Question

A small statue has a height of $1{\text{ cm}}$ and is placed in front of a spherical mirror. The image of the statue is inverted and is $0.5{\text{ cm}}$ tall and located $10{\text{cm}}$ in front of the mirror. Find the focal length and nature of the mirror.

Answer 1

Hint: For solving such problems we need to recall the mirror characteristics and image formed by it, also the sign hypothesis used for calculation of magnification or the height of the image or the distance of the image from the mirror. We should remember the mirror formula and the magnification formula so that the value of the unknowns can be calculated.

Complete step by step answer:
The height of the object as well the image is given in the question so we can easily calculate the magnification using the following formula,
$m = \dfrac{{ - {h_2}}}{{{h_1}}}$
Where ${h_1}$ is the height of the statue and ${h_2}$ is the height of the image formed.
Therefore by substituting the values we get,
$ \Rightarrow m = \dfrac{{ - 0.5}}{1}$
$ \Rightarrow m = - 0.5{\text{ cm}}$
Now the distance between the mirror and the image formed is given so using the focal length formula to find magnification we get,
$ \Rightarrow m = f - \dfrac{v}{f}$
Where $v$ is the distance between the image and the mirror.
Now substituting the values
$ \Rightarrow - 0.5 = \dfrac{{f + 10}}{f}$
$ \therefore f = \dfrac{{ - 20}}{3}{\text{ cm}}$
The nature of the mirror would be a concave mirror as the focal length is negative.

Note: The most common types of spherical mirror are concave mirror and convex mirror. The key difference between them is that the convex mirror always forms real, inverted and diminished images whereas the concave mirror forms images depending on the position of the object. The negative sign in the magnification denotes that the image formed is inverted that is below the central axis of the mirror. The distance of any point is measured along the central axis by convention, as the direction towards the object in the mirror is taken positive and away from the mirror is taken negatively. The focal length is negative for the concave lens and positive for the convex lens.