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Converging rays are incident on a convex spherical mirror so that their extensions intersect $30\,cm$ behind the mirror on the optical axis. The reflected rays form a diverging beam, so that their extensions intersect the optical axis $1.2\,m$ from the mirror. The focal length of the mirror is 
A. $40\,cm$
B. $60\,cm$
C. $30\,cm$
D. $24\,cm$

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Last updated date: 19th Jul 2024
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Answer
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Hint: Reflected rays get diverged or converged to form an image while incident rays are converged to form an object. Thus image distance and object distance can be obtained from the question and we must find the focal length of the mirror by using the mirror formula in this scenario.

Complete step by step answer:
If the surface of the sphere is silvered such that it can reflect light, then the mirror is claimed to be convex. The middle of that original sphere is called the center of curvature (C) and therefore the line that passes from the mirror's surface through the sphere's center called the optic axis. All the incident and refracted rays pass through a point on the principal axis and the point is known as the focus of the mirror. Note that the center of curvature and therefore the focus are located on the side of the mirror that is behind the convex mirror. 

Since the focus is found behind the convex mirror, such a mirror is claimed to possess a negative focal distance value. A convex mirror forms a virtual and erect image for all the positions of the object.Now here, when the incident rays converge so that their extensions intersect at a point, then that point is our object and the distance of that object from the center of curvature is called object distance. 

When the reflected rays diverge so that their extensions intersect at a point, then that point is our image and the distance of that image from the center of curvature is called image distance. Object distance is denoted as u and image distance is denoted as $v$.The focal length (f) of the mirror can be obtained from the mirror formula as follows:
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u} $
$\Rightarrow \dfrac{1}{f} = \dfrac{1}{{120}} + \dfrac{1}{{30}} $
$\therefore f = 24\,cm $
Hence option D is the correct answer.

Note: Here, in the case of convex mirrors, image distance has been taken negative as it was formed on the opposite side of the mirror, while if it were a concave mirror, the image distance would have been negative. Also, generally the focal length of a convex mirror would be positive as the rays would converge on the opposite side of the mirror.