Question

An object of length $4\,cm$ is placed in front of a concave mirror at a distance of $30\,cm$. The focal length of the mirror is $15\,cm$ . What will be the length of the image?
A. $1\,cm$
B. $2\,cm$
C. $4\,cm$
D. $6\,cm$

Answer 1

here, the object is placed in front of the concave mirror. Firstly, we will calculate the distance of the image from the mirror using the mirror formula. The size of the image can be calculated by using the magnification formula.

Complete step by step answer:
Consider an object which is placed at a distance of $30\,cm$ from the concave mirror. Let the length of the object is $4\,cm$ . Also, the focal length of the mirror is $15\,cm$ .
Therefore, the height of the object, $h = 4\,cm$
The distance of the object from the mirror, $u = 30\,cm$
The focal length of the mirror, $f = - 15\,cm$
Now, we will first calculate the distance of the image from the concave mirror. For this, we will use the mirror formula as shown below
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$
$\Rightarrow \,\dfrac{1}{{ - 15}} = \dfrac{1}{v} + \dfrac{1}{{30}}$
$\Rightarrow \,\dfrac{1}{v} = \dfrac{1}{{ - 15}} - \dfrac{1}{{ - 30}}$
$\Rightarrow \,\dfrac{1}{v} = - \dfrac{1}{{15}} + \dfrac{1}{{30}}$
$\Rightarrow \,\dfrac{1}{v} = \dfrac{{ - 2 + 1}}{{30}}$
$\Rightarrow \,\dfrac{1}{v} = \dfrac{{ - 1}}{{30}}$
$\Rightarrow \,v = - 30\,cm$
Therefore, the distance of the image from the concave mirror is $- 30\,cm$ .
Now, the magnification is the ratio of the distance of the image from the mirror to the distance of the object from the mirror. Also, magnification is the ratio of the size of the image to the size of the object. The magnification of the concave mirror is negative and is given below
$m = \dfrac{{h'}}{h} = - \dfrac{v}{u}$
$\Rightarrow \,h' = \dfrac{{ - v}}{u}h$
$h' = \dfrac{{ - \left( { - 30} \right)}}{{ - 30}} \times 4$
$h' = - 4\,cm$
Here, the negative sign shows that the image is on the same side as that of the object and it is of the same size as that of the object.
Therefore, the size of the image is $4\,cm$ and it is inverted in nature.

So, the correct answer is “Option C”.

Note:
The magnification of the concave mirror is negative.
The magnification of the mirror depends upon the nature of the image.
In the case of a concave mirror, the image formed is inverted or erect.
Therefore, in the above example, the image formed is inverted.
That is why we have taken the negative magnification.