question_answer

    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)  \[\frac{{{f}_{1}}{{f}_{2}}}{{{f}_{1}}-{{f}_{2}}}\]

Correct Answer: C

Solution :

                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                 \[\frac{1}{F}=\frac{1}{{{f}_{1}}}+\frac{1}{{{f}_{2}}}=\frac{1}{v}-\frac{1}{u}\] \[\Rightarrow \]               \[F=\frac{{{f}_{1}}{{f}_{2}}}{{{f}_{1}}+{{f}_{2}}}\]