A convex lens is in contact with a concave lens. The magnitude of the ratio of their focal lengths is $\dfrac{2}{3}$. Their equivalent focal length is 30 cm. What are their individual focal lengths?
A. -15, 10
B.  10,-15
C.   75, 50
D. -75, 50

Last updated date: 29th Jan 2023
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Answer
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Hint: The equivalent focal length of two thin lenses in contact is similar to the equivalent resistance of two resistors connected in parallel connection. Now, using the ratio of the focal lengths of two lenses, find individual focal lengths.

Formula Used:
Equivalent focal length of two thin lens is given by:
$\dfrac{1}{F} = \dfrac{1}{{{f_1}}} + \dfrac{1}{{{f_2}}}$ -(1)
Where,
$F$ is the equivalent focal length of the combination,
${f_1}$ is the focal length of the first lens,
${f_2}$ is the focal length of the second lens.

Complete answer:
Given:
Ratio of the focal length for convex and concave lenses is $\dfrac{2}{3}$.
Equivalent focal length of the combination is 30 cm.
To find: focal length of individual lens.

Step 1:
Let the focus length of the convex lens be ${f_1} = f$. Hence, using the given information you’ll get the focal length of the concave lens as ${f_2} =  - \dfrac{3f}{2}$. Here, negative sign comes because the focal length of a concave lens is always negative. 
Now, use the value of F along with these two in eq.(1) to get f:
$\dfrac{1}{30}=\dfrac{1}{f}+\dfrac{1}{-\dfrac{3f}{2}}$
$\Rightarrow \dfrac{1}{30} = \dfrac{1}{f} \left(1 - \dfrac{2}{3} \right)$
$\Rightarrow f = \dfrac{1}{3} \times 30 = 10cm $
Step 2:
Hence, you get the focal length of a convex lens as 10 cm.
So, the focal length of the concave lens will be ${f_2} = - \dfrac{3}{2} \times 10 =  - 15cm$.

Hence, Individual focal lengths are 10,-15 for convex mirror and concave mirror respectively.
Therefore option (B) is correct.

Note: While using this combination formula you need to keep in mind that this formula is an approximated formula only for thin lenses. While deriving this formula, the thickness of each lens has been neglected. So, this formula will give significant error if the thickness of the lens is not negligible compared to their focal length.