Question

An object $3\;{\rm{cm}}$ high is placed at a distance of $10\;{\rm{cm}}$ in front of concave mirror of focal length $20\;{\rm{cm}}$. Find the nature and size of the image formed.

Answer 1

The above problem can be resolved by using the mirror formula. The mirror formula is that mathematical relation that numerically relates the focal lengths, the image distance, and the object distance. The value of object distance and the focal length are given in the problem, and by substituting the values, the image distance is obtained. Moreover, after calculating the image distance, the value of the image's size is calculated by using the magnification formula.

Complete step by step answer:
Given:
The sizer of the object is, $h = 3\;{\rm{cm}}$.
The object distance is, $u = 10\;{\rm{cm}}$.
The focal length is, $f = 20\;{\rm{cm}}$.
Apply the mirror formula to calculate the image distance as,
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$
Here, v is the distance of the image formed. As the mirror is concave by mature, therefore, the sign convention for the focal length and the object distance will be negative.
Then substitute the values as,

\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}\\\ \dfrac{1}{{\left( { - 20\;{\rm{cm}}} \right)}} = \dfrac{1}{v} + \dfrac{1}{{\left( { - 10\;{\rm{cm}}} \right)}}\\\ \dfrac{1}{v} = - \dfrac{1}{{\left( {20\;{\rm{cm}}} \right)}} + \dfrac{1}{{10\;{\rm{cm}}}}\\\ v = 20\;{\rm{cm}} \end{array}$$ The size of the image is calculated by the magnification formula as, $$\dfrac{v}{u} = \dfrac{I}{O}$$ Here, I is the image size and O is the object size. Substitute the value as, $$\begin{array}{l} \dfrac{v}{u} = \dfrac{I}{O}\\\ I = \dfrac{{20\;{\rm{cm}}}}{{10\;{\rm{cm}}}} \times 3\;{\rm{cm}}\\\ I = 6\;{\rm{cm}} \end{array}$$ Therefore, the size of the image is $$6\;{\rm{cm}}$$. **Note:** To resolve the given problem, one must be clear about the mathematical formula of the mirror formula. This formula is applied to calculate the unknown identities like image distance. One should go through the sign convention for the concave mirrors and the convex mirrors along with this.