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}}} $ ?. (i) $ \frac{1}{v}-\frac{1}{v}=\frac{1}{{{f}_{2}}} $ ?.(ii) 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}}} $