Question

Find the location of the object which when placed in front of a convex mirror produces an image half the size of the object. Given the focal length of a convex mirror is 20 cm. (Hint: Size of the image is half in a convex mirror means $(m=1/2)$).

Answer 1

Hint: In this solution, we will be using the formula for magnification to determine the relationship between the image and the object distance for the situation given to us. Then we will use the mirror formula to solve for the location of the object that will satisfy the conditions given to us.

Formula used : In this solution, we will use the following formula:
Mirror formula: ${\displaystyle \frac{1}{v}}+{\displaystyle \frac{1}{u}}={\displaystyle \frac{1}{f}}$ where $v$ is the image distance, $u$ is the object distance, and $f$ is the focal length of the mirror
Magnification of a mirror: $m=-{\displaystyle \frac{v}{u}}$ .

Complete step by step answer:
We’ve been given that the convex mirror produces an image half the size of the object. So, we can say that
 $m=-{\displaystyle \frac{v}{u}}={\displaystyle \frac{1}{2}}$
Which gives us $2v=-u$
Hence the image is located twice as far away from the mirror as the object and is formed at the opposite side of the mirror.
Now, using the mirror formula, we can write that
 ${\displaystyle \frac{1}{v}}+{\displaystyle \frac{1}{u}}={\displaystyle \frac{1}{f}}$
Or alternatively,
 ${\displaystyle \frac{2}{-u}}+{\displaystyle \frac{1}{u}}={\displaystyle \frac{1}{f}}$
Solving for the object distance by substituting $f=20cm$ , we get
 ${\displaystyle \frac{1}{u}}=-{\displaystyle \frac{1}{20}}cm$
On taking the inverse, we get
 $u=-20cm$
Hence the position of the object which when placed in front of a convex mirror produces an image half the size of the object will be $-20cm$ . The negative sign implies that the object is to be placed on the left of the mirror.

Note:
While using the mirror formula and the formula for magnification, we should be careful in taking the sign convention of the object image and the focal length of the system. Since the object is placed to the left of the mirror, the reflective surface of the mirror should be facing the object and for a convex mirror, the focal length will be positive.