Question

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

• A. $-30\text{cm}$
• B. $-20\text{cm}$
• C. $-40\text{cm}$
• D. $-60\text{cm}$

Answer 1

The correct option is B $-20\text{cm}$

Given :
Object size $\left(h_0\right)=10\text{mm}$,
Image size $\left(h_i\right)=-5\text{mm}$,
Image distance $\left(v\right)=-30\text{cm}$

From the magnification formula
$m=\frac{h_i}{h_0}=\frac{-v}{u}$

Substituting the given data we get,

$\frac{-5}{10}=\frac{+30}{u}\Rightarrow u=-60\text{cm}$

Now, focal length can be calculated as follows:

From mirror formula we know that,
$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

Substituting the data we get,

$\frac{1}{-30}+\frac{1}{-60}=\frac{1}{f}\Rightarrow \frac{-2-1}{60}=\frac{1}{f}$

$\Rightarrow f=-20\text{cm}$

[The negative sign confirms that the calculated focal length is of a concave mirror]

Hence, option (b) is the correct answer.