Question

A convex lens forms a real and inverted image of a needle at a distance of 50cm from it. Where is the needle placed in front of the convex lens if the image is equal to the size of the object? Also find the power of the lens.

Answer 1

Hint: It is given that the image of the needle is formed at a distance of 50cm from the lens and the image is real and inverted. In addition, the sizes of the image and the needle are equal. So use the magnification formula. Then use the lens formula and find the focal length. With the focal length calculate the power of the lens.
Formula used:
$\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$
$m=\dfrac{v}{u}$

Complete answer:
In this solution we will use the lens formula $\dfrac{1}{v}-\dfrac{1}{u}=\dfrac{1}{f}$ …… (i).
Here, v and u are the positions of the image and the object with respect to the lens, according to the sign convection. f is the focal length of the lens.
It is said that when the needle is placed in front of the lens, a real and inverted image is formed and the size of the image is equal to the size of the needle.
The magnification is defined as the ratio of the size of the image to the size of the object. The value of magnification is given as $m=\dfrac{v}{u}$.
In this case, m= -1 because the image is inverted.
$\Rightarrow -1=\dfrac{v}{u}$
$\Rightarrow v=-u$.
It is given that the image is formed at a distance of 50cm from the lens. And since the image is real, v = +50cm.
This means that u = -50cm.
Therefore, the needle must be placed at a distance of 50cm from the lens.
Substitute the values of v and u in (i).
 $\Rightarrow \dfrac{1}{50}-\dfrac{1}{-50}=\dfrac{1}{f}$
$\Rightarrow \dfrac{1}{f}=\dfrac{2}{50}=\dfrac{1}{25}$
$\Rightarrow f=25cm$.
Therefore, the focal length of the convex lens is 25cm.
Power of a lens is given as $P=\dfrac{1}{f}$.
$\Rightarrow P=\dfrac{1}{25}c{{m}^{-1}}$
$\Rightarrow P=\dfrac{1}{25}{{\left( {{10}^{-2}}m \right)}^{-1}}=\dfrac{100}{25}{{m}^{-1}}=4{{m}^{-1}}$.
$1{{m}^{-1}}=1D$
$\Rightarrow P=4D$
This means that power of the given lens is 4D (dioptre).

Note:
Note that the focal length of a convex lens is always positive and the focal length of a concave lens is always negative. Therefore, with this data we can identify an unknown lens.
The solution can be remembered as a point that when the object distance is 2f, the image distance is also 2f and the sizes of the image and the object are the same.