image distance($\mathrm{v}$) is taken as positive as it is formed behind the mirror.)

Step 2: Formula used:

1. Object distance($\mathrm{u}$), image distance($\mathrm{v}$) and focal length($\mathrm{f}$) of a mirror are related as:

$\frac{1}{\mathrm{v}}+\frac{1}{\mathrm{u}}=\frac{1}{\mathrm{f}}$

Step 3: Calculation of focal length($\mathrm{f}$) of the mirror:

$$\begin{gathered} \frac{1}{5}+\frac{1}{-25\mathrm{cm}}=\frac{1}{\mathrm{f}} \\ \frac{1}{5}-\frac{1}{25}=\frac{1}{\mathrm{f}} \\ \mathrm{Or},\frac{1}{\mathrm{f}}=\frac{1}{5}-\frac{1}{25} \\ \frac{1}{\mathrm{f}}=\frac{5-1}{25} \\ \frac{1}{\mathrm{v}}=\frac{4}{25} \\ \mathrm{Or}, \\ \mathrm{v}=+\frac{25}{4}\mathrm{cm} \\ \mathrm{v}=+6.25\mathrm{cm} \end{gathered}$$

Thus, the focal length of the spherical mirror will be $6.25\mathrm{cm}$.

Since the focal length is positive therefore the mirror is a convex mirror in this case.