Question

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

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

Answer 1

The correct option is A $25\text{cm}\text{and}-50\text{cm}$

Given,

$\frac{{\omega }_1}{{\omega }_2}=\frac{1}{2};f_{net}=50cm$

For achromatic combination,

$(\frac{{\omega }_1}{f_1}+\frac{{\omega }_2}{f_2})=0$

$\Rightarrow \frac{f_1}{f_2}=-\frac{{\omega }_1}{{\omega }_2}=-\frac{1}{2}$

$\Rightarrow f_2=-2f_1$

Now, focal length of combination is,

$\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}$

$\frac{1}{50}=\frac{1}{f_1}+\frac{1}{-2f_1}$

$\frac{1}{50}=\frac{-2+1}{-2f_1}=\frac{1}{2f_1}$

$\Rightarrow 2f_1=50$

$\Rightarrow f_1=25\text{cm}\text{and}$

$f_2=-2f_1=-2\times 25=-50\text{cm}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(A\right)$ is the correct answer.