Question

It is desired to make a converging achromatic combination of mean focal length $50 \, \text{cm}$ by using two lenses of materials $A$ and $B$ . If the dispersive power of $A$ and $B$ are in ratio $1 : 2 ,$ the focal lengths of the convex and the concave lenses are respectively

• A. $\text{25} \, \text{cm}$ and $\text{50} \, \text{cm}$
• B. $50 \, \text{cm}$ and $\text{25} \, \text{cm}$
• C. $50 \, \text{cm}$ and $\text{100} \, \text{cm}$
• D. $100 \, \text{cm}$ and $\text{50} \, \text{cm}$

Answer 1

Answer: (A)

$\frac{\omega _{1}}{\omega _{2}}=\frac{1}{2}$
Now, $ \, \frac{f_{1}}{f_{2}}=-\frac{\omega _{1}}{\omega _{2}}=-\frac{1}{2}$
Or $ \, f_{2}=-2f_{1}$
Now, $\frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}}$
$\frac{1}{50}=\frac{1}{f_{1}}+\frac{1}{- 2 f_{1}}$
Or $50f=\frac{- 2 + 1}{- 2 f_{1}}=\frac{1}{2 f_{1}}$
Or $2f_{1}=50 \, $ or $ \, f_{1}=25 \, cm$
Again $f_{2}=-2\times 25 \, \text{cm}$
$f_{2}=-50 \, \text{cm}$