Question

A tank of height 33.25 is completely filled with liquid $(\mu =1.33)$. An object is placed at the bottom of the tank on the axis of the concave mirror as shown in the figure. Image of the object is formed 25 cm below the surface of the liquid, then focal length of the mirror is

A.10 cm
B.15 cm
C.20 cm
D.25 cm

Answer 1

Firstly, you need to calculate the distance of image from the mirror. Then, calculate the distance of the object from the mirror. Then use the relation between object distance, image distance and focal length. By substituting the values in it, you can get the focal length of the mirror.
Formula used:
$\dfrac{1}{v} + \dfrac{1}{u}= \dfrac{1}{f}$

Complete answer:
We know the relation,
${D}_{app}= \dfrac {{D}_{real}}{\mu}$ ...(1)
Where, ${D}_{app}$ is the apparent depth
${D}_{real}$ is the real depth
Substituting values in above equation we get,
${D}_{app}= \dfrac{33.25}{1.33}$
$\therefore {D}_{app}= 25 cm$
Thus, the image of the object is at a depth of 25 cm.
Therefore, the distance of the object from the mirror is 25 +15 cm= 40 cm.
Depth of the final image is 25 cm
Thus, substituting this value in the equation. (1) we get,
${D}_{app}= \dfrac{25}{1.33}$
$\therefore {D}_{app}= 18.80 cm$
Thus, the image of the object is at a depth of 18.80 cm.
Therefore, the distance of the image from the mirror is 18.80 +15 cm= 33.80cm.
We know, Lens formula for mirror is given by,
$\dfrac{1}{v} + \dfrac{1}{u}= \dfrac{1}{f}$
Where, u is the distance of object
v is the distance of image
f is the focal length
Substituting the values in above equation we get,
$\dfrac{1}{-33.80} + \dfrac{1}{-40}= \dfrac{1}{f}$
$\therefore \dfrac{1}{f}= -0.0296 - 0.025$
$\therefore \dfrac{1}{f}= -0.0546$
$\therefore f= -18.32$
Thus, the focal length of the mirror is approximately 20 cm.

Hence, the correct answer is option C i.e. 20 cm.

Note:
Idea of Lens equation or mirror formula is essential for such type of problems. Students must know the relation between focal length, object distance, image distance and radius of curvature of every lens and mirror.
Remember focal length of a concave mirror is always taken positive while the focal length of a convex mirror is always considered to be negative. Focal length of a plane mirror is infinity. If the object image is formed beyond the focal length then the concave mirror will produce only a virtual image.