Question

A rod of length $10cm$ lies along the principal axis of a concave mirror of focal length $10cm$ in such a way that its end closer to the pole is $20cm$ away from the mirror. The length of the image is:
(A) $10cm$
(B) $15cm$
(C) $2.5cm$
(D) $5cm$

Answer 1

From the question, we can understand that we need to find the length of the image. This can be done by using the mirror formula. Concave mirror can be used to produce both real and virtual images.

Complete Step-By-Step Solution:
We know using the mirror formula, we can find the image distance, object distance, height of the image and focal length of the mirror and the height of the object provided, and any one of the remaining is unknown.
We know the mirror formula is given as:
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
Where,
$v$ is the image distance
$u$is the object distance
$f$ is the focal length of the mirror.
Since, the object lies in the left of the mirror, object distance is always negative.
Thus, position of image at the nearer end of the mirror can be calculated by putting the values, we get:
$\dfrac{1}{{{v_1}}} + \dfrac{1}{{ - 20}} = \dfrac{1}{{ - 10}}$
On solving the above equation, we get:
${v_1} = - 20$
Now, finding position of the image for the farther end of the mirror can be found by adding the focal length to the distance of the image:
$\dfrac{1}{{{v_2}}} = \dfrac{1}{{ - (10 + 20)}} - \dfrac{1}{{ - 10}}$
On solving the equation, we get:
${v_2} = - 15$
Hence, we can find the height of the image as $20 - 15cm = 5cm$
Thus, this is the height of the image.

Hence, the correct answer is option (D).

Note:
There are mainly three types of mirror, these are concave mirror, convex mirror and plane mirror. In the case of a concave mirror the image formed is real and inverted, that is the light rays meet on the same side as the mirror. The convex mirror on the other hand forms images that are virtual in nature, that is the light rays meet behind the mirror and are erect.