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, $\frac{1}{F}=\frac{1}{f_1}+\frac{1}{f_2}$
$\frac{1}{50}=\frac{1}{f_1}+\frac{1}{-2f_1}$
Or $50f=\frac{-2+1}{-2f_1}=\frac{1}{2f_1}$
Or $2f_1=50$ or $f_1=25cm$
Again $f_2=-2\times 25\text{cm}$
$f_2=-50\text{cm}$