Question

A convex lens has a focal length of 10 cm. At what distance from the lens should the object be placed so that forms a real and inverted image 20 cm away from the lens? What would be the size of the image formed if the object is 2 cm high? Show the formation of an image by drawing a ray diagram.

Answer 1

The lens formula should be used to solve this problem. Using the given values of the focal length and the image distance, we have to find the object distance. If the object distance equals the image distance and has twice the value of the focal length, then, the image is said to be formed by 2F.
Formula used:
$\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}$

Complete answer:
From the given information we have the data as follows.
The focal length of the convex lens is, $f=+10\,cm$
The image distance is, $v=+20\,cm$
The height of the object, $O=2\,cm$
As the image distance is positive, thus, the image formed will be real and inverted.
Consider the formula that relates the focal length of the lens, image distance and object distance.
$\dfrac{1}{f}=\dfrac{1}{v}-\dfrac{1}{u}$
Where f is the focal length, v is the image distance and u is the object distance.
Substitute the values in the above equation.

& \dfrac{1}{10}=\dfrac{1}{20}-\dfrac{1}{u} \\\ & \Rightarrow \dfrac{1}{u}=\dfrac{1}{20}-\dfrac{1}{10} \\\ & \therefore \dfrac{1}{u}=\dfrac{1}{20} \\\ \end{aligned}$$ Therefore, the object distance is 20 cm. The value of the object distance is twice that of the focal length of the convex lens. So, the object is placed at 2F and the image also forms at 2F on the other side of the lens, as the image distance equals the object distance. Thus, the height of the image will be 2 cm. The formation of the image by drawing a ray diagram is given as follows.  $$\therefore $$ The distance from the lens at which the object is placed so that forms a real and inverted image 20 cm away from the lens is 20 cm. The size of the image formed if the object is 2 cm high is the same as that of the size of the object. **Note:** If the object is placed at distant, then the image will form at focus, if the object is placed at 2F, then, the image will form at 2F, if the object is placed between 2F and F, then the image will form beyond 2F, if the object is placed at F, then, the image will form parallel rays and if the object is placed between F and lens, then, the image will form behind the object on the same side of the lens.