Question

An object $1 cm$ tall is placed $4 cm$ in front of a mirror. In order to produce an upright image of 3 cm height, one needs a:
(A) Convex mirror of radius of curvature 12 cm
(B) Concave mirror of radius of curvature 12 cm
(C) Concave mirror of radius of curvature 4 cm
(D) Plane mirror of height 12 cm

Answer 1

Hint:Use the formula for magnification of mirror in terms of height of image and height of object to determine the image distance. Use the mirror equation to determine the focal length of the mirror using the obtained quantities. If the focal length turns out positive, then the mirror is convex and if the focal length is negative then the mirror is concave. Also, the radius of curvature is twice the focal length.

Formula used:
-Magnification, \[m = \dfrac{{{h_i}}}{{{h_o}}}\]
Here, \[{h_i}\] is the height of the image and \[{h_o}\] is the height of the object.
-Mirror equation, \[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\]
Here, f is the focal length of the mirror, v is the image distance and u is the object distance.

Complete step by step answer:
We have given the object height 1 cm and image height 3 cm. The object distance \[u\] is\[ - 4\,cm\].
We know the formula for magnification for the mirror,
\[m = \dfrac{{{h_i}}}{{{h_o}}}\]
Here, \[{h_i}\] is the height of the image and \[{h_o}\] is the height of the object.
Substituting 1 cm for \[{h_o}\] and 3 cm for \[{h_i}\] in the above equation, we get,
\[m = \dfrac{{3\,cm}}{{1\,cm}}\]
\[ \Rightarrow m = 3\]
We also have other formula for magnification of the mirror in terms of object distance and image distance,
\[m = - \dfrac{v}{u}\]
Here, v is the image distance from the mirror.
We substitute 3 for m in the above equation.
\[3 = - \dfrac{v}{u}\]
\[ \Rightarrow v = - 3u\]
\[ \Rightarrow v = \left( { - 3} \right)\left( { - 4} \right)\]
\[ \Rightarrow v = 12\,cm\]
Now, we have the mirror equation,
\[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\]
Here, f is the focal length of the mirror.
Substituting \[ - 4\,cm\] for u and 12 cm for v in the above equation, we get,
\[\dfrac{1}{f} = \dfrac{1}{{12}} - \dfrac{1}{4}\]
\[ \Rightarrow \dfrac{1}{f} = \dfrac{{ - 2}}{{12}}\]
\[ \Rightarrow f = - 6\,cm\]
We know that the focal length is negative for a concave mirror. Therefore, the mirror should be concave.Now, we know that the radius of curvature of the mirror is twice the focal length. Therefore, we have,
\[R = 2f\]
\[ \Rightarrow R = 2\left( { - 6} \right)\]
\[ \therefore R = - 12\,cm\]

Therefore, the radius of curvature of that mirror is 12 cm.So, the correct answer is option (B).

Note:The lens equation and mirror equation have slight differences. The lens equation is given as, \[\dfrac{1}{f} = \dfrac{1}{v} - \dfrac{1}{u}\]. Therefore, do not get confused between these two. The radius of curvature of the mirror does not have a sign. It’s always a positive quantity. The sign of the radius of curvature can be used to identify the type of the mirror whether it is concave or convex.