Question

A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of 10cm. The separation between the two lenses is 2cm. The focal lengths of the component lenses
A. 18cm, 20cm
B. 10cm, 12cm
C. 12cm, 14cm
D. 16cm, 18cm

Answer 1

In order to solve this problem we need to use the formula of focal length for convergent doublet of separated lenses and get the answers. We will put all the values given in the option and the option which gives the resultant focal length as 10cm that is the right answer. We also need to know that spherical aberration is the type of defect found in optical systems with spherical surfaces. Knowing these things and drawing the diagram will give you the right answer.

Complete answer:
It is given that a convergent doublet of separated lenses, corrected for spherical aberration, has a resultant focal length of 10cm. The separation between the two lenses is 2cm. We need to find the focal lengths of the component lenses.

The figure of the convergent doublet of separated lenses is:

We know that the formula of focal for the convergent doublet of separated lenses is:
$\Rightarrow f = \dfrac{{{f_1}{f_2}}}{{{f_1} + {f_2} - d}}$…….(1)
Let us check the option A in which ${f_1} = 18cm$ and ${f_2} = 20cm$.
On putting the values of ${f_1} = 18cm$ and ${f_2} = 20cm$ in the above equation we get,
$f = \dfrac{{(20)(18)}}{{20 + 18 - 2}} = \dfrac{{360}}{{36}} = 10$
So, the first option itself gives the focal length as 10cm.
As focal length cannot vary so we will not check the other options.

So, the correct answer is “Option A”.

Note:
In this problem we have just used the formula of focal length for separated convergent lenses and checked which option is right. We need to know that lenses cannot have different focal length at the same time unless and until there is something changed in the system. We need to know that the shape of a spherical lens causes a problem called spherical aberration. In spherical aberration, parallel light rays that pass through the central region of the lens focus farther away than light rays that pass through the edges of the lens. The result is many focal points, which produce a blurry image. Knowing this will help you and will solve your problem.