Question

A $10mm$ long awl pin is placed vertically in front of a concave mirror. A $5mm$ long image of the awl pin is formed at $30cm$in front of the mirror. The focal length of this mirror is

• A. $–30cm$
• B. $–20cm$
• C. $–40cm$
• D. $–60cm$

Answer 1

The correct option is B

$–20cm$

Step 1: Given Data

Object height $h_o=10mm$

Image height $h_i=5mm$

Image distance $v=-30cm$

Let the focal length be $f$ and object distance be $u$.

Step 2: Formula Used

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

Step 3: Diagram

Step 3: Calculate the Object Distance

We know that, magnification

$$\begin{gathered} m=\frac{h_i}{h_o}=\frac{v}{u} \\ \Rightarrow \frac{5mm}{10mm}=\frac{-30cm}{u} \\ \Rightarrow \frac{1}{2}=\frac{-30cm}{u} \\ \Rightarrow u=\left(-30\right)\times 2 \\ =-60cm \end{gathered}$$

Step 5: Calculate the Focal Length

We know the formula,

$$\begin{gathered} \frac{1}{f}=\frac{1}{v}+\frac{1}{u} \\ \Rightarrow \frac{1}{f}=\frac{1}{-30}+\frac{1}{-60} \\ \Rightarrow \frac{1}{f}=\frac{-2-1}{60} \\ \Rightarrow \frac{1}{f}=-\frac{1}{20} \\ \Rightarrow f=-20cm \end{gathered}$$

Hence, the correct answer is option (B).