Question

Two thin lenses of focal lengths $ f_1 $ and $ f_2 $ are placed in contact with each other. The focal length of the combination is :

• A. $ \frac{f_{1}+f_{2}}{2} $
• B. $ \sqrt{f_{1}}\,f_{2} $
• C. $ \frac{f_{1}f_{2}}{f_{1}+f_{2}} $
• D. $ \frac{f_{1}f_{2}}{f_{1}-f_{2}} $

Answer 1

Answer: (C)

When $f_{1}$ and $f_{2}$ are focal lengths of lenses combined together, image formation takes place as follows,

From lens formula
$\frac{1}{v'}-\frac{1}{u}=\frac{1}{f_{1}} $
$\frac{1}{v}-\frac{1}{v^{\prime}}=\frac{1}{f_{2}}$
Adding Eqs. (i) and (ii) we get
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f_{1}}+\frac{1}{f_{2}}$
If this lens is replaced by a single lens, then focal length of combination is
$ \frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}}=\frac{1}{v}-\frac{1}{u} $
$\Rightarrow \, F=\frac{f_{1} f_{2}}{f_{1}+f_{2}}$