Question

The size of the image formed by a convex mirror is half the size of the object, when the latter is held at a distance of 30 cm from the mirror. Find the radius of curvature of the mirror.

Answer 1

Hint:Determine the image distance using the formula for magnification of the lens. Using the lens formula, determine the focal length of the convex lens. It should be a positive. The radius of curvature is twice the focal length of the lens.

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.
Lens formula, \[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\]
Here, f is the focal length, u is the object distance and v is the image distance.

Complete step by step answer:
We have given that the size of the image is half the size of the object. We have the expression for the magnification of the lens,
\[m = \dfrac{{{h_i}}}{{{h_o}}}\]
Here, \[{h_i}\] is the height of the image and \[{h_o}\] is the height of the object.
According to the question,
\[\dfrac{{{h_i}}}{{{h_o}}} = \dfrac{1}{2} = m\] …… (1)
We also have the expression for the magnification in terms of image distance and object distance,
\[m = - \dfrac{v}{u}\]
Here, v is the image distance and u is the object distance.
Using equation (1) in the above equation, we get,
\[\dfrac{1}{2} = - \dfrac{v}{u}\]
\[ \Rightarrow \dfrac{1}{2} = - \dfrac{v}{{\left( { - 30} \right)}}\]
\[ \Rightarrow v = 15\,{\text{cm}}\]
We have the lens equation,
\[\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\]
Substituting \[v = 15\,{\text{cm}}\] and \[u = - 30\,{\text{cm}}\] in the above equation, we get,
\[\dfrac{1}{f} = \dfrac{1}{{15}} + \dfrac{1}{{ - 30}}\]
\[ \Rightarrow \dfrac{1}{f} = \dfrac{2}{{30}} - \dfrac{1}{{30}}\]
\[ \Rightarrow f = 30\,{\text{cm}}\]
We know that the radius of curvature of the lens is twice the focal length. Therefore,
\[R = 2f\]
Substituting \[f = 30\,{\text{cm}}\] in the above equation, we get,
\[R = 2\left( {30} \right)\]
\[ \therefore R = 60\,{\text{cm}}\]

Therefore, the radius of curvature of the convex lens is 60 cm.

Note: Students must be able to recognize the nature of the lens using the sign of focal length. The convex lens has a positive focal length while the concave lens has a negative focal length. Also, the object distance is always negative for both convex lens and concave lens while the image distance is positive for real image formed by convex lens.