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

Since the image is real and is formed in front of the mirror,

$v=-30cm$

Height of the object $h_o=10mm=1cm$

Height of the image $h_i=5mm=0.5cm$

Step 2: Calculate the Image Distance

We know that for a concave mirror magnification is given as,

$m=\frac{h_i}{h_o}=-\frac{v}{u}$

Upon substituting the values we get,

$\frac{0.5}{1}=-\frac{\left(-30\right)}{u}$

$\Rightarrow u=\frac{30}{0.5}=60$

Since, the object is in front of the mirror,

$u=-60cm$

Step 3: Calculate the Focal Length

According to the mirror formula, the focal length is given as

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

Upon substituting the values we get,

$\frac{1}{f}=\frac{1}{-30}+\frac{1}{-60}$

$=-\left(\frac{2+1}{60}\right)$

$=-\frac{3}{60}=-\frac{1}{20}$

$\therefore f=-20cm$

Hence, the correct answer is option (B).