Question

A candle flame 3 cm high is placed at a distance of 3 m from a wall. How far from the wall must a concave mirror be placed so that it may form a 9 cm high image of the flame on the same wall? Also find the focal length of the mirror.

Answer 1

Hint: First find the magnification of the mirror using the image and object heights. Using the obtained magnification, find the object distance and image distance. To find the focal length, use the mirror equation by substituting the values of obtained image and object distances.

Formula Used: Magnification m of a mirror is $m = \dfrac{v}{u}$, where v is the image distance and u is the object distance and $m = \dfrac{H}{h}$, where H is the height of the image and h is the height of the object.
Mirror equation is $\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$, where f is the focal length.

Complete step by step answer:We are given that a candle 3 cm high is placed 3 m from the wall and we have to find the distance of candle from a concave mirror so that it forms an image of 9 cm high on the wall.
We also have to calculate the focal length of the mirror.

Magnification of a mirror is the ratio of height of image to the height of object.
The image height (H) is 9 cm and object height (h) is 3 cm.
$
  m = \dfrac{H}{h} \\
   \Rightarrow m = \dfrac{9}{3} \\
  \therefore m = 3 \\
 $
The object distance as we can see in the figure is x m and the image distance is $\left( {3 + x} \right)$ m.
Magnification is also the ratio of image distance to the object distance.
$
  m = \dfrac{v}{u} \\
  v = \left( {3 + x} \right),u = x,m = 3 \\
   \Rightarrow 3 = \dfrac{{3 + x}}{x} \\
   \Rightarrow 3x = 3 + x \\
   \Rightarrow 3x - x = 3 \\
   \Rightarrow 2x = 3 \\
   \Rightarrow x = \dfrac{3}{2} \\
  \therefore x = 1.5m \\
 $
Therefore, the object distance is 1.5 cm which means the concave mirror must be placed at a distance of $3 + x = 3 + 1.5 = 4.5m$ from the wall.
Focal length of the mirror can be calculated using the mirror equation.
$
  \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u} \\
  v = 4.5m,u = 1.5m \\
   \Rightarrow \dfrac{1}{f} = \dfrac{1}{{4.5}} + \dfrac{1}{{1.5}} \\
   \Rightarrow \dfrac{1}{f} = \dfrac{{1 + 3}}{{4.5}} \\
   \Rightarrow f = \dfrac{{4.5}}{4} = 1.125m \\
 $
Focal length of the given concave mirror is 1.125 m, object distance of the candle is 1.5 m and the image distance is 4.5 m.

Note:Do not confuse concave mirrors with convex. The object distance of a convex mirror is negative whereas it is positive for a concave mirror. Concave mirror forms only virtual and upright images if the object is placed in front of the focal point and else real images whereas a convex mirror forms only virtual and erect images.